IPS7e SuppExer

SUPPLEMENTARY
EXERCISES
for
INTRODUCTION TO THE
PRACTICE OF STATISTICS
Seventh Edition
David S. Moore
George P. McCabe
and
Bruce A. Craig
These exercises appeared in the third, fourth, fifth, or sixth edi-

tions of Introduction to the Practice of Statistics. They do not
appear in the seventh edition, but they remain high-quality ex-
ercises that supplement those in the text.
CHAPTER 1
Section 1.1
1.1 Data from a medical study contain values of many variables for each of the
people who were the subjects of the study. Which of the following variables are
categorical and which are quantitative?
(a) Gender (female or male)
(b) Age (years)
(c) Race (Asian, black, white, or other)
(d) Smoker (yes or no)
(e) Systolic blood pressure (millimeters of mercury)
(f) Level of calcium in the blood (micrograms per milliliter)
1.2 Political party preference in the United States depends in part on the age, in-
come, and gender of the voter. A political scientist selects a large sample of registered
voters. For each voter, she records gender, age, household income, and whether they
voted for the Democratic or for the Republican candidate in the last Congressional
election. Which of these variables are categorical and which are quantitative?
1.3 Here is a small part of a data set that describes mutual funds available to the
public:
Net assets Year-to-date Largest

Fund Category (millions of $) return holding
..
.
Fidelity Low-Priced Stock Small value 6,189 4.56% Dallas Semiconductor
Price International Stock International stock 9,745 −0.45% Vodafone
Vanguard 500 Index Large blend 89,394 ;45% General Electric
..
.
What individuals does this data set describe? In addition to the fund’s name, how
many variables does the data set contain? Which of these variables are categorical
and which are quantitative?
1.4 Congress wants the medical establishment to show that progress is being made
in fighting cancer. Some variables that might be used are: (a) Total deaths from
cancer. These have risen over time, from 331,000 in 1970 to 505,000 in 1980 to
550,000 in 1999.
(b) The percent of all Americans who die from cancer. The percent of deaths due
to cancer has also risen steadily, from 17.2% in 1970 to 20.9% in 1980 to 23.0% in
1999.
(c) The percent of cancer patients who survive for five years from the time the disease
is discovered. These rates are rising slowly. For whites, the five-year survival rate
was 50.8% in the 1974 to 1979 period and 60.9% from 1989 to 1995.
Discuss the usefulness of each of these variables as a measure of the effectiveness of
cancer treatment. In particular, explain why both (a) and (b) could increase even
1
2 Section 1.1
if treatment is getting more effective, and why (c) could increase even if treatment
is getting less effective.
1.5 The National Highway Traffic and Safety Administration says that an average
of 11 children die each year in school bus accidents, and an average of 600 school-age
children die each year in auto accidents during school hours. These numbers suggest
that riding the bus is safer than driving to school with a parent. The counts aren’t
fully convincing, however. What rates would you like to know to compare the safety
of bus and private auto?
1.6 Popular magazines often rank cities in terms of how desirable it is to live and
work in each city. Describe five variables that you would measure for each city if
you were designing such a study. Give reasons for each of your choices.
1.7 All the members of a physical education class are asked to measure their
pulse rate as they sit in the classroom. The students use a variety of methods.
Method 1: count heart beats for 6 seconds and multiply by 10 to get beats per
minute. Method 2: count heart beats for 30 seconds and multiply by 2 to get beats
per minute.
(a) Which method do you prefer? Why?
(b) One student proposes a third method: starting exactly on a heart beat, measure
the time needed for 50 beats and convert this time into beats per minute. This
method is more accurate than either method in (a). Why?
1.8 Each year Fortune magazine lists the top 500 companies in the United States,
ranked according to their total annual sales in dollars. Describe three other variables
that could reasonably be used to measure the “size” of a company.
1.9 You are writing an article for a consumer magazine based on a survey of the
magazine’s readers on the reliability of their household appliances. Of 13,376 readers
who reported owning Brand A dishwashers, 2942 required a service call during the
past year. Only 192 service calls were reported by the 480 readers who owned Brand
B dishwashers. Describe an appropriate variable to measure the reliability of a make
of dishwasher, and compute the values of this variable for Brand A and for Brand
B.
1.10 In 1997, there were 12,298,000 undergraduate students in U.S. colleges. Ac-
cording to the U.S. Department of Education, there were 127,000 American Indian or
Alaskan Native students, 737,000 Asian or Pacific Islander, 1,380,000 non-Hispanic
black, 1,108,000 Hispanic, and 8,682,000 non-Hispanic white students. In addition,
265,000 foreign undergraduates were enrolled in U.S. colleges.
(a) Each number, including the total, is rounded to the nearest thousand. Separate
rounding may cause roundoff errors, so that the sum of the counts does not equal
the total given. Are roundoff errors present in these data?
(b) Present the data in a graph.
1.11 The number of deaths among persons aged 15 to 24 years in the United States
in 1999 due to the eight leading causes of death for this age group were: accidents,
13,602; homicide, 4989; suicide, 3885; cancer, 1724; heart disease, 1048; congenital
Chapter 1 Exercises 3
defects, 430; respiratory disease, 208; AIDS, 197.

(a) Make a bar graph to display these data.
(b) What additional information do you need to make a pie chart?
1.12 According to the 2000 census, there are 105.5 million households in the United
States. A household consists of people living together in the same residence, re-
gardless of their relationship to each other. Of these, 71.8 million were “family
households” in which at least one other person was related to the householder by
blood, marriage, or adoption. The family households include 54.5 million headed by
a married couple and 17.3 million other families (for example, a single parent with
children). The other 33.7 million households are “nonfamily households.” Of these,
27.2 million contain a person living alone, 5.5 million are unmarried couples living
together, and 1 million consist of other unrelated people living together. Be creative:
make a bar graph that displays these facts, including the distinction between family
and nonfamily households.
1.13 Here are the percents of doctoral degrees in each of several subjects that were
earned by women in 1997–1998: psychology, 67.5%; education, 63.2%; life sciences,
42.5%; business, 31.4%; physical sciences, 25.2%; engineering, 12.2%.
(a) Explain clearly why we cannot use a pie chart to display these data, even if we
knew the percent female for every academic subject.
(b) Make a bar graph of the data. (Comparisons are easier if you order the bars by
height, which is the order in which we give the percents.)
1.14 Three landmarks of baseball achievement are Ty Cobb’s batting average of

.420 in 1911, Ted Williams’s .406 in 1941, and George Brett’s .390 in 1980. These
batting averages cannot be compared directly because the distribution of major
league batting averages has changed over the decades. The distributions are quite
symmetric and (except for outliers such as Cobb, Williams, and Brett) reasonably
normal. Although the mean batting average has been held roughly constant by rule
changes and the balance between hitting and pitching, the standard deviation has
dropped over time. Here are the facts:
Decade Mean Std Dev
1910s .266 .0371
1940s .267 .0326
1970s .261 .0317
Compute the standardized batting averages for Cobb, Williams, and Brett to com-
pare how far each stood above his peers. (Data from Stephen Jay Gould, “Entropic
homogeneity isn’t why no one hits .400 any more,” Discover, August 1986, pp.
60–66. Gould does not standardize but gives a speculative discussion instead.)
1.15 The Survey of Study Habits and Attitudes (SSHA) is a psychological test that
evaluates college students’ motivation, study habits, and attitudes toward school. A
selective private college gives the SSHA to a sample of 18 of its incoming first-year
college women. Their scores are
154 109 137 115 152 140 154 178 101
103 126 126 137 165 165 129 200 148
4 Section 1.1
The college also administers the test to a sample of 20 first-year college men. Their
scores are
108 140 114 91 180 115 126 92 169 146
109 132 75 88 113 151 70 115 187 104
(a) Make a back-to-back stemplot of the men’s and women’s scores. The overall
shapes of the distributions are indistinct, as often happens when only a few obser-
vations are available. Are there any outliers?
(b) Compare the midpoints and the ranges of the two distributions. What is the
most noticeable contrast between the female and male scores?
1.16 Plant scientists have developed varieties of corn that have increased amounts
of the essential amino acid lysine. In a test of the protein quality of this corn, an
experimental group of 20 one-day-old male chicks was fed a ration containing the
new corn. A control group of another 20 chicks was fed a ration that was identical
except that it contained normal corn. Here are the weight gains (in grams) after
21 days (based on G. L. Cromwell et al., “A comparison of the nutritive value of
opaque-2, floury-2 and normal corn for the chick,” Poultry Science, 47 (1968), pp.
840–847):
Control Experimental
380 321 366 356 361 447 401 375
283 349 402 462 434 403 393 426
356 410 329 399 406 318 467 407
350 384 316 272 427 420 477 392
345 455 360 431 430 339 410 326
Make a back-to-back stemplot of these data. Report the approximate midpoints of
both groups. Does it appear that the chicks fed high-lysine corn grew faster? Are
there any outliers or other problems?
1.17 There is some evidence that increasing the amount of calcium in the diet can
lower blood pressure. In a medical experiment one group of men was given a daily
calcium supplement, while a control group received a placebo (a dummy pill). The
seated systolic blood pressure of all the men was measured before the treatments
began and again after 12 weeks. The blood pressure distributions in the two groups
should have been similar at the beginning of the experiment. Here are the initial
blood pressure readings for the two groups:
Calcium group
107 110 123 129 112 111 107 112 136 102
Placebo group
123 109 112 102 98 114 119 112 110 117 130
Make a back-to-back stemplot of these data. Does your plot show any major differ-
ences in the two groups before the treatments began? In particular, are the centers
of the two blood pressure distributions close together?
1.18 The Degree of Reading Power (DRP) test is often used to measure the reading
ability of children. Here are the DRP scores of 44 third-grade students, measured
during research on ways to improve reading performance (data provided by Mari-

beth Cassidy Schmitt, from her PhD dissertation, “The effects of an elaborated
directed reading activity on the metacomprehension skills of third graders,” Purdue
University, 1987).
40 26 39 14 42 18 25 43 46 27 19
47 19 26 35 34 15 44 40 38 31 46
52 25 35 35 33 29 34 41 49 28 52
47 35 48 22 33 41 51 27 14 54 45
Make a stemplot of these data. Then make a histogram. Which display do you
prefer, and why? Describe the main features of the distribution.
1.19 The following table gives the number of medical doctors per 100,000 people in
each state:
State Doctors State Doctors State Doctors

Alabama 198 Louisiana 246 Ohio 235
Alaska 167 Maine 223 Oklahoma 169
Arizona 202 Maryland 374 Oregon 225
Arkansas 190 Massachusetts 412 Pennsylvania 291
California 247 Michigan 224 Rhode Island 338
Colorado 238 Minnesota 249 South Carolina 207
Connecticut 354 Mississippi 163 South Dakota 184
Delaware 234 Missouri 230 Tennessee 246
Florida 238 Montana 190 Texas 203
Georgia 211 Nebraska 218 Utah 200
Hawaii 265 Nevada 173 Vermont 305
Idaho 154 New Hampshire 237 Virginia 241
Illinois 260 New Jersey 295 Washington 235
Indiana 195 New Mexico 212 West Virginia 215
Iowa 173 New York 387 Wisconsin 227
Kansas 203 North Carolina 232 Wyoming 171
Kentucky 209 North Dakota 222 D.C. 737
(a) Why is the number of doctors per 100,000 people a better measure of the avail-
ability of health care than a simple count of the number of doctors in a state?
(b) Make a graph to display the distribution of doctors per 100,000 people. Write a
brief description of the distribution. Are there any outliers? If so, can you explain
them?
1.20 Here are the monthly percent returns on Philip Morris stock for the period
from July 1990 to May 1997 (the return on an investment consists of the change in
its price plus any cash payments made, given here as a percent of its price at the
start of each month):
6 Section 1.1
−5.7 1.2 4.1 3.2 7.3 7.5 18.6 3.7 −1.8 2.4
−6.5 6.7 9.4 −2.0 −2.8 −3.4 19.2 −4.8 0.5 −0.6
2.8 −0.5 −4.5 8.7 2.7 4.1 −10.3 4.8 −2.3 −3.1
−10.2 −3.7 −26.6 7.2 −2.9 −2.3 3.5 −4.6 17.2 4.2
0.5 8.3 −7.1 −8.4 7.7 −9.6 6.0 6.8 10.9 1.6
0.2 −2.4 −2.4 3.9 1.7 9.0 3.6 7.6 3.2 −3.7
4.2 13.2 0.9 4.2 4.0 2.8 6.7 −10.4 2.7 10.3
5.7 0.6 −14.2 1.3 2.9 11.8 10.6 5.2 13.8 −14.7
3.5 11.7 1.3
(a) Make either a histogram or a stemplot of these data. How did you decide which
graph to make?
(b) There is one clear outlier. What is the value of this observation? (It is explained
by news of action against smoking, which depressed this tobacco company stock.)
Describe the shape, center, and spread of the data after you omit the outlier.
(c) The data appear in time order reading from left to right across each row in turn,
beginning with the −5.7% return in July 1990. Make a time plot of the data. This
was a period of increasing action against smoking, so we might expect a trend toward
lower returns. But it was also a period in which stocks in general rose sharply, which
would produce an increasing trend. What does your time plot show?
1.21 The distribution of the ages of a nation’s population has a strong influence on
economic and social conditions. The following table shows the age distribution of
U.S. residents in 1950 and 2050, in millions of people. The 1950 data come from
that year’s census, while the 2050 data are projections made by the Census Bureau.
Age group 1950 2050
Under 10 years 29.3 53.3
10 to 19 years 21.8 53.2
20 to 29 years 24.0 51.2
30 to 39 years 22.8 50.5
40 to 49 years 19.3 47.5
50 to 59 years 15.5 44.8
60 to 69 years 11.0 40.7
70 to 79 years 5.5 30.9
80 to 89 years 1.6 21.7
90 to 99 years 0.1 8.8
100 to 109 years – 1.1
Total 151.1 403.7
(a) Because the total population in 2050 is much larger than the 1950 population,
comparing percents (relative frequencies) in each age group is clearer than comparing
counts. Make a table of the percent of the total population in each age group for
both 1950 and 2050.
(b) Make a relative frequency histogram of the 1950 age distribution. Describe the
main features of the distribution. In particular, look at the percent of children
relative to the rest of the population.
(c) Make a relative frequency histogram of the projected age distribution for the
year 2050. Use the same scales as in (b) for easy comparison. What are the most
important changes in the U.S. age distribution projected for the century between
1950 and 2050?
1.22 (Optional) Sometimes you want to make a histogram from data that are
already grouped into classes of unequal width. A report on the recent graduates of
a large state university includes the following relative frequency table of the first-year
salaries of last year’s graduates. Salaries are in $1000 units, and it is understood
that each class includes its left endpoint but not its right endpoint—for example, a
salary of exactly $20,000 belongs in the second class.
Salary 15–20 20–25 25–30 30–35 35–40 40–50 50–60 60–80

Percent 7 14 29 23 13 9 4 1
The last three classes are wider than the others. An accurate histogram must take
this into account. If the base of each bar in the histogram covers a class and the
height is the percent of graduates with salaries in that class, the areas of the three
rightmost bars will overstate the percent who have salaries in these classes. To make
a correct histogram, the area of each bar must be proportional to the percent in that
class. Most classes are $5000 wide. A class twice as wide ($10,000) should have a
bar half as tall as the percent in that class. This keeps the area proportional to the
percent. How should you treat the height of the bar for a class $20,000 wide? Make
a correct histogram with the heights of the bars for the last three classes adjusted
so that the areas of the bars reflect the percent in each class.
1.23 “Major hurricanes account for just over 20% of the tropical storms and hurri-
canes that strike the United States but cause more than 80% of the damage.” So say
investigators who have shown that major hurricanes (with sustained wind speeds
at least 50 meters per second) are tied to ocean temperature and other variables.
These variables change slowly, so the high level of hurricane activity that began
in 1995 “is likely to persist for an additional 10 to 40 years.” This is bad news
for people with beach houses on the Atlantic coast. Here are the counts of major
hurricanes for each year between 1944 and 2000 (Stanley B. Goldenberg et al., “The
recent increase in Atlantic hurricane activity: causes and implications,” Science, 293
(2001), pp. 474–479):
Year Count Year Count Year Count Year Count Year Count
1944 3 1956 2 1968 0 1980 2 1992 1
1945 2 1957 2 1969 3 1981 3 1993 1
1946 1 1958 4 1970 2 1982 1 1994 0
1947 2 1959 2 1971 1 1983 1 1995 5
1948 4 1960 2 1972 0 1984 1 1996 6
1949 3 1961 6 1973 1 1985 3 1997 1
1950 7 1962 0 1974 2 1986 0 1998 3
1951 2 1963 2 1975 3 1987 1 1999 5
1952 3 1964 5 1976 2 1988 3 2000 3
1953 3 1965 1 1977 1 1989 2
1954 2 1966 3 1978 2 1990 1
1955 5 1967 1 1979 2 1991 2
8 Section 1.1
(a) What is the average number of major hurricanes per year during the period 1944
to 2000?
(b) Make a time plot of the count of major hurricanes. Draw a line across your plot
at the average number of hurricanes per year. This helps divide the plot into three
periods. Describe the pattern you see.
1.24 Treasury bills are short-term borrowing by the U.S. government. They are
important in financial theory because the interest rate for Treasury bills is a “risk-
free rate” that says what return investors can get while taking (almost) no risk.
More risky investments should in theory offer higher returns in the long run. The
following table gives the annual returns on Treasury bills from 1970 to 2000.
(a) Make a time plot of the returns paid by Treasury bills in these years.
(b) Interest rates, like many economic variables, show cycles, clear but irregular up-
and-down movements. In which years did the interest rate cycle reach temporary
peaks?
(c) A time plot may show a consistent trend underneath cycles. When did interest
rates reach their overall peak during these years? Has there been a general trend
downward since that year?
Year Rate Year Rate Year Rate Year Rate

1970 6.52 1978 7.19 1986 6.16 1994 3.91
1971 4.39 1979 10.38 1987 5.47 1995 5.60
1972 3.84 1980 11.26 1988 6.36 1996 5.20
1973 6.93 1981 14.72 1989 8.38 1997 5.25
1974 8.01 1982 10.53 1990 7.84 1998 4.85
1975 5.80 1983 8.80 1991 5.60 1999 4.69
1976 5.08 1984 9.84 1992 3.50 2000 5.69
1977 5.13 1985 7.72 1993 2.90
1.25 Time series data often display the effects of changes in policy. Here are data
on motor vehicle deaths in the United States. For proper year-to-year comparison,
we look at the death rate per 100 million miles driven.
Year Rate Year Rate Year Rate Year Rate

1960 5.1 1970 4.7 1980 3.3 1990 2.1
1962 5.1 1972 4.3 1982 2.8 1992 1.7
1964 5.4 1974 3.5 1984 2.6 1994 1.7
1966 5.5 1976 3.2 1986 2.5 1996 1.7
1968 5.2 1978 3.3 1988 2.3 1998 1.6
(a) Make a time plot of these death rates. During these years, safety requirements
for motor vehicles became stricter and interstate highways replaced older roads.
How does the pattern of your plot reflect these changes?
(b) In 1974 the national speed limit was lowered to 55 miles per hour in an attempt
to conserve gasoline after the 1973 Arab-Israeli War. In the mid-1980s most states
raised speed limits on interstate highways to 65 miles per hour. Some said that the
lower speed limit saved lives. Is the effect of lower speed limits between 1974 and
the mid-1980s visible in your plot?
(c) Does it make sense to make a histogram of these 20 death rates? Explain your
answer.
1.26 The impression that a time plot gives depends on the scales you use on the two
axes. If you stretch the vertical axis and compress the time axis, change appears
to be more rapid. Compressing the vertical axis and stretching the time axis make
change appear slower. Make two more time plots of the data in Exercise 1.25, one
that makes motor vehicle death rates appear to decrease very rapidly and one that
shows only a slow decrease. The moral of this exercise is: pay close attention to the
scales when you look at a time plot.
1.27 Babe Ruth was a pitcher for the Boston Red Sox in the years 1914 to 1917. In
1918 and 1919 he played some games as a pitcher and some as an outfielder. From
1920 to 1934 Ruth was an outfielder for the New York Yankees. He ended his career
in 1935 with the Boston Braves. The following table gives the number of home runs
Ruth hit in each year. Make a time plot and describe its main features.
Year HRs Year HRs Year HRs

1914 0 1921 59 1928 54
1915 4 1922 35 1929 46
1916 3 1923 41 1930 49
1917 2 1924 46 1931 46
1918 11 1925 25 1932 41
1919 29 1926 47 1933 34
1920 54 1927 60 1934 22
1935 6
1.28 The following table gives the times (in minutes, rounded to the nearest minute)
for the winning man in the Boston Marathon in the years 1959 to 2004:
Year Time Year Time Year Time Year Time Year Time
1959 143 1970 131 1981 129 1992 128 2003 130
1960 141 1971 139 1982 129 1993 130 2004 131
1961 144 1972 136 1983 129 1994 127
1962 144 1973 136 1984 131 1995 129
1963 139 1974 134 1985 134 1996 129
1964 140 1975 130 1986 128 1997 131
1965 137 1976 140 1987 132 1998 128
1966 137 1977 135 1988 129 1999 130
1967 136 1978 130 1989 129 2000 130
1968 142 1979 129 1990 128 2001 130
1969 134 1980 132 1991 131 2002 129
Display these data in an appropriate graph. Describe the pattern that you see. Have
times stopped improving in recent years? If so, when did improvement end?
1.29 Here is a small part of a data set that describes the fuel economy (in miles per
gallon) of 2004 model motor vehicles:
10 Section 1.1
Make and Vehicle Transmission Number of City Highway

model type type cylinders MPG MPG
..
.
Acura NSX Two-seater Automatic 6 17 24
BMW 330I Compact Manual 6 20 30
Cadillac Seville Midsize Automatic 8 18 26
Ford F150 2WD Standard pickup truck Automatic 6 16 19
..
.
(a) What are the individuals in this data set?
(b) For each individual, what variables are given? Which of these variables are
1.30 Here are the first lines of a professor’s data set at the end of a statistics course:
NAME MAJOR POINTS GRADE
ADVANI, SURA COMM 397 B

BARTON, DAVID HIST 323 C
BROWN, ANNETTE BIOL 446 A
CHIU, SUN PSYC 405 B
CORTEZ, MARIA PSYC 461 A
What are the individuals and the variables in these data? Which variables are
1.31 How can we help wood surfaces resist weathering, especially when restoring
historic wooden buildings? A study of this question prepared wooden panels and
then exposed them to the weather. Here are some of the variables recorded. Which
of these variables are categorical, and which are quantitative?
(a) Type of wood (yellow poplar, pine, cedar)
(b) Water repellent (solvent-based, water-based)
(c) Paint thickness (millimeters)
(d) Paint color (white, gray, light blue)
(e) Weathering time (months)
1.32 You are preparing to study the television-viewing habits of college students.
Describe two categorical variables and two quantitative variables that you might
measure for each student. Give the units of measurement for the quantitative vari-
ables.
1.33 You want to measure the “physical fitness” of college students. Describe several
variables you might use to measure fitness. What instrument or instruments does
each measurement require?
1.34 Popular magazines rank colleges and universities on their “academic quality”
in serving undergraduate students. Describe five variables that you would like to
see measured for each college if you were choosing where to study. Give reasons for
each of your choices.
1.35 Is driving becoming more dangerous? Traffic deaths declined for years, bot-
toming out at 39,250 killed in 1992, then began to increase again. In 2002, 42,815
people died in traffic accidents. But more vehicles drove more miles in 2002 than in
1992. In fact, the government says that motor vehicles traveled 2247 billion miles in
1992 and 2830 billion miles in 2002. Reports on transportation use deaths per 100
million miles as a measure of risk. Compare the rates for 1992 and 2002. What do
you conclude?
1.36 What color is your car? Here are the most popular colors for vehicles made in
North America during the 2003 model year.
Color Percent of vehicles
Silver 20.1
White 18.4
Black 11.6
Medium/dark gray 11.5
Light brown 8.8
Medium/dark blue 8.5
Medium red 6.9
What percent of vehicles had other colors? Display these data in a bar graph. Is it
also correct to use a pie chart if a category for “Other” is included? If you are using
software, make a pie chart if it is correct to do so.
1.37 Favorite vehicle colors differ among types of vehicle. Here are data on the most
popular colors in 2003 for luxury cars and for SUVs, trucks, and vans. The entry
“–” means “less than 1%.” Be creative: make one bar graph that compares the two
vehicle types as well as comparing colors. Arrange your graph so that it is easy to
compare the two types of vehicle.
Color Luxury car percent SUV/truck/van percent
Black 10.9 11.6
Light brown – 6.3
Medium/dark blue 3.8 9.3
Medium/dark gray 23.3 8.8
Medium/dark green – 7.0
Medium red 3.9 6.2
White 30.4 22.3
Silver 18.8 17.0
1.38 The Department of Education estimates the “average unmet need” for un-
dergraduate students—the cost of school minus estimated family contributions and
financial aid. Here are the averages for full-time students at four types of institution
in the 1999–2000 academic year:
Public 2-year Public 4-year Private nonprofit 4-year Private for-profit
$2747 $2369 $4931 $6548
Make a bar graph of these data. Write a one-sentence conclusion about the unmet
needs of students. Explain clearly why it is incorrect to make a pie chart.
12 Section 1.1
1.39 The J.D. Power Initial Quality Survey polls more than 50,000 buyers of new
motor vehicles 90 days after their purchase. A two-page questionnaire asks about
“things gone wrong.” Here are data on problems per 100 vehicles for vehicles made
by Toyota and by General Motors in recent years. Toyota has been the industry
leader in quality. Make two time plots in the same graph to compare Toyota and
GM. What are the most important conclusions you can draw from your graph?
Year 1998 1999 2000 2001 2002 2003 2004
GM 187 179 164 147 130 134 120
Toyota 156 134 116 115 107 115 101
1.40 Study habits of students. You are planning a survey to collect information
about the study habits of college students. Describe two categorical variables and
two quantitative variables that you might measure for each student. Give the units
of measurement for the quantitative variables.
1.41 Physical fitness of students. You want to measure the “physical fitness” of
college students. Describe several variables you might use to measure fitness. What
instrument or instruments does each measurement require?
1.42 An aging population. The population of the United States is aging, though
less rapidly than in other developed countries. Here is a stemplot of the percents
of residents aged 65 and over in the 50 states, according to the 2000 census. The
stems are whole percents and the leaves are tenths of a percent.
5 7
6
7
8 5
9 679
10 6
11 02233677
12 0011113445789
13 00012233345568
14 034579
15 36
16
17 6
(a) There are two outliers: Alaska has the lowest percent of older residents, and
Florida has the highest. What are the percents for these two states?
(b) Ignoring Alaska and Florida, describe the shape, center, and spread of this
distribution.
1.43 Split the stems. Make another stemplot of the percent of residents aged 65
and over in the states other than Alaska and Florida by splitting stems 8 to 15 in
the plot from the previous exercise. Which plot do you prefer? Why?
1.44 Tornado damage. The states differ greatly in the kinds of severe weather
that afflict them. The following table shows the average property damage caused
by tornadoes per year over the period from 1950 to 1999 in each of the 50 states
and Puerto Rico. (National Climatic Data Center, storm events data base. See
sciencepolicy.colorado.edu/sourcebook/tornadoes.html.) (To adjust for the
changing buying power of the dollar over time, all damages were restated in 1999
dollars.)
(a) What are the top five states for tornado damage? The bottom five?
(b) Make a histogram of the data, by hand or using software, with classes “0 ≤
damage < 10,” “10 ≤ damage < 20,” and so on. Describe the shape, center, and
spread of the distribution. Which states may be outliers? (To understand the
outliers, note that most tornadoes in largely rural states such as Kansas cause little
property damage. Damage to crops is not counted as property damage.)
(c) If you are using software, also display the “default” histogram that your software
makes when you give it no instructions. How does this compare with your graph in
(b)?
Average property damage per year due to tornadoes
Damage Damage Damage
State ($millions) State ($millions) State ($millions)
Alabama 51.88 Louisiana 27.75 Ohio 44.36
Alaska 0.00 Maine 0.53 Oklahoma 81.94
Arizona 3.47 Maryland 2.33 Oregon 5.52
Arkansas 40.96 Massachusetts 4.42 Pennsylvania 17.11
California 3.68 Michigan 29.88 Puerto Rico 0.05
Colorado 4.62 Minnesota 84.84 Rhode Island 0.09
Connecticut 2.26 Mississippi 43.62 South Carolina 17.19
Delaware 0.27 Missouri 68.93 South Dakota 10.64
Florida 37.32 Montana 2.27 Tennessee 23.47
Georgia 51.68 Nebraska 30.26 Texas 88.60
Hawaii 0.34 Nevada 0.10 Utah 3.57
Idaho 0.26 New Hampshire 0.66 Vermont 0.24
Illinois 62.94 New Jersey 2.94 Virginia 7.42
Indiana 53.13 New Mexico 1.49 Washington 2.37
Iowa 49.51 New York 15.73 West Virginia 2.14
Kansas 49.28 North Carolina 14.90 Wisconsin 31.33
Kentucky 24.84 North Dakota 14.69 Wyoming 1.78
1.45 Use an applet for the tormado damage data. The One-Variable Sta-
tistical Calculator applet on the text CD and Web site will make stemplots and
histograms. It is intended mainly as a learning tool rather than as a replacement
for statistical software. The histogram function is particularly useful because you
can change the number of classes by dragging with the mouse. The tornado damage
data from Exercise 1.44 are available in the applet. Choose this data set and go to
the “Histogram” tab.
(a) Sketch the default histogram that the applet first presents. If the default graph
does not have nine classes, drag it to make a histogram with nine classes and sketch
the result. This should agree with your histogram in part (b) of the previous exer-
cise.
(b) Make a histogram with one class and also a histogram with the greatest number
14 Section 1.1
of classes that the applet allows. Sketch the results.

(c) Drag the graph until you find the histogram that you think best pictures the
data. How many classes did you choose? Sketch your final histogram.
1.46 Caliornia temperatures. The following table contains data on the mean
annual temperatures (degrees Fahrenheit) for the years 1951 to 2000 at two locations
in California: Pasadena and Redding. (Data from the U.S. Historical Climatology
Network, archived at www.co2science.org. Despite claims made on this site, tem-
peratures at most U.S. locations show a gradual increase over the past century.)
Make time plots of both time series and compare their main features. You can see
why discussions of climate change often bring disagreement.
Mean annual temperatures (◦ F) in two California cities

Mean Temperature Mean Temperature
Year Pasadena Redding Year Pasadena Redding
1951 62.27 62.02 1976 64.23 63.51
1952 61.59 62.27 1977 64.47 63.89
1953 62.64 62.06 1978 64.21 64.05
1954 62.88 61.65 1979 63.76 60.38
1955 61.75 62.48 1980 65.02 60.04
1956 62.93 63.17 1981 65.80 61.95
1957 63.72 62.42 1982 63.50 59.14
1958 65.02 64.42 1983 64.19 60.66
1959 65.69 65.04 1984 66.06 61.72
1960 64.48 63.07 1985 64.44 60.50
1961 64.12 63.50 1986 65.31 61.76
1962 62.82 63.97 1987 64.58 62.94
1963 63.71 62.42 1988 65.22 63.70
1964 62.76 63.29 1989 64.53 61.50
1965 63.03 63.32 1990 64.96 62.22
1966 64.25 64.51 1991 65.60 62.73
1967 64.36 64.21 1992 66.07 63.59
1968 64.15 63.40 1993 65.16 61.55
1969 63.51 63.77 1994 64.63 61.63
1970 64.08 64.30 1995 65.43 62.62
1971 63.59 62.23 1996 65.76 62.93
1972 64.53 63.06 1997 66.72 62.48
1973 63.46 63.75 1998 64.12 60.23
1974 63.93 63.80 1999 64.85 61.88
1975 62.36 62.66 2000 66.25 61.58
1.47 What do you miss in the histogram? Make a histogram of the mean
annual temperatures at Pasadena for the years 1951 to 2000 (data appear in Exercise
1.46). Describe the distribution of temperatures. Then explain why this histogram
misses very important facts about temperatures in Pasadena.
1.48 Change the scale of the axis. The impression that a time plot gives depends
on the scales you use on the two axes. If you stretch the vertical axis and compress
the time axis, change appears to be more rapid. Compressing the vertical axis and
stretching the time axis make change appear slower. Make two more time plots of
the data for Pasadena in Exercise 1.46, one that makes mean temperature appear
to increase very rapidly and one that shows only a slow increase. The moral of this
exercise is: pay close attention to the scales when you look at a time plot.
1.49 Fish in the Bering Sea. “Recruitment,” the addition of new members to
a fish population, is an important measure of the health of ocean ecosystems. Here
are data on the recruitment of rock sole in the Bering Sea between 1973 and 2000
(National Oceanic and Atmospheric Administration, www.noaa.gov):
Recruitment Recruitment Recruitment Recruitment

Year (millions) Year (millions) Year (millions) Year (millions)
1973 173 1980 1411 1987 4700 1994 505
1974 234 1981 1431 1988 1702 1995 304
1975 616 1982 1250 1989 1119 1996 425
1976 344 1983 2246 1990 2407 1997 214
1977 515 1984 1793 1991 1049 1998 385
1978 576 1985 1793 1992 505 1999 445
1979 727 1986 2809 1993 998 2000 676
(a) Make a graph to display the distribution of rock sole recruitment, then describe
the pattern and any striking deviations that you see.
(b) Make a time plot of recruitment and describe its pattern. As is often the case
with time series data, a time plot is needed to understand what is happening.
1.50 Oil wells. How much oil the wells in a given field will ultimately produce
is key information in deciding whether to drill more wells. Here are the estimated
total amounts of oil recovered from 64 wells in the Devonian Richmond Dolomite
area of the Michigan basin, in thousands of barrels (J. Marcus Jobe and Hutch Jobe,
“A statistical approach for additional infill development,” Energy Exploration and
Exploitation, 18 (2000), pp. 89–103):
21.7 53.2 46.4 42.7 50.4 97.7 103.1 51.9 43.4 69.5
156.5 34.6 37.9 12.9 2.5 31.4 79.5 26.9 18.5 14.7
32.9 196.0 24.9 118.2 82.2 35.1 47.6 54.2 63.1 69.8
57.4 65.6 56.4 49.4 44.9 34.6 92.2 37.0 58.8 21.3
36.6 64.9 14.8 17.6 29.1 61.4 38.6 32.5 12.0 28.3
204.9 44.5 10.3 37.7 33.7 81.1 12.1 20.1 30.5 7.1
10.1 18.0 3.0 2.0
Graph the distribution and describe its main features.
1.51 Guinea pigs. The following table gives the survival times in days of 72
guinea pigs after they were injected with tubercle bacilli in a medical experiment.
(T. Bjerkedal, “Acquisition of resistance in guinea pigs infected with different doses
of virulent tubercle bacilli,” American Journal of Hygiene, 72 (1960), pp. 130–148.)
Make a suitable graph and describe the shape, center, and spread of the distribution
of survival times. Are there any outliers?
16 Section 1.2
Survival times (days) of guinea pigs

43 45 53 56 56 57 58 66 67 73
74 79 80 80 81 81 81 82 83 83
84 88 89 91 91 92 92 97 99 99
100 100 101 102 102 102 103 104 107 108
109 113 114 118 121 123 126 128 137 138
139 144 145 147 156 162 174 178 179 184
191 198 211 214 243 249 329 380 403 511
522 598
Section 1.2
1.52 The mean and median salaries paid to major league baseball players in 1993
were $490,000 and $1,160,000. Which of these numbers is the mean, and which is
the median? Explain your answer.
1.53 The NASDAQ composite index describes the average price of common stock
traded over the counter, that is, not on one of the stock exchanges. In 1991, the
mean capitalization of the companies in the NASDAQ index was $80 million and
the median capitalization was $20 million. (A company’s capitalization is the total
market value of its stock.) Explain why the mean capitalization is much higher than
the median.
1.54 A college rowing coach tests the 10 members of the women’s varsity rowing
team on a Stanford Rowing Ergometer (a stationary rowing machine). The variable
measured is revolutions of the ergometer’s flywheel in a 1-minute session. The data
are
446 552 527 504 450 583 501 545 549 506
(a) Make a stemplot of these data after rounding to two digits. Then find the mean
and the median of the original, unrounded ergometer scores. Explain the similarity
or difference in these two measures in terms of the symmetry or skewness of the
distribution.
(b) The coach used x and s to summarize these data. Find the standard deviation
s. Do you agree that this is a suitable summary?
1.55 Here are the scores on the Survey of Study Habits and Attitudes (SSHA) for
18 first-year college women:
154 109 137 115 152 140 154 178 101

103 126 126 137 165 165 129 200 148
and for 20 first-year college men:
108 140 114 91 180 115 126 92 169 146

109 132 75 88 113 151 70 115 187 104
(a) Make a back-to-back stemplot of these data, or use your result from Exercise
1.15.
(b) Find the mean x and the median M for both sets of SSHA scores. What feature
of each distribution explains the fact that x > M ?
(c) Find the five-number summaries for both sets of SSHA scores. Your plot in (a)
suggests that there is an outlier among the women’s scores. Does the 1.5 × IQR rule
flag this observation? Make side-by-side modified boxplots for the two distributions.
(d) Use your results to write a brief comparison of the two groups. Do women as
a group score higher than men? Which of your descriptions (stemplots, boxplots,
numerical measures) show this? Which group of scores is more spread out when we
ignore outliers? Which of your descriptions shows this most clearly?
1.56 The SSHA data for women given in the previous exercise contain one high
outlier. Calculate the mean x and the median M for these data with and without
the outlier. How does removing the outlier affect x? How does it affect M ? Your
results illustrate the greater resistance of the median.
1.57 Exercise 1.19 gives the number of medical doctors per 100,000 people in each
state. Your graph of the distribution in Exercise 1.19 shows that the District of
Columbia (D.C.) is a high outlier. Because D.C. is a city rather than a state, we
will omit it here.
(a) Calculate both the five-number summary and x and s for the number of doctors
per 100,000 people in the 50 states. Based on your graph, which description do you
prefer?
(b) What facts about the distribution can you see in the graph that the numerical
summaries don’t reveal? Remember that measures of center and spread are not
complete descriptions of a distribution.
1.58 It is usual in the study of investments to use the mean and standard deviation
to summarize and compare investment returns. Exercise 1.20 gives the monthly
returns on one company’s stock for 82 consecutive months.
(a) Find the mean monthly return and the standard deviation of the returns. If you
invested $100 in this stock at the beginning of a month and got the mean return,
how much would you have at the end of the month?
(b) The distribution can be described as “symmetric and unimodal, with one low
outlier.” If you invested $100 in this stock at the beginning of the worst month in
the data (the outlier), how much would you have at the end of the month? Find
the mean and standard deviation again, this time leaving out the low outlier. How
much did this one observation affect the summary measures? Would leaving out
this one observation change the median? The quartiles? How do you know, without
actual calculation? (Returns over longer periods of time, or returns on portfolios
containing several investments, tend to follow a normal distribution more closely
than these monthly returns do. So use of the mean and standard deviation is better
justified for such data.)
1.59 Find the 10th and 90th percentiles of the distribution of doctors per 100,000
population in the states, from Exercise 1.19. Which states are in the top 10%? In
the bottom 10%?
18 Section 1.2
1.60 Here are the percents of the popular vote won by the successful candidate in
each U.S. presidential election from 1948 to 2000:
Year 1948 1952 1956 1960 1964 1968 1972

Percent 49.6 55.1 57.4 49.7 61.1 43.4 60.7
Year 1976 1980 1984 1988 1992 1996 2000
Percent 50.1 50.7 58.8 53.9 43.2 49.2 47.9
(a) Make a graph to display the distribution of winners’ percents. What are the
main features of this distribution?
(b) What is the median percent of the vote won by the successful candidate in
presidential elections?
(c) Call an election a landslide if the winner’s percent falls at or above the third
quartile. Which elections were landslides?
1.61 How much do users pay for Internet service? Here are the monthly fees (in
dollars) paid by a random sample of 50 users of commercial Internet service providers
in August 2000:
20 40 22 22 21 21 20 10 20 20
20 13 18 50 20 18 15 8 22 25
22 10 20 22 22 21 15 23 30 12
9 20 40 22 29 19 15 20 20 20
20 15 19 21 14 22 21 35 20 22
(a) Make a stemplot of these data. Briefly describe the pattern you see. About
how much do you think America Online and its larger competitors were charging in
August 2000?
(b) Which observations are suspected outliers by the 1.5 × IQR rule? Which obser-
vations would you call outliers based on the stemplot? (Data from the August 2000
supplement to the Current Population Survey, from the Census Bureau Web site.)
The following table shows Consumer Reports magazine’s laboratory test results for
calories and milligrams of sodium (mostly due to salt) in a number of major brands
of hot dogs. There are three types: all beef, “meat” (mainly pork and beef, but
government regulations allow up to 15% poultry meat), and poultry. Exercises 1.62
to 1.64 analyze these data (Consumer Reports, June 1986, pp. 366–367).
Beef hot dogs Meat hot dogs Poultry hot dogs

Calories Sodium Calories Sodium Calories Sodium
186 495 173 458 129 430
181 477 191 506 132 375
176 425 182 473 102 396
149 322 190 545 106 383
184 482 172 496 94 387
190 587 147 360 102 542
158 370 146 387 87 359
139 322 139 386 99 357
175 479 175 507 170 528
148 375 136 393 113 513
152 330 179 405 135 426
111 300 153 372 142 513
141 386 107 144 86 358
153 401 195 511 143 581
190 645 135 405 152 588
157 440 140 428 146 522
131 317 138 339 144 545
149 319
135 298
132 253
1.62 Find the five-number summaries of the calorie content of the three types of
hot dogs. Then use the 1.5 × IQR rule to check for suspected outliers. Make
modified boxplots to compare the three distributions. Write a brief discussion of
your findings.
1.63 Make a stemplot of the calorie content of the 17 brands of meat hot dogs.
What is the most important feature of the overall pattern of the distribution? Are
there any outliers? Note that the five-number summary misses the big feature of
this distribution. Routine numerical summaries are never a substitute for looking
at the data.
1.64 Use graphs and numerical summaries to compare the sodium content of the
three types of hot dogs. Write a summary of your findings suitable for readers who
know no statistics. Can we hold down our sodium intake by buying poultry hot
dogs?
1.65 Exercise 1.16 presents data on the growth of chicks fed normal corn (the con-
trol group) and a new variety with better protein quality (the experimental group).
(a) The researchers used x and s to summarize the data and as a basis for further
statistical analysis. Find these measures for both groups.
(b) What kinds of distributions are best summarized by x and s? Do these distri-
butions seem to fit the criteria?
20 Section 1.2
1.66 The weights in the previous exercise are given in grams. There are 28.35 grams
in an ounce. Use the results of part (a) of the previous exercise to find the mean
and standard deviation of the weight gains measured in ounces.
1.67 In each of the following settings, give the values of a and b for the linear trans-
formation xnew = a + bx that expresses the change in units of measurement.
(a) Change a speed x measured in miles per hour into the metric system value xnew
in kilometers per hour. (A kilometer is 0.62 mile.) What is 65 miles per hour in
metric units?
(b) You are writing a report on the power of car engines. Your sources use horse-
power x. Reexpress power in watts xnew . (One horsepower is 746 watts.) What is
the power in watts of a 140-horsepower engine?
1.68 In each of the following settings, give the values of a and b for the linear trans-
formation xnew = a + bx that expresses the change in units of measurement.
(a) You want to restate water temperature x in a swimming pool, measured in
degrees Fahrenheit, as the difference xnew between x and the “normal” body tem-
perature of 98.6◦.
(b) The recommended daily allowance (RDA) for vitamin C was recently increased
to 120 milligrams. You measure milligrams of vitamin C in foods and want to
convert your results to percent of RDA.
1.69 This is a standard deviation contest. You must choose four numbers from the
whole numbers 0 to 10, with repeats allowed.
(a) Choose four numbers that have the smallest possible standard deviation.
(b) Choose four numbers that have the largest possible standard deviation.
(c) Is more than one choice possible in either (a) or (b)? Explain.
1.70 This exercise requires a calculator with a standard deviation button or statis-
tical software on a computer. The observations
10, 001 10, 002 10, 003
have mean x = 10, 002 and standard deviation s = 1. Adding a 0 in the center of
each number, the next set becomes
100, 001 100, 002 100, 003
The standard deviation remains s = 1 as more 0s are added. Use your calculator or
computer to calculate the standard deviation of these numbers, adding extra 0s until
you get an incorrect answer. How soon did you go wrong? This demonstrates that
calculators and computers cannot handle an arbitrary number of digits correctly.
1.71 College tuition and fees. A study examined the tuition and fees charged
by the 56 four-year colleges in the state of Massachusetts. Here are those charges
(in dollars), arranged in increasing order:
4,123 4,186 4,324 4,342 4,557 4,884 5,397 6,129

6,963 6,972 8,232 13,584 13,612 15,500 15,934 16,230
16,696 16,700 17,044 17,500 18,550 18,750 19,145 19,300
19,410 19,700 19,700 19,910 20,234 20,400 20,640 20,875
21,165 21,302 22,663 23,550 24,324 25,840 26,965 27,522
27,544 27,904 28,011 28,090 28,420 28,420 28,900 28,906
28,950 29,060 29,338 29,392 29,600 29,624 29,630 29,875
Find the five-number summary and make a boxplot. What distinctive feature of the
histogram do these summaries miss? Remember that numerical summaries are not
a substitute for looking at the data.
1.72 Outliers in percent of older residents. The stemplot in Exercise 1.42

displays the distribution of the percents of residents aged 65 and over in the 50
states. Stemplots help you find the five-number summary because they arrange the
observations in increasing order.
(a) Give the five-number summary of this distribution.
(b) Does the 1.5 × IQR rule identify Alaska and Florida as suspected outliers? Does
it also flag any other states?
1.73 Tormados and property damage. Exercise 1.44 gives the average prop-
erty damage caused by tornadoes over a 50-year period in each of the states. The
distribution is strongly skewed to the right.
(a) Give the five-number summary. Explain why you can see from these five num-
bers that the distribution is right-skewed.
(b) A histogram or stemplot suggests that a few states are outliers. Show that there
are no suspected outliers according to the 1.5 × IQR rule. You see once again that
a rule is not a substitute for plotting your data.
(c) Find the mean property damage. Explain why the mean and median differ so
greatly for this distribution.
1.74 Mean versus median for oil wells. Exercise 1.50 gives data on the total
oil recovered from 64 wells. Your graph in that exercise shows that the distribution
is clearly right-skewed.
(a) Find the mean and median of the amounts recovered. Explain how the relation-
ship between the mean and the median reflects the shape of the distribution.
(b) Give the five-number summary and explain briefly how it reflects the shape of
the distribution.
1.75 Effects of logging in Borneo. “Conservationists have despaired over de-

struction of tropical rainforest by logging, clearing, and burning.” These words
begin a report on a statistical study of the effects of logging in Borneo. Researchers
compared forest plots that had never been logged (Group 1) with similar plots
nearby that had been logged 1 year earlier (Group 2) and 8 years earlier (Group
3). All plots were 0.1 hectare in area. Here are the counts of trees for plots in
each group (we thank Ethan J. Temeles of Amherst College for providing the data.
His work is described in Ethan J. Temeles and W. John Kress, “Adaptation in a
plant-hummingbird association,” Science, 300 (2003), pp. 630–633):
22 Section 1.3
Group 1: 27 22 29 21 19 33 16 20 24 27 28 19
Group 2: 12 12 15 9 20 18 17 14 14 2 17 19
Group 3: 18 4 22 15 18 19 22 12 12
Give a complete comparison of the three distributions, using both graphs and nu-
merical summaries. To what extent has logging affected the count of trees? The
researchers used an analysis based on x̄ and s. Explain why this is reasonably well
justified.
1.76 Running and heart rate. How does regular running affect heart rate? The
RUNNERS data set contains heart rates for four groups of people:
Sedentary females
Sedentary males
Female runners (at least 15 miles per week)
Male runners (at least 15 miles per week)
The heart rates were measured after 6 minutes of exercise on a treadmill. There are
200 subjects in each group. Give a complete comparison of the four distributions,
using both graphs and numerical summaries. How would you describe the effect of
running on heart rate? Is the effect different for men and women?
1.77 Guinea pigs. Exercise 1.51 gives the survival times of 72 guinea pigs in a
medical study. Survival times—whether of cancer patients after treatment or of car
batteries in everyday use—are almost always right-skewed. Make a graph to verify
that this is true of these survival times. Then give a numerical summary that is
appropriate for such data. Explain in simple language, to someone who knows no
statistics, what your summary tells us about the guinea pigs.
1.78 Weight gain. A study of diet and weight gain deliberately overfed 16 vol-
unteers for eight weeks. The mean increase in fat was x̄ = 2.39 kilograms and the
standard deviation was s = 1.14 kilograms. What are x̄ and s in pounds? (A
kilogram is 2.2 pounds.)
1.79 Guinea pigs. Find the quintiles (the 20th, 40th, 60th, and 80th percentiles)
of the guinea pig survival times in Exercise 1.51. For quite large sets of data, the
quintiles or the deciles (10th, 20th, 30th, etc. , percentiles) give a more detailed
summary than the quartiles.
1.80 Changing units from inches to centimeters. Changing the unit of length
from inches to centimeters multiplies each length by 2.54 because there are 2.54
centimeters in an inch. This change of units multiplies our usual measures of spread
by 2.54. This is true of IQR and the standard deviation. What happens to the
variance when we change units in this way?
Section 1.3
1.81 The Environmental Protection Agency requires that the exhaust of each model
of motor vehicle be tested for the level of several pollutants. The level of oxides of
nitrogen (NOX) in the exhaust of one light truck model was found to vary among
individual trucks according to a Normal distribution with mean µ = 1.45 grams per
mile driven and standard deviation σ = 0.40 grams per mile. Sketch the density
curve of this Normal distribution, with the scale of grams per mile marked on the
horizontal axis.
1.82 A study of elite distance runners found a mean weight of 63.1 kilograms (kg),
with a standard deviation of 4.8 kg. Assuming that the distribution of weights is
Normal, sketch the density curve of the weight distribution with the horizontal axis
marked in kilograms. (Based on M. L. Pollock et al., “Body composition of elite
class distance runners,” in P. Milvy (ed.), The Marathon: Physiological, Medical,
Epidemiological, and Psychological Studies, New York Academy of Sciences, 1977.)
1.83 Give an interval that contains the middle 95% of NOX levels in the exhaust of
trucks using the model described in Exercise 1.81.
1.84 Use the 68–95–99.7 rule to find intervals centered at the mean that will include
68%, 95%, and 99.7% of the weights of the elite runners described in Exercise 1.82.
1.85 Eleanor scores 680 on the mathematics part of the SAT examination. The
distribution of SAT scores in a reference population is Normal with mean 500 and
standard deviation 100. Gerald takes the ACT mathematics test and scores 27.
ACT scores are normally distributed with mean 18 and standard deviation 6. Find
the z-scores for both students. Assuming that both tests measure the same kind of
ability, who has the higher score?
1.86 Three landmarks of baseball achievement are Ty Cobb’s batting average of

.420 in 1911, Ted Williams’s .406 in 1941, and George Brett’s .390 in 1980. These
batting averages cannot be compared directly because the distribution of major
league batting averages has changed over the decades. The distributions are quite
symmetric and (except for outliers such as Cobb, Williams, and Brett) reasonably
Normal. While the mean batting average has been held roughly constant by rule
changes and the balance between hitting and pitching, the standard deviation has
dropped over time. Here are the facts (Stephen Jay Gould, “Entropic homogeneity
isn’t why no one hits .400 anymore,” Discover, August 1986, pp. 60–66):
Decade Mean Std. dev.

1910s .266 .0371
1940s .267 .0326
1970s .261 .0317
Compute the standardized batting averages for Cobb, Williams, and Brett to com-
pare how far each stood above his peers.
1.87 It is possible to score higher than 800 on the SAT, but scores above 800 are
reported as 800. (That is, a student can get a reported score of 800 without a perfect
paper.) In 2000, the scores of men on the math part of the SAT approximately
followed a Normal distribution with mean 533 and standard deviation 115. What
percent of scores were above 800 (and so reported as 800)?
24 Section 1.3
1.88 Scores on the Wechsler Adult Intelligence Scale for the 20 to 34 age group
are approximately Normally distributed with mean 110 and standard deviation 25.
Scores for the 60 to 64 age group are approximately Normally distributed with mean
90 and standard deviation 25.
Sarah, who is 30, scores 135 on this test. Sarah’s mother, who is 60, also takes
the test and scores 120. Who scored higher relative to her age group, Sarah or her
mother? Who has the higher absolute level of the variable measured by the test?
At what percentile of their age groups are Sarah and her mother? (That is, what
percent of the age group has lower scores?)
1.89 The Graduate Record Examinations (GRE) are widely used to help predict
the performance of applicants to graduate schools. The range of possible scores on
a GRE is 200 to 900. The psychology department at a university finds that the
scores of its applicants on the quantitative GRE are approximately Normal with
mean µ = 544 and standard deviation σ = 103. Find the proportion of applicants
whose score X satisfies each of the following conditions:
(a) X > 700
(b) X < 500
(c) 500 < X < 800
1.90 Using either Table A or your calculator or software, find the proportion of
observations from a standard Normal distribution that satisfies each of the following
statements. In each case, sketch a standard Normal curve and shade the area under
the curve that is the answer to the question.
(a) Z < 2.85
(b) Z > 2.85
(c) Z > −1.66
(d) −1.66 < Z < 2.85
observations on a standard Normal distribution for each of the following events. In
each case, sketch a standard Normal curve with the area representing the proportion
shaded.
(a) Z ≤ −2.25
(b) Z ≥ −2.25
(c) Z > 1.77
(d) −2.25 < Z < 1.77
1.92 Find the value z of a standard Normal variable Z that satisfies each of the
following conditions. (If you use Table A, report the value of z that comes closest to
satisfying the condition.) In each case, sketch a standard Normal curve with your
value of z marked on the axis.
(a) The point z with 25% of the observations falling below it.
(b) The point z with 40% of the observations falling above it.
1.93 The variable Z has a standard Normal distribution.

(a) Find the number z such that the event Z < z has proportion 0.8.
(b) Find the number z such that the event Z > z has proportion 0.35.
1.94 The scores of a reference population on the Wechsler Intelligence Scale for
Children (WISC) are Normally distributed with µ = 100 and σ = 15.
(a) What percent of this population have WISC scores below 100?
(b) Below 80?
(c) Above 140?
(d) Between 100 and 120?
1.95 The distribution of scores on the WISC is described in the previous exercise.
What score will place a child in the top 5% of the population? In the top 1%?
1.96 In 2003, scores on the math part of the SAT approximately followed a Normal
distribution with mean 519 and standard deviation 115.
(a) What proportion of students scored above 500?
(b) What proportion scored between 400 and 600?
1.97 Some companies “grade on a bell curve” to compare the performance of their
managers and professional workers. This forces the use of some low performance
ratings, so that not all workers are graded “above average.” Until the threat of
lawsuits forced a change, Ford Motor Company’s “performance management pro-
cess” assigned 10% A grades, 80% B grades, and 10% C grades to the company’s
18,000 managers. It isn’t clear that the “bell curve” of ratings is really a Normal
distribution. Nonetheless, suppose that Ford’s performance scores are Normally dis-
tributed. One year, managers with scores less than 25 received C’s and those with
scores above 475 received A’s. What are the mean and standard deviation of the
scores?
1.98 The Survey of Study Habits and Attitudes (SSHA) is a common psychological
instrument to evaluate the attitudes of students. The SSHA is used for subjects
from seventh grade through college. Different groups have different distributions.
To prepare to use the SSHA to evaluate future teachers, researchers gave the test
to 238 college juniors majoring in elementary education. Their scores were roughly
Normal with mean 114 and standard deviation 30. Take this as the distribution of
SSHA scores in the population of future elementary school teachers.
A study of Native American education students in Canada found that this rel-
atively disadvantaged group had mean SSHA score 99. (Graham Hurlbut, Eldon
Glade, and John McLaughlin, “Teaching attitudes and study attitudes of Indian
education students,” Journal of American Indian Education, 29, No. 3, (1990), pp.
12–18.) What percentile of the overall distribution is this?
1.99 How high a score on the SSHA test of the previous exercise (mean 114, standard
deviation 30) must an elementary education student obtain to be among the highest-
scoring 30% of the population? What scores make up the lowest 30%?
1.100 The median of any Normal distribution is the same as its mean. We can
use Normal calculations to find the quartiles and related descriptive measures for
Normal distributions.
(a) What is the area under the standard Normal curve to the left of the first quartile?
Use this to find the value of the first quartile for a standard Normal distribution.
Find the third quartile similarly.
26 Section 1.3
(b) Your work in (a) gives the z-scores for the quartiles of any Normal distribution.
Scores on the Wechsler Intelligence Scale for Children (WISC) are Normally dis-
tributed with mean 100 and standard deviation 15. What are the quartiles of WISC
scores?
(c) What is the value of the IQR for the standard Normal distribution?
(d) What percent of the observations in the standard Normal distribution are sus-
pected outliers according to the 1.5 × IQR rule? (This percent is the same for any
Normal distribution.)
1.101 The distribution of Internet access costs in Exercise 1.61 has a compact center
with a long tail on either side. Make a Normal quantile plot of these data. Explain
carefully why the pattern of this plot is typical of a “long-tailed” distribution.
1.102 Is the distribution of monthly returns on Philip Morris stock approximately

Normal with the exception of possible outliers? Make a Normal quantile plot of the
data in Exercise 1.20, and report your conclusions.
1.103 Exercise 1.16 presents data on the weight gains of chicks fed two types of
corn. The researchers use x and s to summarize each of the two distributions. Make
a Normal quantile plot for each group and report your findings. Is use of x and s
justified?
1.104 Sketch density curves that might describe distributions with the following
shapes:
(a) Symmetric, but with two peaks (that is, two strong clusters of observations).
(b) Single peak and skewed to the left.
1.105 Remember that it is areas under a density curve, not the height of the curve,
that give proportions in a distribution. To illustrate this, sketch a density curve that
has its mode (peak point) at 0 on the horizontal axis but has greater area within
0.25 on either side of 1 than within 0.25 on either side of 0.
1.106 If you ask a computer to generate “random numbers” between 0 and 1, you
will get observations from a uniform distribution, whose density curve is 1 between
0 and 1. Use areas under this density curve to answer the following questions.
(a) Why is the total area under this curve equal to 1?
(b) What proportion of the observations lie above 0.75?
(c) What proportion of the observations lie between 0.25 and 0.75?
1.107 Many random number generators allow users to specify the range of the
random numbers to be produced. Suppose that you specify that the outcomes are
to be distributed uniformly between 0 and 2. Then the density curve of the outcomes
has constant height between 0 and 2, and height 0 elsewhere.
(a) What is the height of the density curve between 0 and 2? Draw a graph of the
density curve.
(b) Use your graph from (a) and the fact that areas under the curve are proportions
of outcomes to find the proportion of outcomes that are less than 1.
(c) Find the proportion of outcomes that lie between 0.5 and 1.3.
1.108 One reason that Normal distributions are important is that they describe
how the results of an opinion poll would vary if the poll were repeated many times.
About 40% of adult Americans say they are afraid to go out at night because of
crime. Take many randomly chosen samples of 1050 people. The proportions of
people in these samples who stay home for fear of crime will follow the Normal
distribution with mean 0.4 and standard deviation 0.015. Use this fact and the
68–95–99.7 rule to answer these questions.
(a) In many samples, what percent of samples give results above 0.4? Above 0.43?
(b) In a large number of samples, what range contains the central 95% of proportions
of people who stay home because of crime?
observations from a standard Normal distribution that satisfies each of the following
statements. In each case, sketch a standard Normal curve and shade the area under
the curve that is the answer to the question.
(a) Z < 1.85 (this is a cumulative proportion)
(b) Z > 1.85
(c) Z > −0.66
(d) −0.66 < Z < 1.85
observations from a standard Normal distribution for each of the following events.
In each case, sketch a standard Normal curve and shade the area representing the
proportion.
(a) Z ≤ −2 (this is a cumulative proportion)
(b) Z ≥ −2
(c) Z > 1.67
(d) −2 < Z < 1.67
1.111 Find the value z of a standard Normal variable Z that satisfies each of the
following conditions. (If you use Table A, report the value of z that comes closest to
satisfying the condition.) In each case, sketch a standard Normal curve with your
value of z marked on the axis.
(a) 20% of the observations fall below z.
(b) 30% of the observations fall above z.
1.112 The variable Z has a standard Normal distribution.

(a) Find the number z that has cumulative proportion 0.8.
(b) Find the number z such that the event Z > z has proportion 0.45.
There are two major tests of readiness for college, the ACT and the SAT. ACT
scores are reported on a scale from 1 to 36. The distribution of ACT scores for
more than 1 million students in a recent high school graduating class was roughly
Normal with mean µ = 20.8 and standard deviation σ = 4.8. SAT scores are reported
on a scale from 400 to 1600. The SAT scores for 1.4 million students in the same
graduating class were roughly Normal with mean µ = 1026 and standard deviation
σ = 209. Exercises 1.113 to 1.122 are based on this information.
28 Section 1.3
1.113 Tonya scores 1318 on the SAT. Jermaine scores 27 on the ACT. Assuming
that both tests measure the same thing, who has the higher score?
1.114 Jacob scores 16 on the ACT. Emily scores 670 on the SAT. Assuming that
both tests measure the same thing, who has the higher score?
1.115 Jose scores 1287 on the SAT. Assuming that both tests measure the same
thing, what score on the ACT is equivalent to Jose’s SAT score?
1.116 Maria scores 28 on the ACT. Assuming that both tests measure the same
thing, what score on the SAT is equivalent to Maria’s ACT score?
1.117 Reports on a student’s ACT or SAT usually give the percentile as well as the
actual score. The percentile is just the cumulative proportion stated as a percent:
the percent of all scores that were lower than this one. Tonya scores 1318 on the
SAT. What is her percentile?
1.118 Reports on a student’s ACT or SAT usually give the percentile as well as the
actual score. The percentile is just the cumulative proportion stated as a percent:
the percent of all scores that were lower than this one. Jacob scores 16 on the ACT.
What is his percentile?
1.119 It is possible to score higher than 1600 on the SAT, but scores 1600 and above
are reported as 1600. What proportion of SAT scores are reported as 1600?
1.120 It is possible to score higher than 36 on the ACT, but scores 36 and above
are reported as 36. What proportion of ACT scores are reported as 36?
1.121 What SAT scores make up the top 10% of all scores?
1.122 How well must Abigail do on the ACT in order to place in the top 20% of all
students?
1.123 Changing the mean of a Normal distribution by a moderate amount can

greatly change the percent of observations in the tails. Suppose that a college is
looking for applicants with SAT Math scores 750 and above.
(a) In 2003, the scores of men on the math SAT followed a Normal distribution with
mean 537 and standard deviation 116. What percent of men scored 750 or better?
(b) Women’s scores that year had a Normal distribution with mean 503 and standard
deviation 110. What percent of women scored 750 or better? You see that the
percent of men above 750 is more than two and one-half times the percent of women
with such high scores.
1.124 The yearly rate of return on stock indexes (which combine many individual
stocks) is approximately Normal. Between 1900 and 2002, U.S. common stocks had
a mean yearly return of 8.3%, with a standard deviation of about 20.3%. Take this
Normal distribution to be the distribution of yearly returns over a long period.
(a) In what range do the middle 95% of all yearly returns lie?
(b) The market is down for the year if the return is less than zero. In what percent
of years is the market down?

(c) In what percent of years does the index gain 25% or more?
1.125 Binge drinking survey. One reason that Normal distributions are im-
portant is that they describe how the results of an opinion poll would vary if the
poll were repeated many times. About 20% of college students say they are frequent
binge drinkers. Think about taking many randomly chosen samples of 1600 students.
The proportions of college students in these samples who say they are frequent binge
drinkers will follow the Normal distribution with mean 0.20 and standard deviation
0.01. Use this fact and the 68–95–99.7 rule to answer these questions.
(a) In many samples, what percent of samples give results above 0.2? Above 0.22?
(b) In a large number of samples, what range contains the central 95% of proportions
of students who say they are frequent binge drinkers?
1.126 Heights of women. The heights of women aged 20 to 29 are approximately

Normal with mean 64 inches and standard deviation 2.7 inches. Men the same age
have mean height 69.3 inches with standard deviation 2.8 inches. What are the
z-scores for a woman 6 feet tall and a man 6 feet tall? What information do the
z-scores give that the actual heights do not?
1.127 Proportion of women with high cholesterol. Too much cholesterol in

the blood increases the risk of heart disease. Young women are generally less afflicted
with high cholesterol than other groups. The cholesterol levels for women aged 20
to 34 follow an approximately Normal distribution with mean 185 milligrams per
deciliter (mg/dl) and standard deviation 39 mg/dl. (Results for 1988 to 1991 from
a large sample survey, reported in National Center for Health Statistics, Health,
United States, 1995, 1996.) (a) Cholesterol levels above 240 mg/dl demand medical
attention. What percent of young women have levels above 240 mg/dl?
(b) Levels above 200 mg/dl are considered borderline high. What percent of young
women have blood cholesterol between 200 and 240 mg/dl?
1.128 Proportion of men with high cholesterol. Middle-aged men are more
susceptible to high cholesterol than the young women of the previous exercise. The
blood cholesterol levels of men aged 55 to 64 are approximately Normal with mean
222 mg/dl and standard deviation 37 mg/dl. What percent of these men have high
cholesterol (levels above 240 mg/dl)? What percent have borderline high cholesterol
(between 200 and 240 mg/dl)?
1.129 Logging in Borneo. The study of the effects of logging on tree counts
in the Borneo rain forest (Exercise 1.75) used statistical methods that are based
on Normal distributions. Make Normal quantile plots for each of the three groups
of forest plots. Are the three distributions roughly Normal? What are the most
prominent deviations from Normality that you see?
Chapter 1 Review Exercises
1.130 Here is a stemplot of the percents of residents aged 25 to 34 in each of the 50

states. The stems are whole percents and the leaves are tenths of a percent.
30 Review Exercises
10 9
11 0
12 1344677889
13 0012455566789999
14 11222344445789
15 24478999
(a) Montana and Wyoming have the smallest percents of young adults, perhaps
because they lack job opportunities. What are the percents for these two states?
(b) Ignoring Montana and Wyoming, describe the shape, center, and spread of this
distribution.
1.131 Stemplots help you find the five-number summary because they arrange the
observations in increasing order. The previous exercise gives a stemplot of the
percent of residents aged 25 to 34 in each of the 50 states.
(a) Find the five-number summary of this distribution.
(b) Does the 1.5 × IQR criterion flag Montana and Wyoming as suspected outliers?
(c) How much does the median change if you omit Montana and Wyoming?
1.132 The American Housing Survey provides data on all housing units in the
United States—houses, apartments, mobile homes, and so on. Here are the years
in which a random sample of 100 housing units were built. The survey does not
produce exact dates for years before 1990. Years before 1920 are given as 1919.
Dates between 1920 and 1970 are given in ten-year blocks, so that a unit built in
1956 appears as 1950. Dates between 1970 and 1990 are given in five-year blocks,
so that 1987 appears as 1985.
1960 1920 1991 1919 1985 1985 1975 1980 1975 1985
1930 1993 1985 1975 1970 1970 1975 1980 1940 1940
1980 1919 1980 1950 1940 1950 1993 1985 1975 1960
1919 1950 1960 1975 1950 1919 1920 1985 1970 1975
1930 1975 1960 1920 1940 1950 1985 1990 1950 1970
1985 1920 1950 1980 1975 1950 1950 1919 1919 1985
1985 1991 1980 1960 1940 1960 1930 1998 1994 1960
1919 1975 1919 1950 1975 1930 1919 1970 1920 1930
1950 1975 1970 1985 1919 1960 1930 1980 1960 1950
1996 1940 1950 1998 1930 1919 1930 1950 1950 1920
(a) Make a histogram of these dates, using classes 10 years wide beginning with
1910 to 1919. The first class will contain all housing units built before 1920. In
which decades after 1920 were most housing units that still exist built?
(b) Give the five-number summary of these data. Write a brief warning on how to
interpret your results. For example, what does the fact that the median is 1960 tell
us about the age of American housing?
1.133 The following table shows the salaries paid to the members of the New York
Yankees baseball team as of opening day of the 2001 season. Display this distribution
with a graph and describe its main features. Find the mean and median salary and
explain how the pattern of the distribution explains the relationship between these
two measures of center. Find the standard deviation and the quartiles. Do you prefer
the five-number summary or x and s as a quick description of this distribution?
Player Salary Player Salary Player Salary

Jeter $12,600,000 Posada $4,050,000 Spencer $320,000
B. Williams 12,357,143 Stanton 2,450,000 T. Williams 320,000
Clemens 10,300,000 Hernandez 2,050,000 Almanzar 270,000
Mussina 10,000,000 Watson 1,700,000 Bellinger 230,000
Rivera 9,150,000 Mendoza 1,600,000 Einertson 206,000
Justice 7,000,000 Oliver 1,100,000 Choate 204,750
Pettitte 7,000,000 Rodriguez 850,000 Coleman 204,000
O’Neill 6,500,000 Soriano 630,000 Jimenez 200,000
Knoblauch 6,000,000 Sojo 500,000 Parker 200,000
Martinez 6,000,000 Boehringer 350,000 Seabol 200,000
Brosius 5,250,000
1.134 At the time the salaries in the previous exercise were announced, one U.S.
dollar was worth 1.72 Swiss francs. Answer these questions without doing any
calculations in addition to those you did in the previous exercise.
(a) What transformation converts a salary in dollars into the same salary in Swiss
francs?
(b) What are the mean, median, and quartiles of the distribution in francs?
(c) What are the standard deviation and interquartile range of the distribution in
francs?
1.135 The Internal Revenue Service reports that in 1998 about 124 million individ-
ual income tax returns showed adjusted gross income (AGI) greater than 0. The
mean and median AGI on these tax returns were $25,491 and $44,186. Which of
these numbers is the mean and which is the median? How do you know?
1.136 The Bureau of Justice Statistics says that in 1999, 51% of homicides were
committed with handguns, 14% with other firearms, 13% with knives, and 6% with
blunt objects. Make a graph to display these data. Do you need an “other methods”
category?
1.137 You are planning a sample survey of households in California. You decide to
select households separately within each county and to choose more households from
the more populous counties. To aid in the planning, the following table gives the
populations of California counties from the 2000 census. Examine the distribution
of county populations both graphically and numerically, using whatever tools are
most suitable. Write a brief description of the main features of this distribution.
Sample surveys often select households from all of the most populous counties but
from only some of the less populous. How would you divide California counties into
three groups according to population, with the intent of including all of the first
group, half of the second, and a smaller fraction of the third in your survey?
32 Review Exercises
County Population County Population County Population

Alameda 1,443,741 Marin 247,289 San Mateo 707,161
Alpine 1,208 Mariposa 17,130 Santa Barbara 399,347
Amador 35,100 Mendocino 86,265 Santa Clara 1,682,585
Butte 203,171 Merced 210,554 Santa Cruz 255,602
Calaveras 40,554 Modoc 9,449 Shasta 163,256
Colusa 18,804 Mono 12,853 Sierra 3,555
Contra Costa 948,816 Monterey 401,762 Siskiyou 44,301
Del Norte 27,507 Napa 124,279 Solano 394,542
El Dorado 156,299 Nevada 92,033 Sonoma 458,614
Fresno 799,407 Orange 2,846,289 Stanislaus 446,997
Glenn 26,453 Placer 248,399 Sutter 78,930
Humboldt 126,518 Plumas 20,824 Tehama 56,039
Imperial 142,361 Riverside 1,545,387 Trinity 13,022
Inyo 17,945 Sacramento 1,223,499 Tulare 368,021
Kern 661,645 San Benito 53,234 Tuolumne 54,501
Kings 129,461 San Bernardino 1,709,434 Ventura 753,197
Lake 58,309 San Diego 2,813,833 Yolo 168,660
Lassen 33,828 San Francisco 776,733 Yuba 60,219
Los Angeles 9,519,338 San Joaquin 563,598
Madera 123,109 San Luis Obispo 246,681
1.138 The Chapin Social Insight Test evaluates how accurately the subject ap-
praises other people. In the reference population used to develop the test, scores are
approximately Normally distributed with mean 25 and standard deviation 5. The
range of possible scores is 0 to 41.
(a) What proportion of the population has scores below 20 on the Chapin test?
(b) What proportion has scores below 10?
(c) How high a score must you have in order to be in the top quarter of the popu-
lation in social insight?
1.139 The Chapin Social Insight Test described in the previous exercise has a mean
of 25 and a standard deviation of 5. You want to rescale the test using a linear
transformation so that the mean is 100 and the standard deviation is 20. Let x
denote the score in the original scale and xnew be the transformed score.
(a) Find the linear transformation required. That is, find the values of a and b in
the equation xnew = a + bx.
(b) Give the rescaled score for someone who scores 30 in the original scale.
(c) What are the quartiles of the rescaled scores?
1.140 The Florida State University Seminoles have been among the more success-
ful teams in college football. The following table gives the weights in pounds and
positions of the players on the 2000–2001 football team, which was defeated in the
national title game by the University of Oklahoma. The positions are quarter-
back (QB), running back (RB), offensive line (OL), receiver (R), tight end (TE),
kicker (K), defensive back (DB), linebacker (LB), and defensive line (DL). (From
the Florida State University athletics Web site, seminoles.fansonly.com.)
(a) Make side-by-side modified boxplots of the weights for running backs, receivers,
offensive linemen, defensive linemen, linebackers, and defensive backs.
(b) Briefly compare the weight distributions. Which position has the heaviest play-
ers overall? Which has the lightest?
(c) Are any individual players outliers within their position?
QB 235 QB 220 QB 215 QB 228 K 175 K 205

K 220 K 185 RB 200 RB 188 RB 190 RB 215
RB 190 RB 225 RB 225 RB 240 RB 237 R 205
R 185 R 185 R 190 R 195 R 201 R 190
R 195 R 180 OL 291 OL 280 OL 300 OL 320
OL 325 OL 285 OL 305 OL 305 OL 290 OL 310
OL 315 OL 285 OL 290 OL 325 OL 310 OL 256
OL 305 OL 300 DB 170 DB 207 DB 185 DB 175
DB 180 DB 190 DB 210 DB 200 DB 180 DB 195
DB 185 DB 170 DB 180 DB 190 DB 190 LB 220
LB 212 LB 233 LB 190 LB 195 LB 215 LB 220
LB 240 LB 230 LB 220 LB 225 LB 200 DL 260
DL 245 DL 264 DL 254 DL 215 DL 250 DL 295
DL 240 DL 275 DL 255 DL 285 DL 245 DL 270
DL 250 DL 250 TE 255 TE 245 TE 260 TE 245
1.141 Distance-learning courses. The 222 students enrolled in distance-learning

courses offered by a college ranged from 18 to 64 years of age. The mode of their
ages was 19. The median age was 31. (Julie Reinhart and Paul Schneider, “Student
satisfaction, self-efficacy, and the perception of the two-way audio/visual distance
learning environment,” Quarterly Review of Distance Education, 2 (2001), pp. 357–
365.) Explain how this can happen.
1.142 By-products from DDT. By-products from the pesticide DDT were major
threats to the survival of birds of prey until use of DDT was banned at the end of
1972. Can time plots show the effect of the ban? Here are two sets of data for bald
eagles nesting in the forests of northwestern Ontario. (James W. Grier, “Ban of
DDT and subsequent recovery of reproduction of bald eagles,” Science, 218 (1982),
pp. 1232–1235.) The following data set gives the mean number of young per breeding
area:
Year 1966 1967 1968 1969 1970 1971 1972 1973

Young 1.26 0.73 0.89 0.84 0.54 0.60 0.54 0.78
Year 1974 1975 1976 1977 1978 1979 1980 1981
Young 0.46 0.77 0.86 0.96 0.82 0.98 0.93 1.12
The following data are measurements of the chemical DDE (the by-product of DDT
that most threatens birds of prey) from bald eagle eggs in the same area of Canada.
These are in parts per million (ppm). There are often several measurements per
year.
Year 1967 1967 1968 1971 1971 1972 1976 1976 1976 1976
DDE 44 95 121 125 95 87 13.3 16.4 50.4 59.8
Year 1976 1977 1977 1980 1980 1980 1981 1981 1981
DDE 56.4 0.6 23.8 16.6 14.5 24.0 7.8 48.2 53.4
Make time plots of eagle young and of mean DDE concentration in eggs. How does
the effect of banning DDT appear in your plots?
1.143 Damage caused by tornadoes. The average damage caused by tornadoes

in the states (see Exercise 1.44) and the estimated amount of oil recovered from
different oil wells (Exercise 1.50) both have right-skewed distributions. Choose one
of these data sets. Make a Normal quantile plot. How is the skewness of the
distribution visible in the plot? Based on the plot, which observations (if any)
would you call outliers?
1.144 Proportions older than 65. We know that the distribution of the percents
of state residents over 65 years of age has a low outlier (Alaska) and a high outlier
(Florida). The stemplot in Exercise 1.42 looks unimodal and roughly symmetric.
(a) Sketch what a Normal quantile plot would look like for a distribution that is
Normal except for two outliers, one in each direction.
(b) If your software includes Normal quantile plots, make a plot of the percent-over-
65 data and discuss what you see.
1.145 SAT Mathematics scores and grade point averages. The CSDATA
data set contains information on 234 computer science students. We are interested in
comparing the SAT Mathematics scores and grade point averages of female students
with those of male students. Make two sets of side-by-side boxplots to carry out these
comparisons. Write a brief discussion of the male-female comparisons. Then make
Normal quantile plots of grade point averages and SAT Math scores separately for
men and women. Which students are clear outliers? Which of the four distributions
are approximately Normal if we ignore outliers?
34
CHAPTER 2
Section 2.1
2.1 How well does a child’s height at age 6 predict height at age 16? To find out,
measure the heights of a large group of children at age 6, wait until they reach
age 16, then measure their heights again. What are the explanatory and response
variables here? Are these variables categorical or quantitative?
2.2 Here are the golf scores of 12 members of a college women’s golf team in two
rounds of tournament play. (A golf score is the number of strokes required to
complete the course, so low scores are better.)
Player 1 2 3 4 5 6 7 8 9 10 11 12
Round 1 89 90 87 95 86 81 102 105 83 88 91 79
Round 2 94 85 89 89 81 76 107 89 87 91 88 80
(a) Make a scatterplot of the data, taking the first-round score as the explanatory
variable.
(b) Is there an association between the two scores? If so, is it positive or negative?
Explain why you would expect scores in two rounds of a tournament to have an
association like that which you observed.
(c) The plot shows one outlier. Circle it. The outlier may occur because a good
golfer had an unusually bad round or because a weaker golfer had an unusually good
round. Can you tell from the data given whether the outlier is from a good player
or from a poor player? Explain your answer.
2.3 There is some evidence that drinking moderate amounts of wine helps prevent
heart attacks. Here are data on yearly wine consumption (liters of alcohol from
drinking wine, per person) and yearly deaths from heart disease (deaths per 100,000
people) in 19 developed nations. (M. H. Criqui, University of California, San Diego,
reported in the New York Times, December 28, 1994.)
Alcohol Heart disease Alcohol Heart disease

Country from wine deaths Country from wine deaths
Australia 2.5 211 Netherlands 1.8 167
Austria 3.9 167 New Zealand 1.9 266
Belgium 2.9 131 Norway 0.8 227
Canada 2.4 191 Spain 6.5 86
Denmark 2.9 220 Sweden 1.6 207
Finland 0.8 297 Switzerland 5.8 115
France 9.1 71 United Kingdom 1.3 285
Iceland 0.8 211 United States 1.2 199
Ireland 0.7 300 West Germany 2.7 172
Italy 7.9 107
(a) Make a scatterplot that shows how national wine consumption helps explain
heart disease death rates.
35
36 Chapter 2 Exercises
(b) Describe the form of the relationship. Is there a linear pattern? How strong is
the relationship?
(c) Is the direction of the association positive or negative? Explain in simple lan-
guage what this says about wine and heart disease. Do you think these data give
good evidence that drinking wine causes a reduction in heart disease deaths? Why?
2.4 The National Assessment of Educational Progress (NAEP) assesses what stu-
dents know in several subject areas based on large representative samples. The
following table reports some findings of the NAEP year 2000 Mathematics Assess-
ment for fourth-graders in the 40 states that participated. For each state we give
the mean NAEP math score (out of 500) and also the percent of students who were
at least “proficient” in the sense of being able to use math skills to solve real-world
problems. Nationally, about 25% of students are “proficient” by NAEP’s standards.
We expect that average performance and percent of proficient performers will be
strongly related.
Mean NAEP Percent Percent Mean NAEP Percent Percent

State score proficient poverty State score proficient poverty
Alabama 218 14 21.8 Missouri 229 24 14.4
Arizona 219 17 23.6 Montana 230 25 21.2
Arkansas 217 14 13.1 Nebraska 226 24 14.8
California 214 15 22.3 Nevada 220 16 12.8
Connecticut 234 32 13.4 New Mexico 214 12 23.5
Georgia 220 18 24.7 New York 227 22 28.9
Hawaii 216 14 14.5 North Carolina 232 28 21.3
Idaho 227 21 17.4 North Dakota 231 25 17.2
Illinois 225 22 12.2 Ohio 231 26 16.0
Indiana 234 31 12.6 Oklahoma 225 17 19.9
Iowa 233 28 14.2 Oregon 227 24 19.4
Kansas 232 30 13.3 Rhode Island 225 23 20.5
Kentucky 221 17 16.7 South Carolina 220 18 17.6
Louisiana 218 14 29.8 Tennessee 220 13 14.5
Maine 231 24 12.0 Texas 233 27 20.1
Maryland 222 22 8.1 Utah 227 24 11.8
Massachusetts 235 33 15.0 Vermont 232 30 12.2
Michigan 231 29 14.8 Virginia 230 25 7.9
Minnesota 235 34 12.6 West Virginia 225 18 25.7
Mississippi 211 9 19.3 Wyoming 229 25 13.0
(a) Make a scatterplot, using mean NAEP score as the explanatory variable. Notice
that there are several pairs of states with identical values. Use a different symbol
for points that represent two states.
(b) Describe the form, direction, and strength of the relationship.
(c) Circle your home state’s point in the scatterplot. Although there are no clear
outliers, there are some points that you may consider interesting, perhaps because
they are on the edge of the pattern. Choose one such point: Which state is this,
and in what way is it interesting?
2.5 Water flowing across farmland washes away soil. Researchers released water
across a test bed at different flow rates and measured the amount of soil washed
away. The following table gives the flow (in liters per second) and the weight (in
kilograms) of eroded soil. (G. R. Foster, W. R. Ostercamp, and L. J. Lane, “Effect
Section 2.1 37
of discharge rate on rill erosion,” paper presented at the 1982 Winter Meeting of
the American Society of Agricultural Engineers.)
Flow rate 0.31 0.85 1.26 2.47 3.75

Eroded soil 0.82 1.95 2.18 3.01 6.07
(a) Plot the data. Which is the explanatory variable?

(b) Describe the pattern that you see. Would it be reasonable to describe the overall
pattern by a straight line? Is the association positive or negative?
2.6 Here are data on a group of people who contracted botulism, a form of food
poisoning that can be fatal. The variables recorded are the person’s age in years,
the incubation period (the time in hours between eating the infected food and the
first signs of illness), and whether the person survived (S) or died (D). (Modified
from data provided by Dana Quade, University of North Carolina.)
Person 1 2 3 4 5 6 7 8 9
Age 29 39 44 37 42 17 38 43 51
Incubation 13 46 43 34 20 20 18 72 19
Outcome D S S D D S D S D
Person 10 11 12 13 14 15 16 17 18
Age 30 32 59 33 31 32 32 36 50
Incubation 36 48 44 21 32 86 48 28 16
Outcome D D S D D S D S D
(a) Make a scatterplot of incubation period against age, using different symbols for
people who survived and those who died.
(b) Is there an overall relationship between age and incubation period? If so, describe
it.
(c) More important, is there a relationship between either age or incubation period
and whether the victim survived? Describe any relations that seem important here.
(d) Are there any unusual observations that may require individual investigation?
2.7 The presence of harmful insects in farm fields is detected by erecting boards
covered with a sticky material and examining the insects trapped on the boards.
Some colors are more attractive to insects than others. In an experiment aimed at
determining the best color for attracting cereal leaf beetles, six boards of each of
four colors were placed in a field of oats in July. The following table gives data on
the number of cereal leaf beetles trapped. (M. C. Wilson and R. E. Shade, “Relative
attractiveness of various luminescent colors to the cereal leaf beetle and the meadow
spittlebug,” Journal of Economic Entomology, 60 (1967), pp. 578–580.)
Board color Insects trapped

Lemon yellow 45 59 48 46 38 47
White 21 12 14 17 13 17
Green 37 32 15 25 39 41
Blue 16 11 20 21 14 7
(a) Make a plot of the counts of insects trapped against board color (space the four
colors equally on the horizontal axis). Compute the mean count for each color, add
the means to your plot, and connect the means with line segments.
(b) Based on the data, state your conclusions about the attractiveness of these colors
to the beetles.
(c) Does it make sense to speak of a positive or negative association between board
color and insect count?
2.8 In each of the following situations, is it more reasonable to simply explore

the relationship between the two variables or to view one of the variables as an
explanatory variable and the other as a response variable? In the latter case, which
is the explanatory variable and which is the response variable?
(a) The amount of time spent studying for a statistics exam and the grade on the
exam.
(b) The weight in kilograms and height in centimeters of a person.
(c) Inches of rain in the growing season and the yield of corn in bushels per acre.
(d) A student’s scores on the SAT math exam and the SAT verbal exam.
(e) A family’s income and the years of education their eldest child completes.
2.9 City and highway gas mileage. The following table gives the city and
highway gas mileages for minicompact and two-seater cars. We expect a positive
association between the city and highway mileages of a group of vehicles. We have
already seen that the Honda Insight is a different type of car, so omit it as you work
with these data.
(a) Make a scatterplot that shows the relationship between city and highway mileage,
using city mileage as the explanatory variable. Use different plotting symbols for
the two types of cars.
(b) Interpret the plot. Is there a positive association? Is the form of the plot roughly
linear? Is the form of the relationship similar for the two types of cars? What is
the most important difference between the two types?
Section 2.1 39
Fuel economy (miles per gallon) for 2004 model vehicles

Two-Seater Cars Minicompact Cars
Model City Highway Model City Highway
Acura NSX 17 24 Aston Martin Vanquish 12 19
Audi TT Roadster 20 28 Audi TT Coupe 21 29
BMW Z4 Roadster 20 28 BMW 325CI 19 27
Cadillac XLR 17 25 BMW 330CI 19 28
Chevrolet Corvette 18 25 BMW M3 16 23
Dodge Viper 12 20 Jaguar XK8 18 26
Ferrari 360 Modena 11 16 Jaguar XKR 16 23
Ferrari Maranello 10 16 Lexus SC 430 18 23
Ford Thunderbird 17 23 Mini Cooper 25 32
Honda Insight 60 66 Mitsubishi Eclipse 23 31
Lamborghini Gallardo 9 15 Mitsubishi Spyder 20 29
Lamborghini Murcielago 9 13 Porsche Cabriolet 18 26
Lotus Esprit 15 22 Porsche Turbo 911 14 22
Maserati Spyder 12 17
Mazda Miata 22 28
Mercedes-Benz SL500 16 23
Nissan 350Z 20 26
Porsche Boxster 20 29
Porsche Carrera 911 15 23
Toyota MR2 26 32
2.10 Biological clocks. Many plants and animals have “biological clocks” that
coordinate activities with the time of day. When researchers looked at the length
of the biological cycles in the plant Arabidopsis by measuring leaf movements, they
found that the length of the cycle is not always 24 hours. The researchers suspected
that the plants adapt their clocks to their north-south position. Plants don’t know
geography, but they do respond to light, so the researchers looked at the relationship
between the plants’ cycle lengths and the length of the day on June 21st at their
locations. The data file includes data on cycle length and day length, both in
hours, for 146 plants. (We thank C. Robertson McClung of Dartmouth College for
supplying the data. The study is reported in Todd P. Michael et al., “Enhanced
fitness conferred by naturally occurring variation in the circadian clock,” Science,
302 (2003), pp. 1049–1053.) Plot cycle length as the response variable against day
length as the explanatory variable. Does there appear to be a positive association?
Is it a strong association? Explain your answers.
2.11 Two problems with feet. Metatarsus adductus (call it MA) is a turning
in of the front part of the foot that is common in adolescents and usually corrects
itself. Hallux abducto valgus (call it HAV) is a deformation of the big toe that is
not common in youth and often requires surgery. Perhaps the severity of MA can
help predict the severity of HAV. The following table gives data on 38 consecutive
patients who came to a medical center for HAV surgery. (Alan S. Banks et al.,
“Juvenile hallux abducto valgus association with metatarsus adductus,” Journal of
the American Podiatric Medical Association, 84 (1994), pp. 219–224.) Using X-rays,
doctors measured the angle of deformity for both MA and HAV. They speculated
that there is a positive association—more serious MA is associated with more serious
HAV.
(a) Make a scatterplot of the data in the following table. (Which is the explanatory
variable?)
(b) Describe the form, direction, and strength of the relationship between MA angle
and HAV angle. Are there any clear outliers in your graph?
(c) Do you think the data confirm the doctors’ speculation?
Two measurements of foot deformities
HAV MA HAV MA HAV MA

angle angle angle angle angle angle
28 18 21 15 16 10
32 16 17 16 30 12
25 22 16 10 30 10
34 17 21 7 20 10
38 33 23 11 50 12
26 10 14 15 25 25
25 18 32 12 26 30
18 13 25 16 28 22
30 19 21 16 31 24
26 10 22 18 38 20
28 17 20 10 32 37
13 14 18 15 21 23
20 20 26 16
2.12 Fuel consumption and speed. How does the fuel consumption of a car
change as its speed increases? Here are data for a British Ford Escort. Speed
is measured in kilometers per hour, and fuel consumption is measured in liters of
gasoline used per 100 kilometers traveled. (Based on T. N. Lam, “Estimating fuel
consumption from engine size,” Journal of Transportation Engineering, 111 (1985),
pp. 339–357. The data for 10 to 50 km/h are measured; those for 60 and higher are
calculated from a model given in the paper and are therefore smoothed.)
Speed Fuel used Speed Fuel used
(km/h) (liters/100 km) (km/h) (liter/100 km)
10 21.00 90 7.57
20 13.00 100 8.27
30 10.00 110 9.03
40 8.00 120 9.87
50 7.00 130 10.79
60 5.90 140 11.77
70 6.30 150 12.83
80 6.95
(a) Make a scatterplot. (Which variable should go on the x axis?)
(b) Describe the form of the relationship. In what way is it not linear? Explain why
Section 2.1 41
the form of the relationship makes sense.

(c) It does not make sense to describe the variables as either positively associated
or negatively associated. Why not?
(d) Is the relationship reasonably strong or quite weak? Explain your answer.
2.13 Worms and plant growth. To demonstrate the effect of nematodes (micro-
scopic worms) on plant growth, a botanist introduces different numbers of nematodes
into 16 planting pots. He then transplants a tomato seedling into each pot. Here
are data on the increase in height of the seedlings (in centimeters) 14 days after
planting: (Data provided by Matthew Moore.)
Nematodes Seedling growth
0 10.8 9.1 13.5 9.2
1,000 11.1 11.1 8.2 11.3
5,000 5.4 4.6 7.4 5.0
10,000 5.8 5.3 3.2 7.5
(a) Make a scatterplot of the response variable (growth) against the explanatory vari-
able (nematode count). Then compute the mean growth for each group of seedlings,
plot the means against the nematode counts, and connect these four points with
line segments.
(b) Briefly describe the conclusions about the effects of nematodes on plant growth
that these data suggest.
2.14 Mutual funds. Fidelity Investments, like other large mutual funds companies,
offers many “sector funds” that concentrate their investments in narrow segments
of the stock market. These funds often rise or fall by much more than the market
as a whole. We can group them by broader market sector to compare returns. Here
are percent total returns for 23 Fidelity “Select Portfolios” funds for the year 2003,
a year in which stocks rose sharply: (Compiled from Fidelity data in the Fidelity
Insight newsletter, 20 (2004), No. 1.)
Market sector Fund returns (percent)
Consumer 23.9 14.1 41.8 43.9 31.1
Financial services 32.3 36.5 30.6 36.9 27.5
Technology 26.1 62.7 68.1 71.9 57.0 35.0 59.4
Natural resources 22.9 7.6 32.1 28.7 29.5 19.1
(a) Make a plot of total return against market sector (space the four market sectors
equally on the horizontal axis). Compute the mean return for each sector, add the
means to your plot, and connect the means with line segments.
(b) Based on the data, which of these market sectors were the best places to invest
in 2003? Hindsight is wonderful.
(c) Does it make sense to speak of a positive or negative association between market
sector and total return?
2.15 Mutual funds in another year. The data for 2003 in the previous exercise
make sector funds look attractive. Stocks rose sharply in 2003, after falling sharply
in 2002 (and also in 2001 and 2000). Let’s look at the percent returns for both 2003
and 2002 for these same 23 funds. Here they are:
2002 2003 2002 2003 2002 2003

return return return return return return
−17.1 23.9 −0.7 36.9 −37.8 59.4
−6.7 14.1 −5.6 27.5 −11.5 22.9
−21.1 41.8 −26.9 26.1 −0.7 36.9
−12.8 43.9 −42.0 62.7 64.3 32.1
−18.9 31.1 −47.8 68.1 −9.6 28.7
−7.7 32.3 −50.5 71.9 −11.7 29.5
−17.2 36.5 −49.5 57.0 −2.3 19.1
−11.4 30.6 −23.4 35.0
Do a careful graphical analysis of these data: side-by-side comparison of the dis-
tributions of returns in 2002 and 2003 and also a description of the relationship
between the returns of the same funds in these two years. What are your most
important findings? (The outlier is Fidelity Gold Fund.)
Section 2.2
2.16 Archaeopteryx is an extinct beast having feathers like a bird but teeth and a
long bony tail like a reptile. Only six fossil specimens are known. Here are data
on the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in
the upper arm) for the five specimens that preserve both bones: (Marilyn A. Houck
et al., “Allometric scaling in the earliest fossil bird, Archaeopteryx lithographica,”
Science, 247 (1990), pp. 195–198.)
Femur 38 56 59 64 74
Humerus 41 63 70 72 84
(a) Make a scatterplot with femur length on the horizontal axis. There is a strong
positive linear relationship.
(b) Find the correlation r step-by-step. That is, find the mean and standard devia-
tion of the femur lengths and of the humerus lengths. Then find the five standardized
values for each variable and use the formula for r.
(b) Now enter these data into your calculator or software and use the correlation
function to find r. Check that you get the same result as in (a).
2.17 Changing the units of measurement can dramatically alter the appearance of a
scatterplot. Return to the fossil data from the previous exercise. The measurements
are in centimeters. Suppose a deranged scientist measured the femur in meters and
the humerus in millimeters. The data would then be
Femur 0.38 0.56 0.59 0.64 0.74
Humerus 410 630 700 720 840
(a) Draw an x axis extending from 0 to 75 and a y axis extending from 0 to 850.
Plot the original data on these axes. Then plot the new data on the same axes in a
different color. The two plots look very different.
(b) Nonetheless, the correlation is exactly the same for the two sets of measurements.
Why do you know that this is true without doing any calculations? Find the two
correlations to verify that they are the same.
Section 2.2 43
2.18 Here are the golf scores of 11 members of a women’s golf team in two rounds
of college tournament play.
Player 1 2 3 4 5 6 7 8 9 10 11
Round 1 89 90 87 95 86 81 105 83 88 91 79
Round 2 94 85 89 89 81 76 89 87 91 88 80
Make a scatterplot of the data. Find the correlation between the Round 1 and
Round 2 scores. Remove Player 7’s scores and find the correlation for the remaining
10 players. Explain carefully why removing this single case substantially increases
the correlation.
2.19 A mutual fund company’s newsletter says, “A well-diversified portfolio includes

assets with low correlations.” The newsletter includes a table of correlations between
the annual returns on various classes of investments. For example, the correlation
between municipal bonds and large-cap stocks is 0.50 and the correlation between
municipal bonds and small-cap stocks is 0.21.
(a) Rachel invests heavily in municipal bonds. She wants to diversify by adding an
investment whose returns do not closely follow the returns on her bonds. Should she
choose large-cap stocks or small-cap stocks for this purpose? Explain your answer.
(b) If Rachel wants an investment that tends to increase when the return on her
bonds drops, what kind of correlation should she look for?
2.20 Many mutual funds compare their performance with that of a benchmark,
an index of the returns on all securities of the kind the fund buys. The Vanguard
International Growth Fund, for example, takes as its benchmark the Morgan Stanley
EAFE (Europe, Australasia, Far East) index of overseas stock market performance.
Here are the percent returns for the fund and for the EAFE from 1982 (the first
full year of the fund’s existence) to 2000. (From the performance data for the fund
presented at the Vanguard Group Web site, personal.vanguard.com.)
Year Fund EAFE Year Fund EAFE

1982 5.27 −0.86 1992 −5.79 −11.85
1983 43.08 24.61 1993 44.74 32.94
1984 −1.02 7.86 1994 0.76 8.06
1985 56.94 56.72 1995 14.89 11.55
1986 56.71 69.94 1996 14.65 6.36
1987 12.48 24.93 1997 4.12 2.06
1988 11.61 28.59 1998 16.93 20.33
1989 24.76 10.80 1999 26.34 27.30
1990 −12.05 −23.20 2000 −8.60 −13.96
1991 4.74 12.50 2001 −18.92 −21.44
Make a scatterplot suitable for predicting fund returns from EAFE returns. Is
there a clear straight-line pattern? How strong is this pattern? (Give a numerical
measure.) Are there any extreme outliers from the straight-line pattern?
2.21 You are going to use the Correlation and Regression applet to make different
scatterplots with 10 points that have correlation close to 0.7. Many patterns can
have the same correlation. Always plot your data before you trust a correlation.
(a) Stop after adding the first two points. What is the value of the correlation?
Why does it have this value no matter where the two points are located?
(b) Make a lower-left to upper-right pattern of 10 points with correlation about
r = 0.7. (You can drag points up or down to adjust r after you have 10 points.)
Make a rough sketch of your scatterplot.
(c) Make another scatterplot with nine points in a vertical stack at the left of the
plot. Add one point far to the right and move it until the correlation is close to 0.7.
Make a rough sketch of your scatterplot.
(d) Make yet another scatterplot with 10 points in a curved pattern that starts at
the lower left, rises to the right, then falls again at the far right. Adjust the points
up or down until you have a quite smooth curve with correlation close to 0.7. Make
a rough sketch of this scatterplot also.
2.22 Go to the Correlation and Regression applet. Click on the scatterplot to create
a group of 10 points in the lower-left corner of the scatterplot with a strong straight-
line pattern (correlation about 0.9).
(a) Add one point at the upper right that is in line with the first 10. How does the
correlation change?
(b) Drag this last point down until it is opposite the group of 10 points. How small
can you make the correlation? Can you make the correlation negative? A single
outlier can greatly strengthen or weaken a correlation. Always plot your data to
check for outlying points.
2.23 Coffee prices and deforestation. Coffee is a leading export from several
developing countries. When coffee prices are high, farmers often clear forest to
plant more coffee trees. Here are data for five years on prices paid to coffee growers
in Indonesia and the rate of deforestation in a national park that lies in a coffee-
producing region: (Data from a plot in James A. Levine, Norman L. Eberhardt,
and Michael D. Jensen, “Role of nonexercise activity thermogenesis in resistance to
fat gain in humans,” Science, 283 (1999), pp. 212–214.)
Price Deforestation
(cents per pound) (percent)
29 0.49
40 1.59
54 1.69
55 1.82
72 3.10
(a) Make a scatterplot. Which is the explanatory variable? What kind of pattern
does your plot show?
(b) Find the correlation r step-by-step. That is, find the mean and standard devi-
ation of the two variables. Then find the five standardized values for each variable
and use the formula for r. Explain how your value for r matches your graph in (a).
(c) Now enter these data into your calculator or software and use the correlation
function to find r. Check that you get the same result as in (b).
Section 2.2 45
2.24 Coffee prices in dollars or euros. Coffee is currently priced in dollars. If

it were priced in euros, and the dollar prices in the previous exercise were translated
into the equivalent prices in euros, would the correlation between coffee price and
percent deforestation change? Explain your answer.
2.25 Mutual funds. Mutual fund reports often give correlations to describe how
the prices of different investments are related. You look at the correlations between
three Fidelity funds and the Standard & Poor’s 500 stock index, which describes
stocks of large U.S. companies. The three funds are Dividend Growth (stocks of large
U.S. companies), Small Cap Stock (stocks of small U.S. companies), and Emerging
Markets (stocks in developing countries). For 2003, the three correlations are r =
0.35, r = 0.81, and r = 0.98. (Compiled from Fidelity data in the Fidelity Insight
newsletter, 20 (2004), No. 1.)
(a) Which correlation goes with each fund? Explain your answer.
(b) The correlations of the three funds with the index are all positive. Does this tell
you that stocks went up in 2003? Explain your answer.
2.26 Mutual funds. Exercise 2.15 gives data on the returns from 23 Fidelity
“sector funds” in 2002 (a down-year for stocks) and 2003 (an up-year).
(a) Make a scatterplot if you did not do so in the previous exercise. Fidelity Gold
Fund, the only fund with a positive return in both years, is an extreme outlier.
(b) To demonstrate that correlation is not resistant, find r for all 23 funds and then
find r for the 22 funds other than Gold. Explain from Gold’s position in your plot
why omitting this point makes r more negative.
2.27 Gas mileage and speed. The table below gives data on gas mileage against
speed for a small car. Make a scatterplot and find the correlation r. Explain why r
is close to zero despite a strong relationship between speed and gas used.

10 21.00 90 7.57
20 13.00 100 8.27
30 10.00 110 9.03
40 8.00 120 9.87
50 7.00 130 10.79
60 5.90 140 11.77
70 6.30 150 12.83
80 6.95
2.28 Effect of a change in units. Consider again the correlation r between the
speed of a car and its gas consumption from the data in the previous exercise.
(a) Transform the data so that speed is measured in miles per hour and fuel con-
sumption in gallons per mile. (There are 1.609 kilometers in a mile and 3.785 liters
in a gallon.) Make a scatterplot and find the correlation for both the original and
the transformed data. How did the change of units affect your results?
(b) Now express fuel consumption in miles per gallon. (So each value is 1/x if x is
gallons per mile.) Again make a scatterplot and find the correlation. How did this
change of units affect your results?

(Lesson: The effects of a linear transformation of the form xnew = a + bx are simple.
The effects of a nonlinear transformation are more complex.)
2.29 City and highway gas mileage. The table below gives the city and highway
gas mileages for 21 two-seater cars, including the Honda Insight gas-electric hybrid
car.

Mazda Miata 22 28
Nissan 350Z 20 26
Toyota MR2 26 32
(a) Make a scatterplot of highway mileage y against city mileage x for all 21 cars.
There is a strong positive linear association. The Insight lies far from the other
points. Does the Insight extend the linear pattern of the other cars, or is it far from
the line they form?
(b) Find the correlation between city and highway mileages both without and with
the Insight. Based on your answer to (a), explain why r changes in this direction
when you add the Insight.
Section 2.3
2.30 (Review of straight lines) Fred keeps his savings in his mattress. He begins
with $500 from his mother and adds $100 each year. His total savings y after x
years are given by the equation
y = 500 + 100x
Section 2.3 47
(a) Draw a graph of this equation. (Hint: Choose two values of x, such as 0 and
10. Find the corresponding values of y from the equation. Plot these two points on
graph paper and draw the straight line joining them.)
(b) After 20 years, how much will Fred have in his mattress?
(c) If Fred adds $200 instead of $100 each year to his initial $500, what is the
equation that describes his savings after x years?
2.31 (Review of straight lines) Sound travels at a speed of 1500 meters per second
in sea water. You dive into the sea from your yacht. Give an equation for the
distance y at which a shark can hear your splash in terms of the number of seconds
x since you hit the water.
2.32 (Review of straight lines) During the period after birth, a male white rat gains
40 grams (g) per week. (This rat is unusually regular in his growth, but 40 g per
week is a realistic rate.)
(a) If the rat weighed 100 g at birth, give an equation for his weight after x weeks.
What is the slope of this line?
(b) Draw a graph of this line between birth and 10 weeks of age.
(c) Would you be willing to use this line to predict the rat’s weight at age 2 years?
Do the prediction and think about the reasonableness of the result. (There are 454
g in a pound. To help you assess the result, note that a large cat weighs about 10
pounds.)
2.33 (Review of straight lines) A cellular telephone company offers two plans. Plan
A charges $20 a month for up to 75 minutes of air time and $0.45 per minute above
75 minutes. Plan B charges $30 a month for up to 250 minutes and $0.40 per minute
above 250 minutes.
(a) Draw a graph of the Plan A charge against minutes used from 0 to 250 minutes.
(b) How many minutes a month must the user talk in order for Plan B to be less
expensive than Plan A?
2.34 Concrete road pavement gains strength over time as it cures. Highway builders
use regression lines to predict the strength after 28 days (when curing is complete)
from measurements made after 7 days. Let x be strength after 7 days (in pounds per
square inch) and y the strength after 28 days. One set of data gives this least-squares
regression line:
ŷ = 1389 + 0.96x
(a) Draw a graph of this line, with x running from 3000 to 4000 pounds per square
inch.
(b) Explain what the slope b = 0.96 in this equation says about how concrete gains
strength as it cures.
(c) A test of some new pavement after 7 days shows that its strength is 3300 pounds
per square inch. Use the equation of the regression line to predict the strength of
this pavement after 28 days.
2.35 Researchers studying acid rain measured the acidity of precipitation in an

isolated wilderness area in Colorado for 150 consecutive weeks. The acidity of a
solution is measured by pH, with lower pH values indicating that the solution is
more acid. The acid rain researchers observed a linear pattern over time. They
reported that the least-squares line
pH = 5.43 − (0.0053 × weeks)
fit the data well. (William M. Lewis and Michael C. Grant, “Acid precipitation in
the western United States,” Science, 207 (1980), pp. 176–177.)
(a) Draw a graph of this line. Note that the linear change is decreasing rather than
increasing.
(b) According to the fitted line, what was the pH at the beginning of the study
(weeks = 1)? At the end (weeks = 150)?
(c) What is the slope of the fitted line? Explain clearly what this slope says about
the change in the pH of the precipitation in this wilderness area.
2.36 Manatees are large, gentle sea creatures that live along the Florida coast.
Many manatees are killed or injured by powerboats. Here are data on powerboat
registrations (in thousands) and the number of manatees killed by boats in Florida
in the years 1977 to 1990.
Boats Manatees Boats Manatees
Year (thousands) killed Year (thousands) killed
1977 447 13 1984 559 34
1978 460 21 1985 585 33
1979 481 24 1986 614 33
1980 498 16 1987 645 39
1981 513 24 1988 675 43
1982 512 20 1989 711 50
1983 526 15 1990 719 47
(a) Make a scatterplot of these data. Describe the form and direction of the rela-
tionship.
(b) Find the correlation. What fraction of the variation in manatee deaths can be
explained by the number of boats registered? Does it appear that the number of
manatees killed can be predicted accurately from powerboat registrations?
(c) Find the least-squares regression line. Predict the number of manatees that will
be killed by boats in a year when 716,000 powerboats are registered.
(d) Suppose that in some far future year, 2 million powerboats are registered in
Florida. Use the regression line to predict manatees killed. Explain why this pre-
diction is very unreliable.
(e) Here are four more years of manatee data, in the same form as in previous
exercise:
1991 716 53 1993 716 35
1992 716 38 1994 735 49
Add these points to your scatterplot. Florida took stronger measures to protect
manatees during these years. Do you see any evidence that these measures suc-
ceeded?
(f) In part (c) you predicted manatee deaths in a year with 716,000 powerboat reg-
istrations. In fact, powerboat registrations remained at 716,000 for the next three
Section 2.3 49
years. Compare the mean manatee deaths in these years with your prediction from
part (c). How accurate was your prediction?
2.37 The number of people living on American farms has declined steadily during
the past century. Here are data on the farm population (millions of persons) from
1935 to 1980:
Year 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980
Population 32.1 30.5 24.4 23.0 19.1 15.6 12.4 9.7 8.9 7.2
(a) Make a scatterplot of these data and find the least-squares regression line using
year to predict farm population.
(b) According to the regression line, how much did the farm population decline each
year on the average during this period? What percent of the observed variation in
farm population is accounted for by linear change over time?
(c) Use the regression equation to predict the number of people living on farms in
1990. Is this result reasonable? Why?
2.38 Keeping water supplies clean requires regular measurement of levels of pol-
lutants. The measurements are indirect—a typical analysis involves forming a dye
by a chemical reaction with the dissolved pollutant, then passing light through the
solution and measuring its “absorbance.” To calibrate such measurements, the labo-
ratory measures known standard solutions and uses regression to relate absorbance
to pollutant concentration. This is usually done every day. Here is one series of
data on the absorbance for different levels of nitrates. Nitrates are measured in mil-
ligrams per liter of water. (From a presentation by Charles Knauf, Monroe County
(New York) Environmental Health Laboratory.)
Nitrates 50 50 100 200 400 800 1200 1600 2000 2000

Absorbance 7.0 7.5 12.8 24.0 47.0 93.0 138.0 183.0 230.0 226.0
(a) Chemical theory says that these data should lie on a straight line. If the cor-
relation is not at least 0.997, something went wrong and the calibration procedure
is repeated. Plot the data and find the correlation. Must the calibration be done
again?
(b) What is the equation of the least-squares line for predicting absorbance from con-
centration? If the lab analyzed a specimen with 500 milligrams of nitrates per liter,
what do you expect the absorbance to be? Based on your plot and the correlation,
do you expect your predicted absorbance to be very accurate?
2.39 You have data on an explanatory variable x and a response variable y, and
have found the least-squares regression line of y on x. Add one more data point,
with x equal to the mean x for the existing points and y greater than the mean
y for the existing points. Show that the new least-squares line is parallel to the
existing line. (To do this, you must show that the slope does not change when you
add the new point. Start with the fact that the slope is b = rsy /sx and substitute
the definition of the correlation r.)
Studies of disease often ask people about their diet in years past in order to discover
links between diet and disease. How well do people remember their past diet? Can
we predict actual past diet as well or better from what subjects eat now as from their
memory of past habits? Data on actual past diets are available for 91 people who
were asked about their diet when they were 18 years old and again when they were
30. Researchers asked them at about age 55 to describe their eating habits at ages
18 and 30 and also their current diet. The study report says:
The first study aim, to determine how accurately this group of partic-
ipants remembered past consumption, was addressed by correlations be-
tween recalled and historical consumption in each time period. To eval-
uate the second study aim, that is, whether recalled intake or current
intake more accurately predicts historical intake of food groups at age 30
years, we performed regression analysis.
The following three exercises ask you to interpret the results of this paper to a group
of people who know no statistics. (J. T. Dwyer et al., “Memory of food intake in the
distant past,” American Journal of Epidemiology, 130 (1989), pp. 1033–1046.)
2.40 Explain in nontechnical language what “correlation” means, why correlation

suits the first aim of the study, what “regression” means, and why regression fits
the second study aim. Be sure to point out the distinction between correlation and
regression.
2.41 The study looked at the correlations between actual intake of many foods at
age 18 and the intake the subjects now remember for age 18. The median correlation
was r = 0.217. The authors say, “We conclude that memory of food intake in the
distant past is fair to poor.” Explain to your audience why r = 0.217 points to this
conclusion.
2.42 The authors used regression to predict the intake of a number of foods at age
30 from current intake of those foods and from what the subjects now remember
about their intake at age 30. They conclude that “recalled intake more accurately
predicted historical intake at age 30 years than did current diet.” As evidence,
they present r2-values for the regressions. Explain to your audience why comparing
r2-values is one way to compare how well different explanatory variables predict a
response.
2.43 The following table gives the U.S. resident population of voting age and the
votes cast for president, both in thousands, for presidential elections between 1960
and 2000:
Year Population Votes Year Population Votes
1960 109,672 68,838 1984 173,995 92,653
1964 114,090 70,645 1988 181,956 91,595
1968 120,285 73,212 1992 189,524 104,425
1972 140,777 77,719 1996 196,511 96,456
1976 152,308 81,556 2000 209,128 105,363
1980 163,945 86,515
(a) For each year compute the percent of people who voted. Make a time plot of the
percent who voted. Describe the change over time in participation in presidential
Section 2.3 51
elections.
(b) Before proposing political explanations for this change, we should examine pos-
sible lurking variables. The minimum voting age in presidential elections dropped
from 21 to 18 years in 1970. Use this fact to propose a partial explanation for the
trend you saw in (a).
2.44 First test and final exam. Here are data for eight students from anelemen-
tary statistics course:
First test score 153 144 162 149 127 118 158 153
Final exam score 145 140 145 170 145 175 170 160
(a) Plot the data with the first test scores on the x axis and the final exam scores
on the y axis.
(b) Find the least-squares regression line for predicting the final exam score using
the first test score.
(c) Graph the least-squares regression line on your plot.
2.45 Second test and final exam. Refer to the previous exercise. Here are the
data for the second test and the final exam for the same students:
Second test score 158 162 144 162 136 158 175 153
Final exam score 145 140 145 170 145 175 170 160
(a) Plot the data with the second test scores on the x axis and the final exam scores
on the y axis.
(b) Find the least-squares regression line for predicting the final exam score using
the second test score.
(c) Graph the least-squares regression line on your plot.
2.46 The effect of an outlier. Refer to the previous two exercises. Add a ninth
student whose scores on the second test and final exam would lead you to classify
the additional data point as an outlier. Recalculate the least-squares regression
line with this additional case and summarize the effect it has on the least-squares
regression line.
2.47 The effect of a different point. Examine the data in Exercise 2.45 and
add a diferent ninth student who has low scores on the second test and the final
exam, and fits the overall pattern of the other scores in the data set. Recalculate
the least-squares regression line with this additional case and summarize the effect
it has on the least-squares regression line.
2.48 Problems with feet. Metatarsus adductus (call it MA) is a turning in

of the front part of the foot that is common in adolescents and usually corrects
itself. Hallux abducto valgus (call it HAV) is a deformation of the big toe that is
not common in youth and often requires surgery. Perhaps the severity of MA can
help predict the severity of HAV. The following table gives data on 38 consecutive
patients who came to a medical center for HAV surgery. (Alan S. Banks et al.,
“Juvenile hallux abducto valgus association with metatarsus adductus,” Journal
of the American Podiatric Medical Association, 84 (1994), pp. 219–224.) Using
X-rays, doctors measured the angle of deformity for both MA and HAV. They
speculated that there is a positive association—more serious MA is associated with
more serious HAV. A scatterplot the severity of the mild foot deformity called MA
can help predict the severity of the more serious deformity called HAV.
Two measurements of foot deformities
HAV MA HAV MA HAV MA

angle angle angle angle angle angle
28 18 21 15 16 10
32 16 17 16 30 12
25 22 16 10 30 10
34 17 21 7 20 10
38 33 23 11 50 12
26 10 14 15 25 25
25 18 32 12 26 30
18 13 25 16 28 22
30 19 21 16 31 24
26 10 22 18 38 20
28 17 20 10 32 37
13 14 18 15 21 23
20 20 26 16
(a) Find the equation of the least-squares regression line for predicting HAV angle
from MA angle. Make a scatterplot of the data with the lease-squares regression
line.
(b) A new patient has MA angle 25 degrees. What do you predict this patient’s
HAV angle to be?
(c) Does knowing MA angle allow doctors to predict HAV angle accurately? Explain
your answer from the scatterplot, then calculate a numerical measure to support
your finding.
2.49 Predict final exam scores. In Professor Friedman’s economics course the
correlation between the students’ total scores before the final examination and their
final examination scores is r = 0.55. The pre-exam totals for all students in the
course have mean 270 and standard deviation 30. The final exam scores have mean
70 and standard deviation 9. Professor Friedman has lost Julie’s final exam but
knows that her total before the exam was 310. He decides to predict her final exam
score from her pre-exam total.
(a) What is the slope of the least-squares regression line of final exam scores on
pre-exam total scores in this course? What is the intercept?
(b) Use the regression line to predict Julie’s final exam score.
(c) Julie doesn’t think this method accurately predicts how well she did on the final
exam. Calculate r2 and use the value you get to argue that her actual score could
have been much higher or much lower than the predicted value.
2.50 Pesticide decay. Fenthion is a pesticide used to control the olive fruit fly.
There are government limits on the amount of pesticide residue that can be present
in olive products. Because the pesticide decays over time, producers of olive oil
Section 2.4 53
might simply store the oil until the fenthion has decayed. The simple exponential
decay model says that the concentration C of pesticide remaining after time t is
C = C0 e−kt
where C0 is the initial concentration and k is a constant that determines the rate of
decay. This model is a straight line if we take the logarithm of the concentration:
log C = log C0 − kt
(The logarithm here is the natural logarithm, not the common logarithm with base
10.) Here are data on the concentration (milligrams of fenthion per kilogram of oil)
in specimens of Greek olive oil: (Data from plots in Chaido Lentza-Rizos, Elizabeth
J. Avramides, and Rosemary A. Roberts, “Persistence of fenthion residues in olive
oil,” Pesticide Science, 40 (1994), pp. 63–69.)
Days stored Concentration

28 0.99 0.99 0.96 0.95 0.93
84 0.96 0.94 0.91 0.91 0.90
183 0.89 0.87 0.86 0.85 0.85
273 0.87 0.86 0.84 0.83 0.83
365 0.83 0.82 0.80 0.80 0.79
(a) Plot the natural logarithm of concentration against days stored. Notice that
there are several pairs of identical data points. Does the pattern suggest that the
model of simple exponential decay describes the data reasonably well, at least over
this interval of time? Explain your answer.
(b) Regress the logarithm of concentration on time. Use your result to estimate the
value of the constant k.
2.51 The decay product is toxic. Unfortunately, the main product of the decay
of the pesticide fenthion is fenthion sulfoxide, which is also toxic. Here are data on
the total concentration of fenthion and fenthion sulfoxide in the same specimens of
olive oil described in the previous exercise:
28 1.03 1.03 1.01 0.99 0.99
84 1.05 1.04 1.00 0.99 0.99
183 1.03 1.02 1.01 0.98 0.98
273 1.07 1.06 1.03 1.03 1.02
365 1.06 1.02 1.01 1.01 0.99
(a) Plot concentration against days stored. Your software may fill the available
space in the plot, which in this case hides the pattern. Try a plot with vertical scale
from 0.8 to 1.2. Be sure your plot takes note of the pairs of identical data points.
(b) What is the slope of the least-squares line for predicting concentration of fenthion
and fenthion sulfoxide from days stored? Explain why this value agrees with the
graph.
(c) What do the data say about the idea of reducing fenthion in olive oil by storing
the oil before selling it?
Section 2.4
2.52 The following table gives the calories and sodium content for each of 17 brands
of meat hot dogs. “Eat Slim Veal Hot Dogs,” with just 107 calories, is a low outlier
in the distribution of calories.
186 495 173 458 129 430
181 477 191 506 132 375
176 425 182 473 102 396
149 322 190 545 106 383
184 482 172 496 ?94 387
190 587 147 360 102 542
158 370 146 387 ?87 359
139 322 139 386 ?99 357
175 479 175 507 170 528
148 375 136 393 113 513
152 330 179 405 135 426
111 300 153 372 142 513
141 386 107 144 ?86 358
153 401 195 511 143 581
190 645 135 405 152 588
157 440 140 428 146 522
131 317 138 339 144 545
149 319
135 298
132 253
(a) Make a scatterplot of sodium content y against calories x. Describe the main
features of the relationship. Is “Eat Slim Veal Hot Dogs” an outlier in this plot?
Circle its point.
(b) Calculate two least-squares regression lines, one using all of the observations and
the other omitting “Eat Slim.” Draw both lines on your plot. Does a comparison
of the two regression lines show that the “Eat Slim” brand is influential? Explain
your answer.
(c) A new brand of meat hot dog (not made with veal) has 150 calories per frank.
How many milligrams of sodium do you estimate that one of these hot dogs contains?
2.53 Research on digestion requires accurate measurements of blood flow through

the lining of the stomach. A promising way to make such measurements easily is
to inject mildly radioactive microscopic spheres into the blood stream. The spheres
lodge in tiny blood vessels at a rate proportional to blood flow; their radioactiv-
ity allows blood flow to be measured from outside the body. Medical researchers
compared blood flow in the stomachs of dogs, measured by use of microspheres,
with simultaneous measurements taken using a catheter inserted into a vein. The
data, in milliliters of blood per minute (ml/minute), appear below. (Based on L. H.
Archibald, F. G. Moody, and M. Simons, “Measurement of gastric blood flow with
radioactive microspheres,” Journal of Applied Physiology, 38 (1975), pp. 1051–1056.)
Section 2.4 55
Spheres 4.0 4.7 6.3 8.2 12.0 15.9 17.4 18.1 20.2 23.9
Vein 3.3 8.3 4.5 9.3 10.7 16.4 15.4 17.6 21.0 21.7
(a) Make a scatterplot of these data, with the microsphere measurement as the ex-
planatory variable. There is a strongly linear pattern.
(b) Calculate the least-squares regression line of venous flow on microsphere flow.
Draw your regression line on the scatterplot.
(c) Predict the venous measurement for microsphere measurements 6, 12, and 18
ml/minute. If the microsphere measurements are within about 10% to 15% of the
predicted venous measurements, the researchers will use the microsphere measure-
ments in future studies. Is this condition satisfied over this range of blood flow?
2.54 The following table gives information on states’ performance on the National
Assessment of Educational Progress (NAEP) year 2000 Mathematics Assessment.
The two measures of performance are closely related. (In fact, the correlation is
about r = 0.95.) The table also gives the percent of children aged 5 to 17 years in
each state who lived in households with incomes below the federal poverty level in
1998. We expect that poverty among children will be related to NAEP performance.
Mean NAEP Percent Percent Mean NAEP Percent Percent

State score proficient poverty State score proficient poverty
Alabama 218 14 21.8 Missouri 229 24 14.4
Arizona 219 17 23.6 Montana 230 25 21.2
Arkansas 217 14 13.1 Nebraska 226 24 14.8
California 214 15 22.3 Nevada 220 16 12.8
Connecticut 234 32 13.4 New Mexico 214 12 23.5
Georgia 220 18 24.7 New York 227 22 28.9
Hawaii 216 14 14.5 North Carolina 232 28 21.3
Idaho 227 21 17.4 North Dakota 231 25 17.2
Illinois 225 22 12.2 Ohio 231 26 16.0
Indiana 234 31 12.6 Oklahoma 225 17 19.9
Iowa 233 28 14.2 Oregon 227 24 19.4
Kansas 232 30 13.3 Rhode Island 225 23 20.5
Kentucky 221 17 16.7 South Carolina 220 18 17.6
Louisiana 218 14 29.8 Tennessee 220 13 14.5
Maine 231 24 12.0 Texas 233 27 20.1
Maryland 222 22 8.1 Utah 227 24 11.8
Massachusetts 235 33 15.0 Vermont 232 30 12.2
Michigan 231 29 14.8 Virginia 230 25 7.9
Minnesota 235 34 12.6 West Virginia 225 18 25.7
Mississippi 211 9 19.3 Wyoming 229 25 13.0
(a) Make a scatterplot suitable for predicting mean NAEP score from poverty
percent. Describe the relationship, using correlation as a measure to complement
your verbal description.
(b) Use software to find the least-squares regression line for predicting mean NAEP
score from poverty rate, and the residuals from this line. We might call states with
large positive residuals overachievers, because their fourth-graders do better than
the state poverty rate would lead us to guess. Similarly, states with large negative
residuals might be called underachievers. What are the three states with the largest
positive residuals and the three states with the largest negative residuals?
2.55 We might expect states with more poverty to have fewer doctors. Here are
data on the percent of each state’s residents living below the poverty line and on
the number of M.D.’s per 100,000 residents in each state.
Poverty M.D.’s per Poverty M.D.’s per

State percent 100,000 State percent 100,000
Alabama 15.1 198 Montana 15.9 190
Alaska 8.6 167 Nebraska 11.0 218
Arizona 15.2 202 Nevada 11.0 173
Arkansas 16.4 190 New Hampshire 8.9 237
California 15.3 247 New Jersey 8.5 295
Colorado 8.6 238 New Mexico 20.8 212
Connecticut 8.4 354 New York 15.7 387
Delaware 10.1 234 North Carolina 13.0 232
D.C. 19.7 737 North Dakota 13.9 222
Florida 13.3 238 Ohio 11.4 235
Georgia 13.7 211 Oklahoma 13.5 169
Hawaii 11.9 265 Oregon 13.1 225
Idaho 13.9 154 Pennsylvania 10.6 291
Illinois 10.4 260 Rhode Island 11.4 338
Indiana 8.3 195 South Carolina 12.8 207
Iowa 8.7 173 South Dakota 11.7 184
Kansas 10.5 203 Tennessee 13.2 246
Kentucky 13.8 209 Texas 15.6 203
Louisiana 18.2 246 Utah 7.9 200
Maine 10.4 223 Vermont 9.6 305
Maryland 7.6 374 Virginia 9.8 241
Massachusetts 10.9 412 Washington 9.2 235
Michigan 10.3 224 West Virginia 16.7 215
Minnesota 9.1 249 Wisconsin 8.5 227
Mississippi 16.8 163 Wyoming 11.9 171
Missouri 11.1 230
(a) Make a scatterplot and calculate a regression line suitable for predicting M.D.’s
per 100,000 from poverty rate. Draw the line on your plot. Surprise: The slope is
positive, so poverty and M.D.’s go up together.
(b) The District of Columbia is an outlier, with both very many M.D.’s and a high
poverty rate. (D.C. is a city rather than a state.) Circle the point for D.C. on your
plot and explain why this point may strongly influence the least-squares line.
(c) Calculate the regression line for the 50 states, omitting D.C. Add the new line
to your scatterplot. Was this point highly influential? Does the number of doctors
now go down with increasing poverty, as we initially expected?
2.56 The Standard & Poor’s 500 stock index is an average of the price of 500 stocks.
There is a moderately strong correlation (roughly r = 0.6) between how much this
index changes in January and how much it changes during the entire year. If we
looked instead at data on all 500 individual stocks, we would find a quite different
correlation. Would the correlation be higher or lower? Why?
Section 2.4 57
2.57 Airborne particles such as dust and smoke are an important part of air pollu-
tion. To measure particulate pollution, a vacuum motor draws air through a filter
for 24 hours. Weigh the filter at the beginning and end of the period. The weight
gained is a measure of the concentration of particles in the air. A study of air pollu-
tion made measurements every 6 days with identical instruments in the center of a
small city and at a rural location 10 miles southwest of the city. Because the prevail-
ing winds blow from the west, we suspect that the rural readings will be generally
lower than the city readings, but that the city readings can be predicted from the
rural readings. Here are readings taken every 6 days over a 7-month period. The
entry NA means that the reading for that date is not available, usually because of
equipment failure. (Data provided by Matthew Moore.)
Day 1 2 3 4 5 6 7 8 9 10 11 12
Rural NA 67 42 33 46 NA 43 54 NA NA NA NA
City 39 68 42 34 48 82 45 NA NA 60 57 NA
Day 13 14 15 16 17 18 19 20 21 22 23 24
Rural 38 88 108 57 70 42 43 39 NA 52 48 56
City 39 NA 123 59 71 41 42 38 NA 57 50 58
Day 25 26 27 28 29 30 31 32 33 34 35 36
Rural 44 51 21 74 48 84 51 43 45 41 47 35
City 45 69 23 72 49 86 51 42 46 NA 44 42
(a) We hope to use the rural particulate level to predict the city level on the same day.
Make a graph to examine the relationship. Does the graph suggest that using the
least-squares regression line for prediction will give approximately correct results
over the range of values appearing in the data? Calculate the least-squares line
for predicting city pollution from rural pollution. What percent of the observed
variation in city pollution levels does this straight-line relationship account for?
(b) Find the residuals from your least-squares fit. Plot the residuals both against x
and against the time order of the observations, and comment on the results.
(c) Which observation appears to be the most influential? Circle this observation
on your plot. Is it the observation with the largest residual?
(d) On the 14th date in the series, the rural reading was 88 and the city reading
was not available. What do you estimate the city reading to be for that date?
(e) Make a normal quantile plot of the residuals. (Make a stemplot or histogram
if your software does not make Normal quantile plots.) Is the distribution of the
residuals approximately Normal?
2.58 Go to the Correlation and Regression applet. Click on the scatterplot to create
a group of 10 points in the lower-left corner of the scatterplot with a strong straight-
line pattern (correlation about 0.9). Now click the “Show least-squares line” box to
display the regression line.
(a) Add one point at the upper right that is far from the other 10 points but exactly
on the regression line. Why does this outlier have no effect on the line even though
it changes the correlation?
(b) Now drag this last point down until it is opposite the group of 10 points. You
see that one end of the least-squares line chases this single point, while the other
end remains near the middle of the original group of 10. What makes the last point
so influential?
2.59 A multimedia statistics learning system includes a test of skill in using the
computer’s mouse. The software displays a circle at a random location on the
computer screen. The subject tries to click in the circle with the mouse as quickly
as possible. A new circle appears as soon as the subject clicks the old one. The
following table gives data for one subject’s trials, 20 with each hand. Distance is
the distance from the cursor location to the center of the new circle, in units whose
actual size depends on the size of the screen. Time is the time required to click in
the new circle, in milliseconds.
(a) We suspect that time depends on distance. Make a scatterplot of time against
distance, using separate symbols for each hand.
(b) Describe the pattern. How can you tell that the subject is right-handed?
(c) Find the regression line of time on distance separately for each hand. Draw
these lines on your plot. Which regression does a better job of predicting time from
distance? Give numerical measures that describe the success of the two regressions.
(d) It is possible that the subject got better in later trials due to learning. It is also
possible that he got worse due to fatigue. Plot the residuals from each regression
against the time order of the trials (down the columns in the following table). Is
either of these systematic effects of time visible in the data?
Time Distance Hand Time Distance Hand
115 190.70 right 240 190.70 left
96 138.52 right 190 138.52 left
110 165.08 right 170 165.08 left
100 126.19 right 125 126.19 left
111 163.19 right 315 163.19 left
101 305.66 right 240 305.66 left
111 176.15 right 141 176.15 left
106 162.78 right 210 162.78 left
96 147.87 right 200 147.87 left
96 271.46 right 401 271.46 left
95 40.25 right 320 40.25 left
96 24.76 right 113 24.76 left
96 104.80 right 176 104.80 left
106 136.80 right 211 136.80 left
100 308.60 right 238 308.60 left
113 279.80 right 316 279.80 left
123 125.51 right 176 125.51 left
111 329.80 right 173 329.80 left
95 51.66 right 210 51.66 left
108 201.95 right 170 201.95 left
2.60 Fuel consumption and speed. The table following this exercise gives data
on the fuel consumption y of a car at various speeds x. The relationship is strongly
curved: Fuel used decreases with increasing speed at low speeds, then increases
again as higher speeds are reached. The equation of the least-squares regression line
Section 2.4 59
for these data is

ŷ = 11.058 − 0.01466x
The residuals, in the same order as the observations, are
10.09 2.24 −0.62 −2.47 −3.33 −4.28 −3.73 −2.94
−2.17 −1.32 −0.42 0.57 1.64 2.76 3.97
(a) Make a scatterplot of the observations and draw the regression line on your plot.
The line is a poor description of the curved relationship.
(b) Check that the residuals have sum zero (up to roundoff error).
(c) Make a plot of the residuals against the values of x. Draw a horizontal line at
height zero on your plot. The residuals show the same pattern about this line as
the data points show about the regression line in the scatterplot in (a).
10 21.00 90 7.57
20 13.00 100 8.27
30 10.00 110 9.03
40 8.00 120 9.87
50 7.00 130 10.79
60 5.90 140 11.77
70 6.30 150 12.83
80 6.95
2.61 Pesticide in olive oil. The table following this exercise gives data on the
concentration of the pesticide fenthion in Greek olive oil that has been stored for
various lengths of time. The exponential decay model used to describe how con-
centration decreases over time proposes a curved relationship between storage time
and concentration. Do the residuals from fitting a regression line show a curved
pattern? The least-squares line for predicting concentration is
ŷ = 0.965 − 0.00045x
(a) The first batch of olive oil was stored for 28 days and had fenthion concentration
0.99 mg/kg. What is the predicted concentration for this batch? What is the
residual?
(b) The residuals, arranged as in the data table, are:
Days stored Residual
28 0.0378 0.0378 0.0078 −0.0022 −0.0222
84 0.0329 0.0129 −0.0171 −0.0171 −0.0271
183 0.0072 −0.0128 −0.0228 −0.0328 −0.0328
273 0.0275 0.0175 −0.0025 −0.0125 −0.0125
365 0.0286 0.0186 −0.0014 −0.0014 −0.0114
Check that your residual from (a) agrees (up to roundoff error) with the value 0.0378
given here. Verify that the residuals sum to zero (again up to roundoff error).
(c) Make a residual plot. Is a curved pattern visible? Is the curve very strong?
(Software often makes the pattern hard to see because it fills the entire plot area.
Try a plot with vertical scale from −0.1 to 0.1.)

28 0.99 0.99 0.96 0.95 0.93
84 0.96 0.94 0.91 0.91 0.90
183 0.89 0.87 0.86 0.85 0.85
273 0.87 0.86 0.84 0.83 0.83
365 0.83 0.82 0.80 0.80 0.79
2.62 City and highway gas mileage. The following table gives the city and
highway gas mileages for 21 two-seater cars, including the Honda Insight gas-electric
hybrid car.

Mazda Miata 22 28
Nissan 350Z 20 26
Toyota MR2 26 32
(a) Make a scatterplot of highway mileage (response) against city mileage (explana-
tory) for all 21 cars.
(b) Use software or a graphing calculator to find the regression line for predicting
highway mileage from city mileage and also the 21 residuals for this regression. Make
a residual plot with a horizontal line at zero. (The “stacks” in the plot are due to
the fact that mileage is measured only to the nearest mile per gallon.)
(c) Which car has the largest positive residual? The largest negative residual?
(d) The Honda Insight, an extreme outlier, does not have the largest residual in
either direction. Why is this not surprising?
2.63 City and highway gas mileage. Continue your work in the previous exer-
cise. Find the regression line for predicting highway mileage from city mileage for
Section 2.5 61
the 20 two-seater cars other than the Honda Insight. Draw both regression lines on
your scatterplot. Is the Insight very influential for the least-squares line? (Look at
the position of the lines for city mileages between 10 and 30 mpg, values that cover
most cars.) What explains your result?
2.64 Stride rate of runners. Runners are concerned about their form when
racing. One measure of form is the stride rate, the number of steps taken per
second. As running speed increases, the stride rate should also increase. In a study
of 21 of the best American female runners, researchers measured the stride rate for
different speeds. The following table gives the speeds (in feet per second) and the
mean stride rates for these runners: (R. C. Nelson, C. M. Brooks, and N. L. Pike,
“Biomechanical comparison of male and female distance runners,” in P. Milvy (ed.),
The Marathon: Physiological, Medical, Epidemiological, and Psychological Studies,
New York Academy of Sciences, 1977, pp. 793–807.)
Speed 15.86 16.88 17.50 18.62 19.97 21.06 22.11
Stride rate 3.05 3.12 3.17 3.25 3.36 3.46 3.55
(a) Plot the data with speed on the x axis and stride rate on the y axis. Does a
straight line adequately describe these data?
(b) Find the equation of the regression line of stride rate on speed. Draw this line
on your plot.
(c) For each of the speeds given, obtain the predicted value of the stride rate and
the residual. Verify that the residuals add to zero.
(d) Plot the residuals against speed. Describe the pattern. What does the plot
indicate about the adequacy of the linear fit? Are there any potentially influential
observations?
2.65 Stride rate and running speed. The previous exercise gives data on the
mean stride rate of a group of 21 elite female runners at various running speeds. Find
the correlation between speed and stride rate. Would you expect this correlation to
increase or decrease if we had data on the individual stride rates of all 21 runners
at each speed? Why?
Section 2.5
For exercises on data analysis for two-way tables, use the exercises for Chapter 9
and ignore the parts that ask questions about statistical inference.
Section 2.6
2.66 A study of grade school children aged 6 to 11 years found a high positive
correlation between reading ability y and shoe size x. Explain why common response
to a lurking variable z accounts for this correlation.
2.67 There is a negative correlation between the number of flu cases y reported
each week through the year and the amount of ice cream x sold that week. It is
unlikely that ice cream prevents flu. What is a more plausible explanation for this
correlation?
2.68 Members of a high school language club believe that study of a foreign language
improves a student’s command of English. From school records, they obtain the
scores on an English achievement test given to all seniors. The mean score of seniors
who had studied a foreign language for at least two years is much higher than the
mean score of seniors who studied no foreign language. The club’s advisor says that
these data are not good evidence that language study strengthens English skills.
Identify the explanatory and response variables in this study. Then explain what
lurking variable prevents the conclusion that language study improves students’
English scores.
2.69 CEO compensation and layoffs. “Based on an examination of twenty-two

companies that announced large layoffs during 1994, Downs found a strong (.31)
correlation between the size of the layoffs and the compensation of the CEOs.”
(Kevin Phillips, Wealth and Democracy, Broadway Books, 2002, p. 151.) This
correlation is probably explained by common response to a lurking variable, the size
of the company as measured by its number of employees. Explain how common
response could create the observed correlation. Use a diagram to illustrate your
explanation.
2.70 Health and income. An article entitled “The Health and Wealth of Nations”
says: “The positive correlation between health and income per capita is one of the
best-known relations in international development. This correlation is commonly
thought to reflect a causal link running from income to health. . . . Recently, however,
another intriguing possibility has emerged: that the health-income correlation is
partly explained by a causal link running the other way—from health to income.”
(David E. Bloom and David Canning, “The health and wealth of nations,” Science,
287 (2000), pp. 1207–1208.)
2.71 Self-esteem and work performance. People who do well tend to feel
good about themselves. Perhaps helping people feel good about themselves will
help them do better in their jobs and in life. Raising self-esteem became for a
time a goal in many schools and companies. Can you think of explanations for the
association between high self-esteem and good performance other than “self-esteem
causes better work”?
The following three exercises concern these data on the total returns on U.S. and
overseas common stocks over a 30-year period. (The total return is change in price
plus any dividends paid, converted into U.S. dollars. Both returns are averages over
many individual stocks.)
Review Exercises 63
Overseas U.S. Overseas U.S.

Year % return % return Year % return % return
1971 29.6 14.6 1986 69.4 18.6
1972 36.3 18.9 1987 24.6 5.1
1973 −14.9 −14.8 1988 28.5 16.8
1974 −23.2 −26.4 1989 10.6 31.5
1975 35.4 37.2 1990 −23.0 −3.1
1976 2.5 23.6 1991 12.8 30.4
1977 18.1 −7.4 1992 −12.1 7.6
1978 32.6 6.4 1993 32.9 10.1
1979 4.8 18.2 1994 6.2 1.3
1980 22.6 32.3 1995 11.2 37.6
1981 −2.3 −5.0 1996 6.4 23.0
1982 −1.9 21.5 1997 2.1 33.4
1983 23.7 22.4 1998 20.3 28.6
1984 7.4 6.1 1999 27.2 21.0
1985 56.2 31.6 2000 −14.0 −9.1
2.72 (a) Make a scatterplot suitable for predicting overseas returns from U.S. re-
turns.
(b) Find the correlation and r2. Describe the relationship between U.S. and overseas
returns in words, using r and r2 to make your description more precise.
(c) Find the least-squares regression line of overseas returns on U.S. returns. Draw
the line on the scatterplot. What are the predicted return ŷ and the observed return
y for 1993?
(d) Are you confident that predictions using the regression line will be quite accu-
rate? Why?
2.73 Return to the scatterplot and regression line in the previous exercise.
(a) Circle the point that has the largest residual (either positive or negative). What
year is this? Redo the regression without this point and add the new regression line
to your plot. Was this observation very influential?
(b) Whenever we regress two variables that both change over time, we should plot
the residuals against time as a check for time-related lurking variables. Make this
plot for the stock returns data. Are there any suspicious patterns in the residuals?
2.74 Investors also want to know what typical returns are and how much year-to-
year variability (called volatility in finance) there is. Regression and correlation
don’t answer questions about center and spread.
(a) Find the five-number summaries for both U.S. and overseas returns, and make
side-by-side boxplots to compare the two distributions.
(b) Were returns generally higher in the United States or overseas during this period?
Explain your answer.
(c) Were returns more volatile (more variable) in the United States or overseas
during this period? Explain your answer
There are different ways to measure the amount of money spent on education. Av-
erage salary paid to teachers and expenditures per pupil are two possible measures.
The table at the top of the next page gives the 1995 values for these variables by
state. The states are classified according to region: NE (New England), MA (Mid-
dle Atlantic), ENC (East North Central), WNC (West North Central), SA (South
Atlantic), ESC (East South Central), WSC (West South Central), MN (Mountain),
and PA (Pacific). The following three exercises are based on these data.
State Region Pay Spend State Region Pay Spend

Me. NE 32.0 6.41 N.H. NE 29.0 6.13
Vt. NE 35.4 7.37 Mass. NE 42.2 6.17
R.I. NE 40.7 7.36 Conn. NE 50.0 8.50
N.Y. MA 47.6 9.45 N.J. MA 46.1 9.86
Pa. MA 44.5 7.20 Ohio ENC 36.8 5.62
Ind. ENC 36.8 6.00 Ill. ENC 39.4 5.26
Mich. ENC 47.4 6.93 Wis. ENC 37.7 7.00
Minn. WNC 35.9 5.11 Iowa WNC 31.5 5.56
Mo. WNC 31.2 4.97 N.Dak. WNC 26.3 4.60
S.Dak. WNC 26.0 4.84 Nebr. WNC 30.9 5.38
Kans. WNC 34.7 5.76 Del. SA 39.1 7.17
Md. SA 40.7 6.72 D.C. SA 43.7 8.21
Va. SA 34.0 5.66 W.Va. SA 31.9 6.52
N.C. SA 30.8 4.95 S.C. SA 30.3 4.93
Ga. SA 32.6 5.40 Fla. SA 32.6 5.72
Ky. ESC 32.3 5.61 Tenn. ESC 32.5 4.54
Ala. ESC 31.1 4.46 Miss. ESC 26.8 4.12
Ark. WSC 28.9 4.26 La. WSC 26.5 4.71
Okla. WSC 28.2 4.38 Tex. WSC 31.2 5.42
Mont. MN 28.8 5.83 Idaho MN 29.8 6.03
Wyo. MN 31.3 6.07 Colo. MN 34.6 5.50
N.Mex. MN 28.5 5.42 Ariz. MN 32.2 4.25
Utah MN 29.1 3.67 Nev. MN 34.8 5.13
Wash. PA 36.2 5.81 Oreg. PA 38.6 6.25
Calif. PA 41.1 4.73 Alaska PA 48.0 9.93
Hawaii PA 38.5 6.16
2.75 Make a stemplot or histogram for teachers’ pay. Is the distribution roughly
symmetric or clearly skewed? Find the five-number summary. Are there any sus-
pected outliers by the 1.5 × IQR criterion? Which states may be outliers? Do the
same for spending per pupil. Are the same states outliers in both distributions?
2.76 (a) Make a scatterplot of teachers’ pay y against spending x. Describe the
pattern of the relationship between pay and spending. Is there a strong association?
If so, is it positive or negative? Explain why you might expect to see an association
of this kind.
(b) Find the least-squares regression line for predicting teachers’ pay from education
spending and draw it on your scatterplot. How much on the average does mean
Review Exercises 65
teachers’ pay increase when spending increases by $1000 per pupil from one state to
another? Give a numerical measure of the success of overall spending on education
in explaining variations in teachers’ pay among states.
(c) On your plot, circle any outlying points found in (a). Label the circled points
with the state identifier. Do these points have large residuals? (You need not
actually calculate the residuals.) The states you have identified lie close together on
the plot. To see if they are influential as a group, find the regression line with all
of these states removed from the calculation. Draw this new line on your plot. Was
this group of states influential?
2.77 Continue the analysis of teachers’ pay and education spending by looking for
regional effects. We will compare these three groups:
Coastal Middle Atlantic, New England, and Pacific
South South Atlantic, East South Central, and West South Central
Midwest East North Central and West North Central
Omit the District of Columbia, which is a city rather than a state.
(a) Make side-by-side boxplots for education spending in the three regions. For each
region, label any outliers (points identified by the 1.5×IQR criterion) with the state
identifier.
(b) Repeat part (a) for teachers’ pay.
(c) Do you see important differences in spending and pay by region? Are the differ-
ences consistent for the two variables? That is, are regions that are high in spending
also high in pay and vice versa?
2.78 Lamb’s-quarter is a common weed that interferes with the growth of corn. An
agriculture researcher planted corn at the same rate in 16 small plots of ground,
then weeded the plots by hand to allow a fixed number of lamb’s-quarter plants to
grow in each meter of corn row. No other weeds were allowed to grow. Here are the
yields of corn (bushels per acre) in each of the plots:
Weeds Corn Weeds Corn Weeds Corn Weeds Corn

per meter yield per meter yield per meter yield per meter yield
0 166.7 1 166.2 3 158.6 9 162.8
0 172.2 1 157.3 3 176.4 9 142.4
0 165.0 1 166.7 3 153.1 9 162.8
0 176.9 1 161.1 3 156.0 9 162.4
(a) What are the explanatory and response variables in this experiment?
(b) Make side-by-side stemplots of the yields, after rounding to the nearest bushel.
What do you conclude about the effect of this weed on corn yield?
(c) Make a scatterplot of corn yield against weeds per meter. Find the least-squares
regression line and add it to your plot. The advantage of regression over the side-
by-side comparison in (b) is that we can use the fitted model to draw conclusions
for counts of weeds other than the ones the researcher actually used. What does the
slope of the fitted line tell us about the effect of lamb’s-quarter on corn yield?
(d) Predict the yield for corn grown under these conditions with six lamb’s-quarter
plants per meter of row.
2.79 Stock prices and earnings. In the long run, the price of a company’s stock
ought to parallel changes in the company’s earnings. The following table gives data
on the annual growth rates in earnings and in stock prices (both in percent) for
major industry groups as set by Standard & Poor’s. (H. Bradley Perry, “Analyzing
growth stocks: what’s a good growth rate?” AAII Journal, 13 (1991), pp. 7–10.)
(a) Make a graph showing how earnings growth explains growth in stock price. Does
it appear to be true that (on the average in the long run) stock price growth parallels
earnings growth?
(b) What percent of the variation in stock price growth among industry groups can
be explained by the linear relationship with earnings growth?
(c) If stock prices exactly followed earnings, the slope of the least-squares line for
predicting price growth from earnings growth would be 1. Explain why. What is
the slope of the least-squares line for these data?
(d) What is the correlation between earnings growth and price growth? If we had
data on all of the individual companies in these 20 industries, would the correlation
be higher or lower? Why?
Percent growth in stock price and earnings for industry groups

Industry Earnings growth (%) Price growth (%)
Auto 3.3 2.9
Banks 8.6 6.5
Chemicals 6.6 3.1
Computers 10.2 5.3
Drugs 11.3 10.0
Electrical equipment 8.5 8.2
Food 7.6 6.5
Household products 9.7 10.1
Machinery 5.1 4.7
Oil: domestic 7.4 7.3
Oil: international 7.7 7.7
Oil equipment/services 10.1 10.8
Railroad 6.6 6.6
Retail: department stores 10.1 9.5
Retail: food 6.9 6.9
Soft drinks 12.7 12.0
Steel −1.0 −1.6
Tobacco 12.3 1.7
Utilities: electric 2.8 1.4
Utilities: gas 5.2 6.2
2.80 Running speed and stride rate. The following table gives data on the
relationship between running speed (feet per second) and stride rate (steps taken
per second) for elite female runners:
Speed 15.86 16.88 17.50 18.62 19.97 21.06 22.11

Stride rate 3.05 3.12 3.17 3.25 3.36 3.46 3.55
Here are the corresponding data from the same source for male runners:
Review Exercises 67
Speed 15.86 16.88 17.50 18.62 19.97 21.06 22.11

Stride rate 2.92 2.98 3.03 3.11 3.22 3.31 3.41
(a) Plot the data for both groups on one graph using different symbols to distinguish
between the points for females and those for males.
(b) Suppose now that the data came to you without identification as to gender.
Compute the least-squares line from all of the data and plot it on your graph.
(c) Compute the residuals from this line for each observation. Make a plot of the
residuals against speed. How does the fact that the data come from two distinct
groups show up in the residual plot?
2.81 Wood flakes as a building material. Wood scientists are interested in

replacing solid wood building material with less expensive products made from wood
flakes. They collected the following data to examine the relationship between the
length (in inches) and the strength (in pounds per square inch) of beams made
from wood flakes: (Data provided by Jim Bateman and Michael Hunt, Purdue
University.)
Length 5 6 7 8 9 10 11 12 13 14
Strength 446 371 334 296 249 254 244 246 239 234
(a) Make a scatterplot that shows how the length of a beam affects its strength.
(b) Describe the overall pattern of the plot. Are there any outliers?
(c) Fit a least-squares line to the entire set of data. Graph the line on your scatter-
plot. Does a straight line adequately describe these data?
(d) The scatterplot suggests that the relation between length and strength can be
described by two straight lines, one for lengths less than 9 inches and another for
lengths 9 inches or greater. Fit least-squares lines to these two subsets of the data,
and draw the lines on your plot. Do they describe the data adequately? What
question would you now ask the wood experts?
2.82 Global investing. One reason to invest abroad is that markets in different
countries don’t move in step. When American stocks go down, foreign stocks may go
up. So an investor who holds both bears less risk. That’s the theory. Now we read:
“The correlation between changes in American and European share prices has risen
from 0.4 in the mid-1990s to 0.8 in 2000.” (“Dancing in step,” Economist, March
22, 2001.) Explain to an investor who knows no statistics why this fact reduces the
protection provided by buying European stocks.
2.83 Stock prices in Europe and the United States. The same article that
claims that the correlation between changes in stock prices in Europe and the United
States was 0.8 in 2000 goes on to say: “Crudely, that means that movements on
Wall Street can explain 80% of price movements in Europe.” Is this true? What is
the correct percent explained if r = 0.8?
2.84 SAT scores and grade point averages. Can we predict college grade point
average from SAT scores and high school grades? The CSDATA data set contains
information on this issue for a large group of computer science students. We will
look only at SAT Mathematics scores as a predictor of later college GPA, using the
variables SATM and GPA from CSDATA. Make a scatterplot, obtain r and r2 , and
draw on your plot the least-squares regression line for predicting GPA from SATM.
Then write a brief discussion of the ability of SATM alone to predict GPA. (In
Chapter 11 we will see how combining several explanatory variables improves our
ability to predict.)
2.85 Sexual imagery in magazine ads. In what ways do advertisers in magazines

use sexual imagery to appeal to youth? One study classified each of 1509 full-page
or larger ads as “not sexual” or “sexual,” according to the amount and style of the
clothing of the male or female model in the ad. The ads were also classified according
to the target readership of the magazine. (Tom Reichert, “The prevalence of sexual
imagery in ads targeted to young adults,” Journal of Consumer Affairs, 37 (2003),
pp. 403–412.) Here is the two-way table of counts:
Magazine readership
Model clothing Women Men General interest Total
Not sexual 351 514 248 1113
Sexual 225 105 66 396
Total 576 619 314 1509
(a) Summarize the data numerically and graphically.

(b) All of the ads were taken from the March, July, and November issues of six
magazines in one year. Discuss how this fact influences your interpretation of the
results.
2.86 Age of the intended readership. The ads in the study described in the
previous exercise were also classified according to the age group of the intended
readership. Here is a summary of the data:
Magazine readership age group

Model clothing Young adult Mature adult
Not sexual (percent) 72.3% 76.1%
Sexual (percent) 27.2% 23.9%
Number of ads 1006 503
Using parts (a) and (b) of the previous exercise as a guide, analyze these data and
write a report summarizing your work.
CHAPTER 3
Section 3.1
3.1 Yvette is a young banker. She and all her friends carry cell phones and use them
heavily. Last year, two of Yvette’s acquaintances developed brain tumors. Yvette
wonders if the tumors are related to use of cell phones. Explain briefly why the
experience of Yvette’s friends does not provide good evidence that cell phones cause
brain tumors.
3.2 There is strong public support for “term limits” that restrict the number of terms
that legislators can serve. One possible explanation for this support is that voters
are dissatisfied with the performance of Congress and other legislative bodies. A
political scientist asks a sample of voters if they support term limits for members of
Congress and also asks several questions that gauge their satisfaction with Congress.
He finds no relationship between approval of Congress and support for term limits.
Is this an observational study or an experiment? Why? What are the explanatory
and response variables?
3.3 There may be a “gender gap” in political party preference in the United States,
with women more likely than men to prefer Democratic candidates. A political
scientist selects a large sample of registered voters, both men and women. She asks
every voter whether they voted for the Democratic or the Republican candidate in
the last congressional election. Is this an observational study or an experiment?
Why? What are the explanatory and response variables?
3.4 Many studies have found that people who drink alcohol in moderation have
lower risk of heart attacks than either nondrinkers or heavy drinkers. Does alco-
hol consumption also improve survival after a heart attack? One study followed
1913 people who were hospitalized after severe heart attacks. In the year before
their heart attack, 47% of these people did not drink, 36% drank moderately, and
17% drank heavily. After four years, fewer of the moderate drinkers had died. (K.
J. Mukamal et al., “Prior alcohol consumption and mortality following acute my-
ocardial infarction,” Journal of the American Medical Association, 285 (2001), pp.
1965–1970.) Is this an observational study or an experiment? Why? What are the
explanatory and response variables?
3.5 A study of the effect of living in public housing on family stability and other
variables in poverty-level households was carried out as follows. The researchers
obtained a list of all applicants for public housing during the previous year. Some
applicants had been accepted, while others had been turned down by the housing
authority. Both groups were interviewed and compared. Was this study an exper-
iment? Why or why not? What are the explanatory and response variables in the
study?
3.6 The National Halothane Study was a major investigation of the safety of anes-
thetics used in surgery. Records of over 850,000 operations performed in 34 major
hospitals showed the following death rates for four common anesthetics. (L. E. Moses
69
and F. Mosteller, “Safety of anesthetics,” in J. M. Tanur et al. (eds.), Statistics: A

Guide to the Unknown, 3rd ed., Wadsworth, 1989, pp. 15–24.)
Anesthetic A B C D
Death rate 1.7% 1.7% 3.4% 1.9%
There is a clear association between the anesthetic used and the death rate of pa-
tients. Anesthetic C appears dangerous.
(a) Explain why we call the National Halothane Study an observational study rather
than an experiment, even though it compared the results of using different anesthet-
ics in actual surgery.
(b) When the study looked at other variables that are confounded with a doctor’s
choice of anesthetic, it found that Anesthetic C was not causing extra deaths. Sug-
gest several variables that are mixed up with what anesthetic a patient receives.
3.7 Some people believe that exercise raises the body’s metabolic rate for as long as
12 to 24 hours, enabling us to continue to burn off fat after our workout has ended.
In a study of this effect, subjects walked briskly on a treadmill for several hours.
Their metabolic rates were measured before, immediately after, and 12 hours after
the exercise. The study was criticized because eating raises the metabolic rate, and
no record was kept of what the subjects ate after exercising. Was this study an
experiment? Why or why not? What are the explanatory and response variables?
3.8 A manufacturer of food products uses package liners that are sealed at the
top by applying heated jaws after the package is filled. The customer peels the
sealed pieces apart to open the package. What effect does the temperature of the
jaws have on the force required to peel the liner? To answer this question, the
engineers prepare 20 pairs of pieces of package liner. They seal five pairs at each
of 250◦F, 275◦F, 300◦F, and 325◦F. Then they measure the peel strength of each
seal. Identify the experimental units or subjects, the factors, the treatments, and
the response variables
3.9 Sickle-cell disease is an inherited disorder of the red blood cells that in the United
States affects mostly blacks. It can cause severe pain and many complications.
Can the drug hydroxyurea reduce the severe pain caused by sickle-cell disease? A
study by the National Institutes of Health gave the drug to 150 sickle-cell sufferers
and a placebo to another 150. The researchers then counted the episodes of pain
reported by each subject. Identify the experimental units or subjects, the factors,
the treatments, and the response variables.
3.10 People who eat lots of fruits and vegetables have lower rates of colon cancer
than those who eat little of these foods. Fruits and vegetables are rich in “antioxi-
dants” such as vitamins A, C, and E. Will taking antioxidants help prevent colon
cancer? A clinical trial studied this question with 864 people who were at risk of
colon cancer. The subjects were divided into four groups: daily beta-carotene, daily
vitamins C and E, all three vitamins every day, and daily placebo. After four years,
the researchers were surprised to find no significant difference in colon cancer among
the groups. (G. Kolata, “New study finds vitamins are not cancer preventers,” New
York Times, July 21, 1994.)
Section 3.1 71
(a) What are the explanatory and response variables in this experiment?
(b) Outline the design of the experiment. Use your judgment in choosing the group
sizes.
(c) Assign labels to the 864 subjects and use Table B starting at line 118 to choose
the first 5 subjects for the beta-carotene group.
(d) The study was double-blind. What does this mean?
(e) What does “no significant difference” mean in describing the outcome of the
study?
(f) Suggest some lurking variables that could explain why people who eat lots of
fruits and vegetables have lower rates of colon cancer. The experiment suggests that
these variables, rather than the antioxidants, may be responsible for the observed
benefits of fruits and vegetables.
3.11 Exercise 3.9 describes a medical study of a new treatment for sickle-cell disease.
(a) Outline the design of this experiment.
(b) Use of a placebo is considered ethical if there is no effective standard treatment to
give the control group. It might seem humane to give all the subjects hydroxyurea in
the hope that it will help them. Explain clearly why this would not provide informa-
tion about the effectiveness of the drug. (In fact, the experiment was stopped ahead
of schedule because the hydroxyurea group had only half as many pain episodes as
the control group. Ethical standards required stopping the experiment as soon as
significant evidence became available.)
3.12 Outline the design of the package liner experiment of Exercise 3.10. Label the
pairs of liner pieces 01 to 20 and carry out the randomization that your design calls
for. (If you use Table B, start at line 120.)
3.13 Surgery patients are often cold because the operating room is kept cool and
the body’s temperature regulation is disturbed by anesthetics. Will warming pa-
tients to maintain normal body temperature reduce infections after surgery? In one
experiment, patients undergoing colon surgery received intravenous fluids from a
warming machine and were covered with a blanket through which air circulated.
For some patients, the fluid and the air were warmed; for others, they were not.
The patients received identical treatment in all other respects.
(a) Outline the design of a randomized comparative experiment for this study.
(b) The following subjects have given consent to participate in this study. Do the
random assignment required by your design. (If you use Table B, begin at line 121.)
Abbott Decker Herrera Lucero Richter
Abdalla Devlin Hersch Masters Riley
Alawi Engel Hurwitz Morgan Samuels
Broden Fuentes Irwin Nelson Smith
Chai Garrett Jiang Nho Suarez
Chuang Gill Kelley Ortiz Upasani
Cordoba Glover Kim Ramdas Wilson
Custer Hammond Landers Reed Xiang
3.14 Will providing child care for employees make a company more attractive to
women, even those who are unmarried? You are designing an experiment to answer
this question. You prepare recruiting material for two fictitious companies, both in
similar businesses in the same location. Company A’s brochure does not mention
child care. There are two versions of Company B’s material, identical except that
one describes the company’s on-site childcare facility. Your subjects are 40 unmar-
ried women who are college seniors seeking employment. Each subject will read
recruiting material for both companies and choose the one she would prefer to work
for. You will give each version of Company B’s brochure to half the women. You
suspect that a higher percentage of those who read the description that includes
child care will choose Company B.
(a) Outline the design of the experiment. Be sure to identify the response variable.
(b) The names of the subjects appear below. Do the randomization required by
your design and list the subjects who will read the version that mentions child care.
(If you use Table B, begin at line 121.)
Abrams Danielson Gutierrez Lippman Rosen
Adamson Durr Howard Martinez Sugiwara
Afifi Edwards Hwang McNeill Thompson
Brown Fluharty Iselin Morse Travers
Cansico Garcia Janle Ng Turing
Chen Gerson Kaplan Quinones Ullmann
Cortez Green Kim Rivera Williams
Curzakis Gupta Lattimore Roberts Wong
3.15 A horticulturist is comparing two methods (call them A and B) of growing

potatoes. Standard potato cuttings will be planted in small plots of ground. The
response variables are number of tubers per plant and fresh weight (weight when
just harvested) of vegetable growth per plant. There are 20 plots available for the
experiment. Sketch the outline of a rectangular field divided into 5 rows of 4 plots
each. Then outline the experimental design and do the required randomization. (If
you use Table B, start at line 145.) Mark on your sketch which growing method you
will use in each plot.
3.16 Once a person has been convicted of drunk driving, one purpose of court-
mandated treatment or punishment is to prevent future offenses of the same kind.
Suggest three different treatments that a court might require. Then outline the
design of an experiment to compare their effectiveness. Be sure to specify the
response variables you will measure.
3.17 Here are some questions about the study of heating surgery patients in Exercise
3.13.
(a) To simplify the setup of the study, we might warm the fluids and air blanket for
one operating team and not for another doing the same kind of surgery. Why might
this design result in bias?
(b) The operating team did not know whether fluids and air blanket were heated,
nor did the doctors who followed the patients after surgery. What is this practice
called? Why was it used here?
3.18 You want to determine the best color for attracting cereal leaf beetles to boards
on which they will be trapped. You will compare four colors: blue, green, white,
Section 3.1 73
and yellow. The response variable is the count of beetles trapped. You will mount
one board on each of 16 poles evenly spaced in a square field, with four poles in
each of four rows. Sketch the field with the locations of the 16 poles. Outline the
design of a completely randomized experiment to compare the colors. Randomly
assign colors to the poles, and mark on your sketch the color assigned to each pole.
(If you use Table B, start at line 115.)
3.19 Continue the discussion of the experiment of the previous exercise. The re-
searchers decide to use two oat fields in different locations and to space eight poles
equally within each field. Outline a randomized block design using the fields as
blocks. Then use Table B, beginning at line 105, to carry out the random assign-
ment of colors to poles. Report your results by means of a sketch of the two fields
with the color at each pole noted.
3.20 A mathematics education researcher is studying where in high school mathe-

matics texts it is most effective to insert questions. She wants to know whether it is
better to present questions as motivation before the text passage or as review after
the passage. The result may depend on the type of question asked: simple fact,
computation, or word problem.
(a) This experiment has two factors. What are they? How many treatments do all
combinations of levels of these factors form? List the treatments.
(b) Because it is disruptive to assign high school students at random to the treat-
ment groups, the researcher will assign two classes of the same grade level to each
treatment. The response variable is score on a mathematics test taken by all the
students in these classes. Outline the design of the experiment. Carry out the
random assignment required.
3.21 A study of the effects of running on personality involved 231 male runners
who each ran about 20 miles a week. The runners were given the Cattell Six-
teen Personality Factor Questionnaire, a 187-item multiple-choice test often used by
psychologists. A news report (New York Times, February 15, 1988) stated, “The
researchers found statistically significant personality differences between the runners
and the 30-year-old male population as a whole.” A headline on the article said,
“Research has shown that running can alter one’s moods.”
(a) Explain carefully, to someone who knows no statistics, what “statistically sig-
nificant” means.
(b) Explain carefully, to someone who knows no statistics, why the headline is mis-
leading.
3.22 Is the right hand generally stronger than the left in right-handed people? You
can crudely measure hand strength by placing a bathroom scale on a shelf with the
end protruding, then squeezing the scale between the thumb below and the four
fingers above it. The reading of the scale shows the force exerted. Describe the
design of a matched pairs experiment to compare the strength of the right and left
hands, using 16 right-handed people as subjects. Use Table B at line 114 to choose
which 8 subjects will try their right hands first.
3.23 Do consumers prefer the taste of Pepsi or Coke in a blind test in which neither
cola is identified? Describe briefly the design of a matched pairs experiment to
investigate this question. How will you use randomization?
3.24 There are several psychological tests available to measure the extent to which
Mexican Americans are oriented toward Mexican/Spanish or Anglo/English culture.
Two such tests are the Bicultural Inventory (BI) and the Acculturation Rating Scale
for Mexican Americans (ARSMA). To study the correlation between the scores on
these two tests, researchers will give both tests to a group of 22 Mexican Americans.
(a) Briefly describe a matched pairs design for this study. In particular, how will
you use randomization in your design?
(b) You have an alphabetized list of the subjects (numbered 1 to 22). Carry out the
randomization required by your design and report the result.
3.25 Will people spend less on health care if their health insurance requires them
to pay some part of the cost themselves? An experiment on this issue asked if the
percent of medical costs that are paid by health insurance has an effect either on
the amount of medical care that people use or on their health. The treatments were
four insurance plans. Each plan paid all medical costs above a ceiling. Below the
ceiling, the plans paid 100%, 75%, 50%, or 0% of costs incurred.
(a) Outline the design of a randomized comparative experiment suitable for this
study.
(b) Describe briefly the practical and ethical difficulties that might arise in such an
experiment.
3.26 A chemical engineer is designing the production process for a new product.
The chemical reaction that produces the product may have higher or lower yield,
depending on the temperature and the stirring rate in the vessel in which the reaction
takes place. The engineer decides to investigate the effects of combinations of two
temperatures (50◦C and 60◦C) and three stirring rates (60 rpm, 90 rpm, and 120
rpm) on the yield of the process. Two batches of the feedstock will be processed at
each combination of temperature and stirring rate.
(a) How many factors are there in this experiment? How many treatments? Identify
each of the treatments. How many experimental units (batches of feedstock) does
the experiment require?
(b) Outline in graphic form the design of an appropriate experiment.
(c) The randomization in this experiment determines the order in which batches of
the feedstock will be processed according to each treatment. Use Table B starting
at line 128 to carry out the randomization and state the result.
3.27 Jamie is a hard-core computer programmer. He and all his friends prefer Jolt
Cola (caffeine equivalent to two cups of coffee) to either Coke or Pepsi (caffeine
equivalent to less than one cup of coffee). Explain why Jamie’s experience is not
good evidence that most young people prefer Jolt to Coke or Pepsi.
3.28 When the discussion turns to the pros and cons of wearing automobile seat
belts, Herman always brings up the case of a friend who survived an accident because
he was not wearing a seat belt. The friend was thrown out of the car and landed
Section 3.1 75
on a grassy bank, suffering only minor injuries, while the car burst into flames and
was destroyed. Explain briefly why this anecdote does not provide good evidence
that it is safer not to wear seat belts.
3.29 Several large observational studies suggested that women who take hormones
such as estrogen after menopause have lower risk of a heart attack than women who
do not take hormones. Hormone replacement became popular. But in 2002, several
careful experiments showed that hormone replacement does not reduce heart attacks.
The National Institutes of Health, after reviewing the evidence, concluded that the
observational studies were wrong. Taking hormones after menopause quickly fell
out of favor.
(a) Explain the difference between an observational study and an experiment to
compare women who do and don’t take hormones after menopause.
(b) Suggest some characteristics of women who choose to take hormones that might
affect the rate of heart attacks. In an observational study, these characteristics are
confounded with taking hormones.
3.30 Moderate use of alcohol is associated with better health. Some studies suggest
that drinking wine rather than beer or spirits confers added health benefits.
(a) Explain the difference between an observational study and an experiment to
compare people who drink wine with people who drink beer.
(b) Suggest some characteristics of wine drinkers that might benefit their health. In
an observational study, these characteristics are confounded with drinking wine.
3.31 Try to find information on this question: what percent of college undergradu-
ates work part-time or full-time while they are taking classes? Start with the Web
site of the National Center for Education Statistics, nces.ed.gov. Keep a log of
your search for this information.
3.32 Doctors identify “chronic tension-type headaches” as headaches that occur al-
most daily for at least six months. Can antidepressant medications or stress manage-
ment training reduce the number and severity of these headaches? Are both together
more effective than either alone? Investigators compared four treatments: antide-
pressant alone, placebo alone, antidepressant plus stress management, and placebo
plus stress management. Outline the design of the experiment. The headache suf-
ferers named below have agreed to participate in the study. Use software or Table B
at line 130 to randomly assign the subjects to the treatments.
Acosta Duncan Han Liang Padilla Valasco

Asihiro Durr Howard Maldonado Plochman Vaughn
Bennett Edwards Hruska Marsden Rosen Wei
Bikalis Farouk Imrani Montoya Solomon Wilder
Chen Fleming James O’Brian Trujillo Willis
Clemente George Kaplan Ogle Tullock Zhang
3.33 How does smoking marijuana affect willingness to work? Canadian researchers
persuaded young adult men who used marijuana to live for 98 days in a “planned
environment.” The men earned money by weaving belts. They used their earnings
to pay for meals and other consumption and could keep any money left over. One
group smoked two potent marijuana cigarettes every evening. The other group
smoked two weak marijuana cigarettes. All subjects could buy more cigarettes but
were given strong or weak cigarettes, depending on their group. Did the weak and
strong groups differ in work output and earnings?
(b) Here are the names of the 20 subjects. Use software or Table B at line 131 to
carry out the randomization your design requires.
Abbott Decker Gutierrez Lucero Rosen

Afifi Engel Hwang McNeill Thompson
Brown Fluharty Iselin Morse Travers
Chen Gerson Kaplan Quinones Ullmann
3.34 Eye cataracts are responsible for over 40% of blindness around the world. Can
drinking tea regularly slow the growth of cataracts? We can’t experiment on people,
so we use rats as subjects. Researchers injected 18 young rats with a substance that
causes cataracts. One group of the rats also received black tea extract; a second
group received green tea extract; and a third got a placebo, a substance with no
effect on the body. The response variable was the growth of cataracts over the next
six weeks. Yes, both tea extracts did slow cataract growth.
(b) Use software or Table B, starting at line 142, to assign rats to treatments.
3.35 Workers who survive a lay-off of other employees at their location may suffer
from “survivor guilt.” A study of survivor guilt and its effects used as subjects
120 students who were offered an opportunity to earn extra course credit by doing
proofreading. Each subject worked in the same cubicle as another student, who was
an accomplice of the experimenters. At a break midway through the work, one of
three things happened:
Treatment 1: The accomplice was told to leave; it was explained that

this was because she performed poorly.
Treatment 2: It was explained that unforeseen circumstances meant
there was only enough work for one person. By “chance,” the accomplice
was chosen to be laid off.
Treatment 3: Both students continued to work after the break.
The subjects’ work performance after the break was compared with performance
before the break.
(a) Outline the design of this completely randomized experiment.
(b) If you are using software, choose the subjects for Treatment 1. If not, use Table B
at line 123 to choose the first four subjects for Treatment 1.
3.36 Twenty-four public middle schools agree to participate in the experiment de-
scribed in Exercise 3.35. Use a diagram to outline a completely randomized design
for this experiment. Then do the randomization required to assign schools to treat-
ments. If you use Table B, start at line 105.
Section 3.1 77
3.37 Stores advertise price reductions to attract customers. What type of price cut
is most attractive? Market researchers prepared ads for athletic shoes announcing
different levels of discounts (20%, 40%, 60%, or 80%). The student subjects who
read the ads were also given “inside information” about the fraction of shoes on sale
(25%, 50%, 75%, or 100%). Each subject then rated the attractiveness of the sale
on a scale of 1 to 7.
(a) There are two factors. Make a sketch that displays the treatments formed by all
combinations of levels of the factors.
(b) Outline a completely randomized design using 80 student subjects. Use software
or Table B at line 133 to choose the subjects for the first treatment.
3.38 We often see players on the sidelines of a football game inhaling oxygen. Their
coaches think this will speed their recovery. We might measure recovery from in-
tense exercise as follows: Have a football player run 100 yards three times in quick
succession. Then allow three minutes to rest before running 100 yards again. Time
the final run. Because players vary greatly in speed, you plan a matched pairs
experiment using 20 football players as subjects. Describe the design of such an
experiment to investigate the effect of inhaling oxygen during the rest period. Why
should each player’s two trials be on different days? Use Table B at line 170 to
decide which players will get oxygen on their first trial.
3.39 Calcium is important to the development of young girls. To study how the
bodies of young girls process calcium, investigators used the setting of a summer
camp. Calcium was given in Hawaiian Punch at either a high or a low level. The
camp diet was otherwise the same for all girls. Suppose that there are 60 campers.
(a) Outline a completely randomized design for this experiment.
(b) Describe a matched pairs design in which each girl receives both levels of calcium
(with a “washout period” between). What is the advantage of the matched pairs
design over the completely randomized design?
(c) The same randomization can be used in different ways for both designs. Label
the subjects 01 to 60. You must choose 30 of the 60. Use Table B at line 160 to
choose just the first 5 of the 30. How are the 30 subjects chosen treated in the
completely randomized design? How are they treated in the matched pairs design?
3.40 Twenty overweight females have agreed to participate in a study of the effec-
tiveness of four reducing regimens, A, B, C, and D. The researcher first calculates
how overweight each subject is by comparing the subject’s actual weight with her
“ideal” weight. The subjects and their excess weights in pounds are
Birnbaum 35 Hernandez 25 Moses 25 Smith 29

Brown 34 Jackson 33 Nevesky 39 Stall 33
Brunk 30 Kendall 28 Obrach 30 Tran 35
Dixon 34 Loren 32 Rodriguez 30 Wilansky 42
Festinger 24 Mann 28 Santiago 27 Williams 22
The response variable is the weight lost after eight weeks of treatment. Because
the initial amount overweight will influence the response variable, a block design is
appropriate.
(a) Arrange the subjects in order of increasing excess weight. Form five blocks by
grouping the four least overweight, then the next four, and so on.
(b) Use Table B to do the required random assignment of subjects to the four
reducing regimens separately within each block. Be sure to explain exactly how you
used the table.
3.41 Fractures of the spine are common and serious among women with advanced
osteoporosis (low mineral density in the bones). Can taking strontium renelate
help? A large medical trial assigned 1649 women to take either strontium renelate
or a placebo each day. All of the subjects had osteoporosis and had had at least
one fracture. All were taking calcium supplements and receiving standard medical
care. The response variables were measurements of bone density and counts of new
fractures over three years. The subjects were treated at 10 medical centers in 10
different countries. Outline an appropriate design for this experiment. Explain why
this is the proper design.
3.42 Final Fu. Your friends are big fans of “Final Fu,” MTV2’s martial arts
competition. To what extent do you think you can generalize your preferences for
this show to all students at your college?
3.43 Compost tea. Compost tea is rich in microorganisms that help plants grow.
It is made by soaking compost in water. (Based on a study conducted by Brent Ladd,
a Water Quality Specialist with the Purdue University Department of Agricultural
and Biological Engineering.) Design a comparative experiment that will provide
evidence about whether or not compost tea works for a particular type of plant that
interests you. Be sure to provide all details regarding your experiment, including
the response variable or variables that you will measure.
3.44 Measuring water quality in streams and lakes. Water quality of streams
and lakes is an issue of concern to the public. Although trained professionals typi-
cally are used to take reliable measurements, many volunteer groups are gathering
and distributing information based on data that they collect. (Based on a study
conducted by Sandra Simonis under the direction of Professor Jon Harbor from the
Purdue University Earth and Atmospheric Sciences Department.) You are part of
a team to train volunteers to collect accurate water quality data. Design an ex-
periment to evaluate the effectiveness of the training. Write a summary of your
proposed design to present to your team. Be sure to include all of the details that
they will need to evaluate your proposal.
3.45 Eye cataracts. Eye cataracts are responsible for over 40% of blindness around
the world. Can drinking tea regularly slow the growth of cataracts? We can’t
experiment on people, so we use rats as subjects. Researchers injected 21 young rats
with a substance that causes cataracts. One group of the rats also received black
tea extract; a second group received green tea extract; and a third got a placebo,
a substance with no effect on the body. The response variable was the growth of
cataracts over the next six weeks. Yes, both tea extracts did slow cataract growth.
(Geetha Thiagarajan et al., “Antioxidant properties of green and black tea, and
their potential ability to retard the progression of eye lens cataract,” Experimental
Section 3.1 79
Eye Research, 73 (2001), pp. 393–401.)

(b) Use software or Table B, starting at line 120, to assign rats to treatments.
3.46 Treatment of clothing fabric. A maker of fabric for clothing is setting

up a new line to “finish” the raw fabric. The line will use either metal rollers or
natural-bristle rollers to raise the surface of the fabric; a dyeing cycle time of either
30 minutes or 40 minutes; and a temperature of either 150◦ or 175◦C. An experi-
ment will compare all combinations of these choices. Four specimens of fabric will
be subjected to each treatment and scored for quality.
(a) What are the factors and the treatments? How many individuals (fabric speci-
mens) does the experiment require?
(b) Outline a completely randomized design for this experiment. (You need not
actually do the randomization.)
3.47 Treatment of pain for cancer patients. Health care providers are giving
more attention to relieving the pain of cancer patients. An article in the journal
Cancer surveyed a number of studies and concluded that controlled-release mor-
phine tablets, which release the painkiller gradually over time, are more effective
than giving standard morphine when the patient needs it. (Carol A. Warfield,
“Controlled-release morphine tablets in patients with chronic cancer pain,” Cancer,
82 (1998), pp. 2299–2306.) The “methods” section of the article begins: “Only those
published studies that were controlled (i.e., randomized, double blind, and compar-
ative), repeated-dose studies with CR morphine tablets in cancer pain patients were
considered for this review.” Explain the terms in parentheses to someone who knows
nothing about medical trials.
3.48 Saint-John’s-wort and depression. Does the herb Saint-John’s-wort re-

lieve major depression? Here are some excerpts from the report of a study of this
issue. (R. C. Shelton et al., “Effectiveness of St. John’s wort in major depression,”
Journal of the American Medical Association, 285 (2001), pp. 1978–1986.) The
study concluded that the herb is no more effective than a placebo.
(a) “Design: Randomized, double-blind, placebo-controlled clinical trial. . . .” Ex-
plain the meaning of each of the terms in this description.
(b) “Participants . . . were randomly assigned to receive either Saint-John’s-wort ex-
tract (n = 98) or placebo (n = 102). . . . The primary outcome measure was the rate
of change in the Hamilton Rating Scale for Depression over the treatment period.”
Based on this information, use a diagram to outline the design of this clinical trial.
3.49 The Monday effect on stock prices. Puzzling but true: stocks tend to
go down on Mondays. There is no convincing explanation for this fact. A recent
study looked at this “Monday effect” in more detail, using data on the daily returns
of stocks on several U.S. exchanges over a 30-year period. Here are some of the
findings:
To summarize, our results indicate that the well-known Monday effect is
caused largely by the Mondays of the last two weeks of the month. The
mean Monday return of the first three weeks of the month is, in general,
not significantly different from zero and is generally significantly higher
than the mean Monday return of the last two weeks. Our finding seems
to make it more difficult to explain the Monday effect.
(K. Wang, Y. Li, and J. Erickson, “A new look at the Monday effect,” Journal
of Finance, 52 (1997), pp. 2171–2186.) A friend thinks that “significantly” in this
article has its plain English meaning, roughly “I think this is important.” Explain
in simple language what “significantly higher” and “not significantly different from
zero” actually tell us here.
3.50 Five-digit zip codes and delivery time of mail. Does adding the five-
digit postal zip code to an address really speed up delivery of letters? Does adding
the four more digits that make up “zip + 4” speed delivery yet more? What about
mailing a letter on Monday, Thursday, or Saturday? Describe the design of an
experiment on the speed of first-class mail delivery. For simplicity, suppose that all
letters go from you to a friend, so that the sending and receiving locations are fixed.
Section 3.2
3.51 A political scientist wants to know how college students feel about the Social
Security system. She obtains a list of the 3456 undergraduates at her college and
mails a questionnaire to 250 students selected at random. Only 104 questionnaires
are returned. What is the population in this study? What is the sample from which
information was actually obtained? What is the rate (percent) of nonresponse?
3.52 Different types of writing can sometimes be distinguished by the lengths of the
words used. A student interested in this fact wants to study the lengths of words
used by Tom Clancy in his novels. She opens a Clancy novel at random and records
the lengths of each of the first 250 words on the page. What is the population in
this study? What is the sample? What is the variable measured?
3.53 A newspaper article about an opinion poll says that “43% of Americans approve
of the president’s overall job performance.” Toward the end of the article, you read:
“The poll is based on telephone interviews with 1210 adults from around the United
States, excluding Alaska and Hawaii.” What variable did this poll measure? What
population do you think the newspaper wants information about? What was the
sample? Are there any sources of bias in the sampling method used?
3.54 A newspaper advertisement for USA Today: The Television Show said:
Should handgun control be tougher? You call the shots in a special call-
in poll tonight. If yes, call 1-900-720-6181. If no, call 1-900-720-6182.
Charge is 50 cents for the first minute.
Explain why this opinion poll is almost certainly biased.
3.55 The students listed below are enrolled in a statistics course taught on television.
Choose an SRS of 6 students to be interviewed in detail about the quality of the
course. (If you use Table B, start at line 139.)
Section 3.2 81
Abate Dubois Hixson Putnam

Anderson Fernandez Klassen Rodriguez
Baxter Frank Liang Rubin
Bowen Fuhrmann Moser Santiago
Bruvold Goel Naber Shen
Casella Gupta Petrucelli Shyr
Choi Hicks Pliego Sundheim
3.56 You want to choose an SRS of 25 of a city’s 440 voting precincts for special
voting-fraud surveillance on election day. How will you label the 440 precincts?
Choose the SRS, and list the precincts you selected. (Use the Simple Random
Sample applet. If you use Table B, enter at line 117 and select only the first 5
precincts in the sample.)
3.57 A firm wants to understand the attitudes of its minority managers toward
its system for assessing management performance. Below is a list of all the firm’s
managers who are members of minority groups. Use Table B at line 139 to choose
6 to be interviewed in detail about the performance appraisal system.
Acosta Dewald Huang Puri

Ali Fleming Kim Richards
Baxter Fonseca Lujan Rodriguez
Bowman Gates Mourning Santiago
Brams Goel Nunez Shen
Cortez Gomez Peters Vega
Cross Hernandez Pliego Watanabe
3.58 An academic department wishes to choose a three-member advisory committee

at random from the members of the department. Use Table B at line 140 to choose
an SRS of size 3 from the 28 faculty listed below. (For convenience, they are labeled
in alphabetical order.)
00 Abate 07 Goodwin 14 Pillotte 21 Theobald

01 Cicirelli 08 Haglund 15 Raman 22 Vader
02 Cuellar 09 Johnson 16 Riemann 23 Wang
03 Dunsmore 10 Keegan 17 Rodriguez 24 Wieczorek
04 Engle 11 Luo 18 Rowe 25 Williams
05 Fitzpatrick 12 Martinez 19 Salazar 26 Wilson
06 Garcia 13 Nguyen 20 Stone 27 Wong
3.59 A university has 2000 male and 500 female faculty members. The equal oppor-
tunity employment officer wants to poll the opinions of a random sample of faculty
members. In order to give adequate attention to female faculty opinion, he decides
to choose a stratified random sample of 200 males and 200 females. He has alpha-
betized lists of female and male faculty members. Explain how you would assign
labels and use random digits to choose the desired sample. Enter Table B at line
122 and give the labels of the first 5 females and the first 5 males in the sample.
3.60 A labor organization wants to study the attitudes of college faculty members
toward collective bargaining. These attitudes appear to be different depending on
the type of college. The American Association of University Professors classifies
colleges as follows:
Class I: Offer doctorate degrees and award at least 15 per year.
Class IIA: Award degrees above the bachelor’s but are not in Class I.
Class IIB: Award no degrees beyond the bachelor’s.
Class III: Two-year colleges.
Discuss the design of a sample of faculty from colleges in your state, with total
sample size about 200 faculty.
3.61 Here are two wordings for the same question. The first question was asked by
presidential candidate Ross Perot, and the second by a Time/CNN Poll, both in
March 1993.
A. Should laws be passed to eliminate all possibilities of special interests

giving huge sums of money to candidates?
B. Should laws be passed to prohibit interest groups from contributing

to campaigns, or do groups have a right to contribute to the candidates
they support?
One of these questions drew 40% favoring banning contributions; the other drew
80% with this opinion. Which question produced the 40% and which got 80%?
Explain why the results were so different. (W. Mitofsky, “Mr. Perot, you’re no
pollster,” New York Times, March 27, 1993.)
3.62 A committee on community relations in a college town plans to survey local

businesses about the importance of students as customers. From telephone book
listings, the committee chooses 150 businesses at random. Of these, 73 return the
questionnaire mailed by the committee. What is the population for this sample
survey? What is the sample? What is the rate (percent) of nonresponse?
3.63 A Gallup Poll asked, “Do you think the U.S. should take the leading role
in world affairs, take a major role but not the leading role, take a minor role, or
take no role at all in world affairs?” Gallup’s report said, “ Results are based on
telephone interviews with 1,002 national adults, aged 18 and older, conducted Feb.
9-12, 2004.”
(a) What is the population for this sample survey? What was the sample size?
(b) Gallup notes that the order of the four possible responses was rotated when the
question was read over the phone. Why was this done?
3.64 For each of the following sampling situations, identify the population as exactly
as possible. That is, say what kind of individuals the population consists of and
say exactly which individuals fall in the population. If the information given is not
complete, complete the description of the population in a reasonable way.
Section 3.2 83
(a) An opinion poll contacts 1161 adults and asks them, “Which political party do
you think has better ideas for leading the country in the twenty-first century?”
(b) A furniture maker buys hardwood in large lots. The supplier is supposed to
dry the wood before shipping—wood that is not dry won’t hold its size and shape.
The furniture maker chooses 5 pieces of wood from each lot and tests their moisture
content. If any piece exceeds 12% moisture content, the entire lot is sent back.
(c) The American Community Survey (ACS) will replace the census “long form”
starting with the 2010 census. The main part of the ACS contacts 250,000 addresses
by mail each month, with follow-up by phone and in person if there is no response.
Each household answers questions about their housing, economic, and social status.
3.65 An opinion poll calls 1800 randomly chosen residential telephone numbers,
then asks to speak with an adult member of the household. The interviewer asks,
“How many movies have you watched in a movie theater in the past 12 months?”
(a) What population do you think the poll has in mind?
(b) In all, 1131 people respond. What is the rate (percent) of nonresponse?
(c) What source of response error is likely for the question asked?
3.66 You want to ask a sample of college students the question “How much do
you trust information about health that you find on the Internet—a great deal,
somewhat, not much, or not at all?” You try out this and other questions on a pilot
group of 10 students chosen from your class. The class members are
Anderson Eckstein Johnson Puri

Arroyo Fernandez Kim Richards
Batista Fullmer Molina Rodriguez
Bell Gandhi Morgan Samuels
Burke Garcia Nguyen Shen
Calloway Glauser Palmiero Velasco
Delluci Helling Percival Washburn
Drasin Husain Prince Zhao
Choose an SRS. If you use Table B, start at line 139.
3.67 Popularity of news personalities. A Gallup Poll conducted telephone

interviews with 1001 U.S. adults aged 18 and over on July 24-27, 2006. One of the
questions asked whether the respondents had a favorable or an unfavorable opinion
of 17 news personalities. Diane Sawyer received the highest rating, with 80% of the
respondents giving her a favorable rating. (From poll.gallup.com on August 8,
2006.) (a) What is the population for this sample survey? What was the sample
size?
(b) The report on the survey states that 8% of the respondents either never heard of
Sawyer or had no opinion about her. When they included only those who provided
an opinion, Sawyer’s approval percent rose to 88% and she was still at the top of the
list. Charles Gibson, on the other hand, was ranked eighth on the original list, with
a 55% favorable rating. When only those providing an opinion were counted, his
rank rose to second, with 87% approving. Discuss the advantages and disadvantages
of the two different ways of reporting the approval percent. State which one you
prefer and why.
3.68 The Excite Poll. The Excite Poll can be found online at poll.excite.com.
The question appears on the screen, and you simply click buttons to vote “Yes,”
“No,” “Not sure,” or “Don’t care.” On July 22, 2006, the question was “Do you
agree or disagree with proposed legislation that would discontinue the U.S. penny
coin?” In all, 631 said “Yes,” another 564 said “No,” and the remaining 65 indicated
that they were not sure.
(a) What is the sample size for this poll?
(b) Compute the percent of responses in each of the possible response categories.
(c) Discuss the poll in terms of the population and sample framework that we have
studied in this chapter.
Section 3.3
In the following four exercises, state whether each boldface number is a parameter
or a statistic.
3.69 Voter registration records show that 68% of all voters in Indianapolis are
registered as Republicans. To test a random digit dialing device, you use the device
to call 150 randomly chosen residential telephones in Indianapolis. Of the registered
voters contacted, 73% are registered Republicans.
3.70 A carload lot of ball bearings has a mean diameter of 2.503 centimeters (cm).
This is within the specifications for acceptance of the lot by the purchaser. The
inspector happens to inspect 100 bearings from the lot with a mean diameter of
2.515 cm. This is outside the specified limits, so the lot is mistakenly rejected.
3.71 A telemarketing firm in Los Angeles uses a device that dials residential tele-
phone numbers in that city at random. Of the first 100 numbers dialed, 43 are
unlisted. This is not surprising, because 52% of all Los Angeles residential phones
are unlisted.
3.72 The Carolina Abecedarian Project investigated the effect of high-quality

preschool programs on children from poor families. Children were randomly as-
signed to two groups. One group participated in a year-round preschool program
from age three months. The control group received social services but no preschool.
At age 21, 35% of the treatment group and 14% of the control group were attending
a four-year college or had already graduated from college.
3.73 Just before a presidential election, a national opinion polling firm increases the
size of its weekly sample from the usual 1500 people to 4000 people. Why do you
think the firm does this?
3.74 A management student is planning to take a survey of student attitudes toward

part-time work while attending college. He develops a questionnaire and plans to
ask 25 randomly selected students to fill it out. His faculty advisor approves the
questionnaire but urges that the sample size be increased to at least 100 students.
Why is the larger sample helpful?
Section 3.3 85
3.75 An entomologist samples a field for egg masses of a harmful insect by placing a
yard-square frame at random locations and examining the ground within the frame
carefully. She wishes to estimate the proportion of square yards in which egg masses
are present. Suppose that in a large field egg masses are present in 20% of all possible
yard-square areas—that is, p = 0.2 in this population.
(a) Use Table B to simulate the presence or absence of egg masses in each square
yard of an SRS of 10 square yards from the field. Be sure to explain clearly which
digits you used to represent the presence and the absence of egg masses. What
proportion of your 10 sample areas had egg masses?
(b) Repeat (a) with different lines from Table B, until you have simulated the results
of 20 SRSs of size 10. What proportion of the square yards in each of your 20 samples
had egg masses? Make a stemplot from these 20 values to display the sampling
distribution of p̂ in this case. What is the mean of this distribution? What is its
shape?
In the following four exercises, state whether each boldface number is a parameter
or a statistic.
3.76 An opinion poll uses random digit dialing equipment to dial 2000 randomly
chosen residential telephone numbers. Of these, 621 are unlisted numbers. This
isn’t surprising, because 35% of all residential numbers are unlisted.
3.77 A study of voting chose 663 registered voters at random shortly after an elec-
tion. Of these, 72% said they had voted in the election. Election records show that
only 56% of registered voters voted in the election.
3.78 The Tennessee STAR experiment randomly assigned children to regular or

small classes during their first four years of school. When these children reached high
school, 40.2% of blacks from small classes took the ACT or SAT college entrance
exams. Only 31.7% of blacks from regular classes took one of these exams.
3.79 How does caffeine affect our bodies? In a matched pairs experiment, subjects
pushed a button as quickly as they could after taking a caffeine pill and again after
taking a placebo pill. The mean pushes per minute were x = 283 for the placebo
and x = 311 for caffeine.
3.80 Opinions of Hispanics. A New York Times News Service article on a poll
concerned with the opinions of Hispanics includes this paragraph:
The poll was conducted by telephone from July 13 to 27, with 3,092 adults
nationwide, 1,074 of whom described themselves as Hispanic. It has a
margin of sampling error of plus or minus three percentage points for
the entire poll and plus or minus four percentage points for Hispanics.
Sample sizes for most Hispanic nationalities, like Cubans or Dominicans,
were too small to break out the results separately.
(Adam Nagourney and Janet Elder, “New York Times CBS Poll: What Hispanics
Believe,” found online at Hispanic.cc.)
(a) Why is the “margin of sampling error” larger for Hispanics than for all 3092
respondents?
(b) Why would a very small sample size prevent a responsible news organization
from breaking out results for Cubans?
3.81 Real estate ownership. An agency of the federal government plans to take
an SRS of residents in each state to estimate the proportion of owners of real estate in
each state’s population. The populations of the states range from less than 500,000
people in Wyoming to about 35 million in California.
(a) Will the variability of the sample proportion vary from state to state if an SRS
of size 2000 is taken in each state? Explain your answer.
(b) Will the variability of the sample proportion change from state to state if an
SRS of 1/10 of 1% (0.001) of the state’s population is taken in each state? Explain
your answer.
Section 3.4
3.82 Serving as an experimental subject for extra credit. Students taking

Psychology 001 are required to serve as experimental subjects. Students in Psy-
chology 002 are not required to serve, but they are given extra credit if they do so.
Students in Psychology 003 are required either to sign up as subjects or to write a
term paper. Serving as an experimental subject may be educational, but current
ethical standards frown on using “dependent subjects” such as prisoners or charity
medical patients. Students are certainly somewhat dependent on their teachers. Do
you object to any of these course policies? If so, which ones, and why?
3.83 The 2000 census. The 2000 census long form asked 53 detailed questions,
for example:
Do you have COMPLETE plumbing facilities in this house, apartment,

or mobile home; that is, 1) hot and and cold piped water, 2) a flush
toilet, and 3) a bathtub or shower?
The form also asked your income in dollars, broken down by source, and whether any
“physical, mental, or emotional condition” causes you difficulty in “learning, remem-
bering, or concentrating.” Some members of Congress objected to these questions,
even though Congress had approved them.
Give brief arguments on both sides of the debate over the long form: the gov-
ernment has legitimate uses for such information, but the questions seem to invade
people’s privacy.
3.84 You read a news report of an experiment that claims to show that a medita-
tion technique lowered the anxiety level of subjects. The experimenter interviewed
the subjects and assessed their levels of anxiety. The subjects then learned how to
meditate and did so regularly for a month. The experimenter reinterviewed them
at the end of the month and assessed whether their anxiety levels had decreased or
not.
(a) There was no control group in this experiment. Why is this a blunder? What
Review Exercises 87
lurking variables might be confounded with the effect of meditation?

(b) The experimenter who diagnosed the effect of the treatment knew that the sub-
jects had been meditating. Explain how this knowledge could bias the experimental
conclusions.
(c) Briefly discuss a proper experimental design, with controls and blind diagnosis,
to assess the effect of meditation on anxiety level.
3.85 A psychologist is interested in the effect of room temperature on the perfor-

mance of tasks requiring manual dexterity. She chooses temperatures of 70◦F and
90◦F as treatments. The response variable is the number of correct insertions, dur-
ing a 30-minute period, in an elaborate peg-and-hole apparatus that requires the use
of both hands simultaneously. Each subject is trained on the apparatus and then
asked to make as many insertions as possible in 30 minutes of continuous effort.
(a) Outline a completely randomized design to compare dexterity at 70◦ and 90◦.
Twenty subjects are available.
(b) Because individuals differ greatly in dexterity, the wide variation in individual
scores may hide the systematic effect of temperature unless there are many subjects
in each group. Describe in detail the design of a matched pairs experiment in which
each subject serves as his or her own control.
3.86 The National Institutes of Mental Health (NIMH) wants to know whether in-
tense education about the risks of AIDS will help change the behavior of people
who now report sexual activities that put them at risk of infection. NIMH investi-
gators screened 38,893 people to identify 3706 suitable subjects. The subjects were
assigned to a control group (1855 people) or an intervention group (1851 people).
The control group attended a one-hour AIDS education session; the intervention
group attended seven single-sex discussion sessions, each lasting 90 to 120 minutes.
After 12 months, 64% of the intervention group and 52% of the control group said
they used condoms. (NIMH Multisite HIV Prevention Trial Group, “The NIMH
multisite HIV prevention trial: reducing HIV sexual risk behavior,” Science, 280
(1998), pp. 1889–1894.)
(a) Because none of the subjects used condoms when the study started, we might
just offer the intervention sessions and find that 64% used condoms 12 months after
the sessions. Explain why this greatly overstates the effectiveness of the interven-
tion.
(b) Outline the design of this experiment.
(c) You must randomly assign 3706 subjects. How would you label them? Use line
119 of Table B to choose the first 5 subjects for the intervention group.
3.87 It is possible to use a computer to make telephone calls over the Internet.
How will the cost affect the behavior of users of this service? You will offer the
service to all 200 rooms in a college dormitory. Some rooms will pay a flat rate.
Others will pay higher rates at peak periods and very low rates off-peak. You are
interested in the amount and time of use and in the effect on the congestion of the
network. Outline the design of an experiment to study the effect of rate structure.
Use Table B, starting at line 125, to assign the first 5 rooms to the flat-rate group.
3.88 What are the most important goals of schoolchildren? Do girls and boys have
different goals? Are goals different in urban, suburban, and rural areas? To find
out, researchers wanted to ask children in the fourth, fifth, and sixth grades this
question:
What would you most like to do at school?
A. Make good grades.

B. Be good at sports.
C. Be popular.
Because most children live in heavily populated urban and suburban areas, an SRS
might contain few rural children. Moreover, it is too expensive to choose children
at random—we must start by choosing schools rather than children. Describe a
suitable sample design for this study, using the ideas of stratified and multistage
samples. Explain the reasoning behind your choice.
3.89 Advice columnist Ann Landers once asked her female readers whether they
would be content with affectionate treatment by men, with no sex ever. Over 90,000
women wrote in, with 72% answering “Yes.” Many of the letters described unfeeling
treatment by men. Explain why this sample is certainly biased. What is the likely
direction of the bias? That is, is that 72% probably higher or lower than the truth
about the population of all adult women?
3.90 A national opinion poll recently estimated that 44% (p̂ = 0.44) of all American
adults agree that parents of school-age children should be given vouchers good for
education at any public or private school of their choice. The polling organization
used a probability sampling method for which the sample proportion has a Normal
distribution with standard deviation about 0.015. The poll therefore announced a
“margin of error” of 0.03 (two standard deviations) for its result. If a sample were
drawn by the same method from the state of New Jersey (population 8 million)
instead of from the entire United States (population 270 million), would this margin
of error be larger or smaller? Explain your answer.
3.91 As the millennium approached, Time magazine launched a Person of the Cen-
tury Poll on the Internet. Web users could choose the person “who most influenced
the course of history” during the twentieth century. The top choice was Elvis Pres-
ley, with 625,045 votes. Further down the list, Linus Torvalds (originator of the
Linux operating system for PCs) beat out Nelson Mandela and Princess Diana.
Time ignored an online campaign to get out the vote for Jesus, leaving him off the
list. Use this example in a brief discussion of the weaknesses of voluntary response
samples.
3.92 Cash bonues for the unemployed. Will cash bonuses speed the return
to work of unemployed people? The Illinois Department of Employment Security
designed an experiment to find out. The subjects were 10,065 people aged 20 to 54
who were filing claims for unemployment insurance. Some were offered $500 if they
found a job within 11 weeks and held it for at least 4 months. Others could tell
potential employers that the state would pay the employer $500 for hiring them.
Review Exercises 89
A control group got neither kind of bonus. (Based on Stephen A. Woodbury and
Robert G. Spiegelman, “Bonuses to workers and employers to reduce unemployment:
randomized trials in Illinois,” American Economic Review, 77 (1987), pp. 513–530.)
(a) Suggest a few response variables of interest to the state and outline the design
of the experiment.
(b) How will you label the subjects for random assignment? Use Table B at line 167
to choose the first 3 subjects for the first treatment.
3.93 Prostate treatment study using Canada’s national health records. A

large observational study used records from Canada’s national health care system to
compare the effectiveness of two ways to treat prostate disease. The two treatments
are traditional surgery and a new method that does not require surgery. The records
described many patients whose doctors had chosen one or the other method. The
study found that patients treated by the new method were significantly more likely
to die within 8 years. (Based on Christopher Anderson, “Measuring what works in
health care,” Science, 263 (1994), pp. 1080–1082.)
(a) Further study of the data showed that this conclusion was wrong. The extra
deaths among patients who received the new treatment could be explained by lurk-
ing variables. What lurking variables might be confounded with a doctor’s choice
of surgical or nonsurgical treatment?
(b) You have 300 prostate patients who are willing to serve as subjects in an ex-
periment to compare the two methods. Use a diagram to outline the design of a
randomized comparative experiment.
3.94 What type of study? What is the best way to answer each of the questions
below: an experiment, a sample survey, or an observational study that is not a
sample survey? Explain your choices.
(a) Are people generally satisfied with how things are going in the country right
now?
(b) Do college students learn basic accounting better in a classroom or using an
online course?
(c) How long do your teachers wait on the average after they ask their class a
question?
3.95 Discolored french fries. Few people want to eat discolored french fries.
Potatoes are kept refrigerated before being cut for french fries to prevent spoiling
and to preserve flavor. But immediate processing of cold potatoes causes discoloring
due to complex chemical reactions. The potatoes must therefore be brought to room
temperature before processing. Design an experiment in which tasters will rate
the color and flavor of french fries prepared from several groups of potatoes. The
potatoes will be fresh picked, stored for a month at room temperature, or stored
for a month refrigerated. They will then be sliced and cooked either immediately
or after an hour at room temperature.
(a) What are the factors and their levels, the treatments, and the response variables?
(b) Describe and outline the design of this experiment.
(c) It is efficient to have each taster rate fries from all treatments. How will you use
randomization in presenting fries to the tasters?
CHAPTER 4
Section 4.1
4.1 Toss a thumbtack on a hard surface 100 times. How many times did it land
with the point up? What is the approximate probability of landing point up?
4.2 In the game of Heads or Tails, Betty and Bob toss a coin four times. Betty wins
a dollar from Bob for each head and pays Bob a dollar for each tail—that is, she
wins or loses the difference between the number of heads and the number of tails.
For example, if there are one head and three tails, Betty loses $2. You can check
that Betty’s possible outcomes are
{−4, −2, 0, 2, 4}
Assign probabilities to these outcomes by playing the game 20 times and using the
proportions of the outcomes as estimates of the probabilities. If possible, combine
your trials with those of other students to obtain long-run proportions that are
closer to the probabilities.
4.3 You read in a book on poker that the probability of being dealt three of a kind
in a five-card poker hand is 1/50. Explain in simple language what this means.
4.4 A recent opinion poll showed that about 73% of married women agree that their
husbands do at least their fair share of household chores. Suppose that this is exactly
true. Choosing a married woman at random then has probability 0.73 of getting one
who agrees that her husband does his share. You can use the Probability applet or
software to simulate choosing many women independently. (In most software, the
key phrase to look for is “Bernoulli trials.” This is the technical term for independent
trials with “Yes/No” outcomes. Our outcomes here are “Agree” and “Disagree.”)
(a) Simulate drawing 20 women, then 80 women, then 320 women. What proportion
agree in each case? We expect (but because of chance variation we can’t be sure)
that the proportion will be closer to 0.73 in longer runs of trials.
(b) Simulate drawing 20 women 10 times and record the percents in each trial who
agree. Then simulate drawing 320 women 10 times and again record the 10 percents.
Which set of 10 results is less variable? We expect the results of larger samples to
be more predictable (less variable) than the results of smaller samples. That is
“long-run regularity” showing itself.
4.5 Continue the exploration begun in the previous exercise. Software allows you
to simulate many independent “Yes/No” trials more quickly if all you want to save
is the count of “Yes” outcomes. The keyword “Binomial” simulates n independent
Bernoulli trials, each with probability p of a “Yes,” and records just the count of
“Yes” outcomes.
(a) Simulate 100 draws of 20 women from this population. Record the number who
say “Agree” on each draw. What is the approximate probability that out of 20
women drawn at random at least 14 agree?
(b) Convert the counts who agree into percents of the 20 women in each trial who
90
Section 4.1 91
agree. Make a histogram of these 100 percents. Describe the shape, center, and
spread of this distribution.
(c) Now simulate drawing 320 women. Do this 100 times and record the percent
who agree on each of the 100 draws. Make a histogram of the percents and describe
the shape, center, and spread of the distribution.
(d) In what ways are the distributions in parts (b) and (c) alike? In what ways do
they differ? (Because regularity emerges in the long run, we expect the results of
drawing 320 women to be less variable than the results of drawing 20 women.)
4.6 Hold a penny upright on its edge under your forefinger on a hard surface, then
snap it with your other forefinger so that it spins for some time before falling. Based
on 50 spins, what is the probability of heads?
4.7 You may feel that it is obvious that the probability of a head in tossing a coin
is about 1/2 because the coin has two faces. Such opinions are not always correct.
The previous exercise asked you to spin a penny rather than toss it—that changes
the probability of a head. Now try another variation. Stand a nickel on edge on a
hard, flat surface. Pound the surface with your hand so that the nickel falls over.
What is the probability that it falls with heads upward? Make at least 50 trials to
estimate the probability of a head.
4.8 Many Internet sites give the probabilities of being dealt various five-card poker
hands. For example, the probability of being dealt two pairs is approximately 1/21.
Explain is simple language what “probability 1/21” means. Also explain why it
does not mean that in 21 deals you will get exactly one two-pair hand.
4.9 About 30% of adult Internet users are between 18 and 29 years of age. Suppose
the probability that a randomly chosen Internet user is in this age group is exactly
0.3. Use the Probability applet to make a study of short-term variability and long-
term regularity as follows.
(a) Set the probability of heads to 0.3. Each head stands for an Internet user who
is between 18 and 29 and each tail is a user who is not. Set the number of tosses to
20. Click “Toss” to make 20 tosses. What was the proportion of heads? Do this 25
times, keep a record of the 25 proportions of heads, and make a stemplot of these
numbers. Lesson: In the short run (20 repetitions) proportions are quite variable
and are often not close to the probability.
(b) With the probability of heads still set to 0.3, make 200 tosses. (Set tosses to
40 and click “Toss” five times without a reset.) What was the proportion of heads?
Do this 25 times and make a stemplot of the 25 proportions of heads. Lesson: More
repetitions make proportions less variable and generally closer to the probability.
Of course, 200 repetitions is not “the long run,” but you see the idea.
4.10 If you use statistical software, you can continue the exploration begun in the
previous exercise. Software allows you to simulate many independent “Yes/No”
trials more quickly if all you want to save is the count of “Yes” outcomes. The
keyword “Binomial” simulates n independent Yes/No trials, each with probability
p of a “Yes,” and records just the count of “Yes” outcomes.
(a) Simulate 100 draws of 20 Internet users from the population in Exercise 4.8.
(That is, ask the software to generate 100 binomial observations, each with n = 20
trials and probability p = 0.3 of a “Yes.”) Record the count in the 18 to 29 age
group on each draw. Convert the counts into percents of the 20 Internet users in
each trial who are 18 to 29. Make a histogram of these 100 percents. Describe the
shape, center, and spread of this distribution.
(b) Now simulate drawing 320 Internet users. (That is, set n = 320 and p = 0.3.)
Do this 100 times and record the percent in the 18 to 29 age group for each of the
100 draws. Make a histogram of the percents and describe the shape, center, and
spread of the distribution.
(c) In what ways are the distributions in parts (a) and (b) alike? In what ways do
they differ? (Because regularity emerges in the long run, we expect the results of
drawing 320 subjects to be less variable than the results of drawing 20 subjects.)
4.11 The color of candy. It is reasonable to think that packages of M&M’s Milk
Chocolate Candies are filled at the factory with candies chosen at random from
the very large number produced. So a package of M&M’s contains a number of
repetitions of a random phenomenon: choosing a candy at random and noting its
color. What is the probability that an M&M’s Milk Chocolate Candy is green? To
find out, buy one or more packs. How many candies did you examine? How many
were green? What is your estimate of the probability that a randomly chosen candy
is green?
Section 4.2
4.12 The percent return on U.S. common stocks in the next year is random. The
following table reports historical data for the years 1971 to 2000. Give a reasonable
sample space for the possible returns next year. Explain how you chose this S.
Year % return Year % return

1971 14.6 1986 18.6
1972 18.9 1987 5.1
1973 −14.8 1988 16.8
1974 −26.4 1989 31.5
1975 37.2 1990 −3.1
1976 23.6 1991 30.4
1977 −7.4 1992 7.6
1978 6.4 1993 10.1
1979 18.2 1994 1.3
1980 32.3 1995 37.6
1981 −5.0 1996 23.0
1982 21.5 1997 33.4
1983 22.4 1998 28.6
1984 6.1 1999 21.0
1985 31.6 2000 −9.1
4.13 In each of the following situations, describe a sample space S for the random
phenomenon. In some cases, you have some freedom in your choice of S.
Section 4.2 93
(a) A seed is planted in the ground. It either germinates or fails to grow.

(b) A patient with a usually fatal form of cancer is given a new treatment. The
response variable is the length of time that the patient lives after treatment.
(c) A student enrolls in a statistics course and at the end of the semester receives a
letter grade.
(d) A basketball player shoots two free throws.
(e) A year after knee surgery, a patient is asked to rate the amount of pain in the
knee. A seven-point scale is used, with 1 corresponding to no pain and 7 corre-
sponding to extreme discomfort.
4.14 In each of the following situations, describe a sample space S for the random
phenomenon. In some cases you have some freedom in specifying S, especially in
setting the largest and smallest value in S.
(a) Choose a student in your class at random. Ask how much time that student
spent studying during the past 24 hours.
(b) The Physicians’ Health Study asked 11,000 physicians to take an aspirin every
other day and observed how many of them had a heart attack in a five-year period.
(c) In a test of a new package design, you drop a carton of a dozen eggs from a
height of 1 foot and count the number of broken eggs.
(d) Choose a student in your class at random. Ask how much cash that student is
carrying.
(e) A nutrition researcher feeds a new diet to a young male white rat. The response
variable is the weight (in grams) that the rat gains in 8 weeks.
4.15 Probability is a measure of how likely an event is to occur. Match one of the
probabilities that follow with each statement about an event. (The probability is
usually a much more exact measure of likelihood than is the verbal statement.)
0, 0.01, 0.3, 0.6, 0.99, 1
(a) This event is impossible. It can never occur.

(b) This event is certain. It will occur on every trial of the random phenomenon.
(c) This event is very unlikely, but it will occur once in a while in a long sequence
of trials.
(d) This event will occur more often than not.
4.16 In each of the following situations, state whether or not the given assignment
of probabilities to individual outcomes is legitimate, that is, satisfies the rules of
probability. If not, give specific reasons for your answer.
(a) When a coin is spun, P (H) = 0.55 and P (T) = 0.45.
(b) When two coins are tossed, P (HH) = 0.4, P (HT) = 0.4, P (TH) = 0.4, and
P (TT) = 0.4.
(c) When a die is rolled, the number of spots on the up-face has P (1) = 1/2,
P (4) = 1/6, P (5) = 1/6, and P (6) = 1/6.
4.17 Here are several assignments of probabilities to the six faces of a die:
Outcome 1 2 3 4 5 6
Probabilities 1 1/3 0 1/6 0 1/6 1/3
Probabilities 2 1/6 1/6 1/6 1/6 1/6 1/6
Probabilities 3 1/7 1/7 1/7 1/7 1/7 1/7
Probabilities 4 1/3 1/3 −1/6 −1/6 1/3 1/3
We can learn which assignment is actually accurate for a particular die only by
rolling the die many times. However, some of the assignments are not legitimate
assignments of probability. That is, they do not obey the rules. Which are legitimate
and which are not? In the case of the illegitimate models, explain what is wrong.
4.18 Chose a student in grades 9 to 12 at random and ask if he or she is studying

a language other than English. Here is the distribution of results:
Language Spanish French German All others None
Probability 0.26 0.09 0.03 0.03 0.59
(a) Explain why this is a legitimate probability model.
(b) What is the probability that a randomly chosen student is studying a language
other than English?
(c) What is the probability that a randomly chosen student is studying French,
German, or Spanish?
4.19 If you draw an M&M candy at random from a bag of the candies, the candy
you draw will have one of six colors. The probability of drawing each color depends
on the proportion of each color among all candies made.
(a) The following table gives the probability of each color for a randomly chosen
plain M&M:
Color Brown Red Yellow Green Orange Blue
Probability 0.3 0.2 0.2 0.1 0.1 ?
What must be the probability of drawing a blue candy?
(b) The probabilities for peanut M&M’s are a bit different. Here they are:
Color Brown Red Yellow Green Orange Blue
Probability 0.2 0.1 0.2 0.1 0.1 ?
What is the probability that a peanut M&M chosen at random is blue?
(c) What is the probability that a plain M&M is any of red, yellow, or orange?
What is the probability that a peanut M&M has one of these colors?
4.20 Wabash Red, when asked to predict the Big Ten Conference men’s basketball
champion, follows the modern practice of giving probabilistic predictions. He says,
“Michigan State has probability 0.3 of winning. Michigan, Minnesota, Northwestern,
and Penn State have no chance. That leaves 6 teams. Iowa, Illinois, and Purdue
all have the same probability of winning. Indiana, Ohio State, and Wisconsin also
have the same probability, but that probability is one-half that of the first 3.” What
probability does Red give to each of the 11 teams?
4.21 Choose an acre of land in Canada at random. The probability is 0.35 that it
is forest and 0.03 that it is pasture.
Section 4.2 95
(a) What is the probability that the acre chosen is not forested?
(b) What is the probability that it is either forest or pasture?
(c) What is the probability that a randomly chosen acre in Canada is something
other than forest or pasture?
4.22 Choose a new car or light truck at random and note its color. Here are the
probabilities of the most popular colors for vehicles made in North America in 2003.
(From the Dupont Automotive North America Color Popularity Survey, reported
at www.dupont.com/automotive/.)
Color Silver White Black Gray Blue Medium red
Probability 0.201 0.184 0.116 0.088 0.085 0.069
(a) What is the probability that the vehicle you choose has any color other than the
six listed?
(b) What is the probability that a randomly chosen vehicle is either silver or white?
(c) Choose two vehicles at random. What is the probability that both are silver or
white?
4.23 A company that offers courses to prepare would-be MBA students for the
GMAT examination has the following information about its customers: 20% are
currently undergraduate students in business; 15% are undergraduate students in
other fields of study; 60% are college graduates who are currently employed; and
5% are college graduates who are not employed.
(a) Is this a legitimate assignment of probabilities to customer backgrounds? Why?
(b) What percent of customers are currently undergraduates?
4.24 Choose an American worker at random and classify his or her occupation into
one of the following classes. These classes are used in government employment data.
A Managerial and professional
B Technical, sales, administrative support
C Service occupations
D Precision production, craft, and repair
E Operators, fabricators, and laborers
F Farming, forestry, and fishing
The following table gives the probabilities that a randomly chosen worker falls into
each of 12 sex-by-occupation classes.
Class A B C D E F
Male 0.14 0.11 0.06 0.11 0.12 0.03
Female 0.09 0.20 0.08 0.01 0.04 0.01
(a) Verify that this is a legitimate assignment of probabilities to these outcomes.
(b) What is the probability that the worker is female?
(c) What is the probability that the worker is not engaged in farming, forestry, or
fishing?
(d) Classes D and E include most mechanical and factory jobs. What is the proba-
bility that the worker holds a job in one of these classes? (e) What is the probability
that the worker does not hold a job in Classes D or E?
4.25 The Pick 4 games in many state lotteries announce a four-digit winning number
each day. The winning number is essentially a four-digit group from a table of
random digits. You win if your choice matches the winning digits. Suppose your
chosen number is 5974.
(a) What is the probability that your number matches the winning number exactly?
(b) What is the probability that your number matches the digits in the winning
number in any order?
4.26 Abby, Deborah, Mei-Ling, Sam, and Roberto work in a firm’s public relations
office. Their employer must choose two of them to attend a conference in Paris. To
avoid unfairness, the choice will be made by drawing two names from a hat. (This
is an SRS of size 2.)
(a) Write down all possible choices of two of the five names. This is the sample
space.
(b) The random drawing makes all choices equally likely. What is the probability
of each choice?
(c) What is the probability that Mei-Ling is chosen?
(d) What is the probability that neither of the two men (Sam and Roberto) is
chosen?
4.27 A general can plan a campaign to fight one major battle or three small battles.
He believes that he has probability 0.6 of winning the large battle and probability
0.8 of winning each of the small battles. Victories or defeats in the small battles are
independent. The general must win either the large battle or all three small battles
to win the campaign. Which strategy should he choose?
4.28 An automobile manufacturer buys computer chips from a supplier. The sup-
plier sends a shipment containing 5% defective chips. Each chip chosen from this
shipment has probability 0.05 of being defective, and each automobile uses 12 chips
selected independently. What is the probability that all 12 chips in a car will work
properly?
4.29 A string of holiday lights contains 20 lights. The lights are wired in series, so
that if any light fails the whole string will go dark. Each light has probability 0.02
of failing during a 3-year period. The lights fail independently of each other. What
is the probability that the string of lights will remain bright for 3 years?
4.30 The most popular game of chance in Roman times was tossing four astragali.
An astragalus is a small six-sided bone from the heel of an animal that comes to
rest on one of four sides when tossed. (The other two sides are rounded.) The
table gives the probabilities of the outcomes for a single astragalus based on modern
experiments. The names “broad convex,” etc. , describe the four sides of the heel
bone. The best throw was the “Venus,” with all four uppermost sides different.
What is the probability of rolling a Venus? (From Florence N. David, Games, Gods
and Gambling, Charles Griffin, London, 1962, p. 7.)
Side broad convex broad concave narrow flat narrow hollow

Probability 0.4 0.4 0.1 0.1
Section 4.2 97
4.31 Government data show that 27% of employed people have at least 4 years of
college and that 14% of employed people work as laborers or operators of machines
or vehicles. Can you conclude that because (0.27)(0.14) = 0.038 about 3.8% of
employed people are college-educated laborers or operators? Explain your answer.
4.32 A randomly chosen subject arrives for a study of exercise and fitness. Describe
a sample space for each of the following. (In some cases, you may have some freedom
in your choice of S.)
(a) The subject is either female or male.
(b) After 10 minutes on an exercise bicycle, you ask the subject to rate his or her
effort on the Rate of Perceived Exertion (RPE) scale. RPE ranges in whole-number
steps from 6 (no exertion at all) to 20 (maximal exertion).
(c) You measure VO2, the maximum volume of oxygen consumed per minute during
exercise. VO2 is generally between 2.5 liters per minute and 6 liters per minute.
(d) You measure the maximum heart rate (beats per minute).
4.33 Choose a student at random from a large statistics class. Give a reasonable
sample space S for answers to each of these questions. (In some cases you may have
some freedom in specifying S.)
(a) Are you right-handed or left-handed?
(b) What is your height in centimeters? (One inch is 2.54 centimeters.)
(c) How much money in coins (not bills) are you carrying?
(d) How many minutes did you study last night?
4.34 Role-playing games like Dungeons & Dragons use many different types of dice.
Suppose that a four-sided die has faces marked 1, 2, 3, 4. The intelligence of a
character is determined by rolling this die twice and adding 1 to the sum of the
spots.
(a) What is the sample space for rolling the die twice (spots on first and second
rolls)?
(b) What is the sample space for the character’s intelligence?
4.35 In each of the following situations, state whether or not the given assignment
of probabilities to individual outcomes is legitimate, that is, satisfies the rules of
probability. If not, give specific reasons for your answer.
(a) Roll a die and record the count of spots on the up-face: P (1) = 0, P (2) = 1/6,
P (3) = 1/3, P (4) = 1/3, P (5) = 1/6, P (6) = 0.
(b) Choose a college student at random and record sex and enrollment status:
P (female full-time) = 0.56, P (female part-time) = 0.24, P (male full-time) = 0.44,
P (male part-time) = 0.17.
(c) Deal a card from a shuffled deck: P (clubs) = 12/52, P (diamonds) = 12/52,
P (hearts) = 12/52, P (spades) = 16/52.
4.36 Role-playing games like Dungeons & Dragons use many different types of
dice. Suppose that a four-sided die has faces marked 1, 2, 3, 4. The intelligence
of a character is determined by rolling this die twice and adding 1 to the sum of
the spots. The faces are equally likely and the two rolls are independent. In the
previous exercise you gave the sample space S for the character’s intelligence. Now
give the assignment of
4.37 Dugout Lou thinks that the probabilities for the American League baseball
champion are as follows. The Yankees have probability 0.6 of winning. The Red
Sox and Angels have equal probabilities of winning. The Athletics and White Sox
also have equal probabilities, but their probabilities are one-third that of the Red Sox
and Angels. No other team has a chance. What is Lou’s assignment of probabilities
to teams?
4.38 When do you study? A student is asked on which day of the week he or
she spends the most time studying. What is the sample space?
4.39 Sample space for heights. You record the height in inches of a randomly
selected student. What is the sample space?
4.40 Phone-related accidents on Monday or Friday. Some states are consid-

ering laws that will ban the use of cell phones while driving because they believe that
the ban will reduce phone-related car accidents. One study classified these types of
accidents by the day of the week when they occurred. (From D. A. Redelmeier and
R. J. Tibshirani, “Association between cellular-telephone calls and motor vehicle
collisions,” New England Journal of Medicine, 336 (1997) pp. 453–458.) For this
example, we use the values from this study as our probability model. Here are the
probabilities:
Day Sun. Mon. Tues. Wed. Thur. Fri. Sat.
Probability 0.03 0.19 0.18 0.23 0.19 0.16 0.02
Find the probability that a phone-related accident occurred on a Monday or a Friday.
4.41 Not on Wednesday. Refer to the previous exercise. Find the probability
that a phone-related accident occurred on a day other than a Wednesday.
4.42 Spam topics. A majority of email messages are now “spam.” Choose a spam
email message at random. Here is the distribution of topics (Robyn Greenspan, “The
deadly duo: spam and viruses, October 2003,” found online at
cyberatlas.internet.com):
Topic Adult Financial Health Leisure Products Scams
Probability 0.145 0.162 0.073 0.078 0.210 0.142
(a) What is the probability that a spam email does not concern one of these topics?
(b) Corinne is particularly annoyed by spam offering “adult” content (that is,
pornography) and scams. What is the probability that a randomly chosen spam
email falls into one or the other of these categories?
4.43 Race in the census. The 2000 census allowed each person to choose from a
long list of races. That is, in the eyes of the Census Bureau, you belong to whatever
race you say you belong to. “Hispanic/Latino” is a separate category; Hispanics
may be of any race. If we choose a resident of the United States at random, the
2000 census gives these probabilities:
Section 4.3 99
Hispanic Not Hispanic

Asian 0.000 0.036
Black 0.003 0.121
White 0.060 0.691
Other 0.062 0.027
Let A be the event that a randomly chosen American is Hispanic, and let B be the
event that the person chosen is white.
(a) Verify that the table gives a legitimate assignment of probabilities.
(b) What is P (A)?
(c) Describe B c in words and find P (B c ) by the complement rule.
(d) Express “the person chosen is a non-Hispanic white” in terms of events A and
B. What is the probability of this event?
4.44 Are the events independent? The previous exercise assigns probabilities
for the ethnic background of a randomly chosen resident of the United States. Let
A be the event that the person chosen is Hispanic, and let B be the event that he
or she is white. Are events A and B independent? How do you know?
Section 4.3
4.45 If a carefully made die is rolled once, it is reasonable to assign probability 1/6
to each of the six faces. What is the probability of rolling a number less than 3?
4.46 A couple plans to have three children. There are 8 possible arrangements of
girls and boys. For example, GGB means the first two children are girls and the
third child is a boy. All 8 arrangements are (approximately) equally likely.
(a) Write down all 8 arrangements of the sexes of three children. What is the
probability of any one of these arrangements?
(b) Let X be the number of girls the couple has. What is the probability that
X = 2?
(c) Starting from your work in (a), find the distribution of X. That is, what values
can X take, and what are the probabilities for each value?
4.47 Choose an American household at random and let the random variable X
be the number of cars (including SUVs and light trucks) they own. Here is the
probability model if we ignore the few households that own more than 5 cars:
Number of cars X 0 1 2 3 4 5
Probability 0.09 0.36 0.35 0.13 0.05 0.02
(a) Verify that this is a legitimate discrete distribution. Display the distribution in
a probability histogram.
(b) Say in words what the event {X ≥ 1} is. Find P (X ≥ 1).
(c) A housing company builds houses with two-car garages. What percent of house-
holds have more cars than the garage can hold?
4.48 A study of social mobility in England looked at the social class reached by the
sons of lower-class fathers. Social classes are numbered from 1 (low) to 5 (high).
Take the random variable X to be the class of a randomly chosen son of a father in
Class 1. The study found that the distribution of X is
Son’s class 1 2 3 4 5
Probability 0.48 0.38 0.08 0.05 0.01
(a) What percent of the sons of lower-class fathers reach the highest class, Class 5?
(b) Check that this distribution satisfies the two requirements for a discrete proba-
bility distribution.
(c) What is P (X ≤ 3)? (Be careful: the event “X ≤ 3” includes the value 3.)
(d) What is P (X < 3)?
(e) Write the event “a son of a lower-class father reaches one of the two highest
classes” in terms of values of X. What is the probability of this event?
4.49 A study of education followed a large group of fifth-grade children to see how
many years of school they eventually completed. Let X be the highest year of school
that a randomly chosen fifth-grader completes. (Students who go on to college are
included in the outcome X = 12.) The study found this probability distribution for
X:
Years 4 5 6 7 8 9 10 11 12
Probability 0.010 0.007 0.007 0.013 0.032 0.068 0.070 0.041 0.752
(a) What percent of fifth graders eventually finished twelfth grade?

(b) Check that this is a legitimate discrete probability distribution.
(c) Find P (X ≥ 6). (Be careful: the event “X ≥ 6” includes the value 6.)
(d) Find P (X > 6).
(e) What values of X make up the event “the student completed at least one year
of high school”? (High school begins with the ninth grade.) What is the probability
of this event?
4.50 An SRS of 400 American adults is asked, “What do you think is the most
serious problem facing our schools?” Suppose that in fact 30% of all adults would
answer “drugs” if asked this question. The proportion p̂ of the sample who answer
“drugs” will vary in repeated sampling. In fact, we can assign probabilities to values
of p̂ using the normal density curve with mean 0.3 and standard deviation 0.023.
Use this density curve to find the probabilities of the following events:
(a) At least half of the sample believes that drugs are the schools’ most serious
problem.
(b) Less than 25% of the sample believes that drugs are the most serious problem.
(c) The sample proportion is between 0.25 and 0.35.
4.51 An opinion poll asks an SRS of 1500 adults, “Do you happen to jog?” Suppose
that the population proportion who jog (a parameter) is p = 0.15. To estimate
p, we use the proportion p̂ in the sample who answer “Yes.” The statistic p̂ is a
random variable that is approximately Normally distributed with mean µ = 0.15
and standard deviation σ = 0.0092. Find the following probabilities:
(a) P (p̂ ≥ 0.16)
(b) P (0.14 ≤ p̂ ≤ 0.16)
Section 4.3 101
4.52 Choose an American household at random and let the random variable Y be
the number of persons living in the household. Here is the distribution of Y :
Number of persons 1 2 3 4 5 6 7
Household probability 0.27 0.33 0.16 0.14 0.06 0.03 0.01
Family probability 0 0.44 0.22 0.20 0.09 0.03 0.02
(a) Express “more than one person lives in this household” in terms of Y . What is
the probability of this event?
(b) What is P (2 < Y ≤ 4)?
(c) What is P (Y 6= 2)?
4.53 Let the random variable X be the number of rooms in a randomly chosen
owner-occupied housing unit in San Jose, California. Here is the distribution of X.
Rooms 1 2 3 4 5 6 7 8 9 10
Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035
Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003
(a) Express “the unit has 5 or more rooms” in terms of X. What is the probability
of this event?
(b) Express the event {X > 5} in words. What is its probability?
(c) What important fact about discrete random variables does comparing your an-
swers to (a) and (b) illustrate?
4.54 Let X be a random number between 0 and 1 produced by a uniform random

number generator. Find the following probabilities:
(a) P (X < 0.5)
(b) P (X ≤ 0.5)
(c) What important fact about continuous random variables does comparing your
answers to (a) and (b) illustrate?
4.55 Let the random variable X be a uniform random number between 0 and 1.
Find the following probabilities:
(a) P (X ≥ 0.27)
(b) P (X = 0.27)
(c) P (0.27 < X < 1.27)
(d) P (0.1 ≤ X ≤ 0.2 or 0.8 ≤ X ≤ 0.9)
(e) The probability that X is not in the interval 0.3 to 0.8.
4.56 Many random number generators allow users to specify the range of the random
numbers to be produced. Suppose that you specify that the range is to be all
numbers between 0 and 2. Call the random number generated Y . Then the density
curve of the random variable Y has constant height between 0 and 2, and height 0
elsewhere.
(a) What is the height of the density curve between 0 and 2? Draw a graph of the
density curve.
(b) Use your graph from (a) and the fact that probability is area under the curve
to find P (Y ≤ 1).
(c) Find P (0.5 < Y < 1.3).

(d) Find P (Y ≥ 0.8).
4.57 Owner-occupied and rented housing units. How do rented housing units
differ from units occupied by their owners? Here are the distributions of the number
of rooms for owner-occupied units and renter-occupied units in San Jose, California:
(From the Census Bureau’s 1998 American Housing Survey.)
Rooms 1 2 3 4 5 6 7 8 9 10
Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035
Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003
Make probability histograms of these two distributions, using the same scales. What
are the most important differences between the distributions for owner-occupied and
rented housing units?
4.58 Find the probabilities. Let the random variable X be the number of rooms
in a randomly chosen owner-occupied housing unit in San Jose, California. The
previous exercise gives the distribution of X.
(a) Express “the unit has 6 or more rooms” in terms of X. What is the probability
of this event?
(b) Express the event {X > 6} in words. What is its probability?
(c) What important fact about discrete random variables does comparing your an-
swers to (a) and (b) illustrate?
4.59 Households and families in government data. In government data, a

household consists of all occupants of a dwelling unit, while a family consists of
two or more persons who live together and are related by blood or marriage. So
all families form households, but some households are not families. Here are the
distributions of household size and of family size in the United States:
Number of persons 1 2 3 4 5 6 7
Household probability 0.27 0.33 0.16 0.14 0.06 0.03 0.01
Family probability 0 0.44 0.22 0.20 0.09 0.03 0.02
Make probability histograms for these two discrete distributions, using the same
scales. What are the most important differences between the sizes of households
and families?
4.60 Select the members of a student advisory board. Weary of the low
turnout in student elections, a college administration decides to choose an SRS of
three students to form an advisory board that represents student opinion. Suppose
that 40% of all students oppose the use of student fees to fund student interest
groups, and that the opinions of the three students on the board are independent.
Then the probability is 0.4 that each opposes the funding of interest groups.
(a) Call the three students A, B, and C. What is the probability that A and B
support funding and C opposes it?
(b) List all possible combinations of opinions that can be held by students A, B,
and C. (Hint: There are eight possibilities.) Then give the probability of each of
Section 4.4 103
these outcomes. Note that they are not equally likely.

(c) Let the random variable X be the number of student representatives who oppose
the funding of interest groups. Give the probability distribution of X.
(d) Express the event “a majority of the advisory board opposes funding” in terms
of X and find its probability.
Section 4.4
4.61 Exercise 4.47 gives the distribution of the number X of cars (including SUVs
and light trucks) owned by American households. What is the average (mean)
number of vehicles owned?
4.62 Keno is a favorite game in casinos, and similar games are popular with the
states that operate lotteries. Balls numbered 1 to 80 are tumbled in a machine
as the bets are placed, then 20 of the balls are chosen at random. Players select
numbers by marking a card. The simplest of the many wagers available is “Mark
1 Number.” Your payoff is $3 on a $1 bet if the number you select is one of those
chosen. Because 20 of 80 numbers are chosen, your probability of winning is 20/80,
or 0.25.
(a) What is the probability distribution (the outcomes and their probabilities) of
the payoff X on a single play?
(b) What is the mean payoff µX ?
(c) In the long run, how much does the casino keep from each dollar bet?
4.63 (a) A gambler knows that red and black are equally likely to occur on each
spin of a roulette wheel. He observes five consecutive reds and bets heavily on red
at the next spin. Asked why, he says that “red is hot” and that the run of reds is
likely to continue. Explain to the gambler what is wrong with this reasoning.
(b) After hearing you explain why red and black remain equally probable after five
reds on the roulette wheel, the gambler moves to a poker game. He is dealt five
straight red cards. He remembers what you said and assumes that the next card
dealt in the same hand is equally likely to be red or black. Is the gambler right or
wrong? Why?
4.64 In an experiment on the behavior of young children, each subject is placed in

an area with five toys. The response of interest is the number of toys that the child
plays with. Past experiments with many subjects have shown that the probability
distribution of the number X of toys played with is as follows:
Number of toys xi 0 1 2 3 4 5
Probability pi 0.03 0.16 0.30 0.23 0.17 0.11
Calculate the mean µX and the standard deviation σX .
4.65 You have two balanced, six-sided dice. The first has 1, 3, 4, 5, 6, and 8 spots
on its six faces. The second die has 1, 2, 2, 3, 3, and 4 spots on its faces.
(a) What is the mean number of spots on the up-face when you roll each of these
dice?
(b) Write the probability model for the outcomes when you roll both dice indepen-
dently. From this, find the probability distribution of the sum of the spots on the
up-faces of the two dice.
(c) Find the mean number of spots on the two up-faces in two ways: from the dis-
tribution you found in (b) and by applying the addition rule to your results in (a).
You should of course get the same answer.
4.66 Laboratory data show that the time required to complete two chemical re-
actions in a production process varies. The first reaction has a mean time of 40
minutes and a standard deviation of 2 minutes; the second has a mean time of 25
minutes and a standard deviation of 1 minute. The two reactions are run in se-
quence during production. There is a fixed period of 5 minutes between them as
the product of the first reaction is pumped into the vessel where the second reaction
will take place. What is the mean time required for the entire process?
4.67 The times for the two reactions in the chemical production process described
in the previous exercise are independent. Find the standard deviation of the time
required to complete the process.
4.68 The academic motivation and study habits of female students as a group are
better than those of males. The Survey of Study Habits and Attitudes (SSHA) is
a psychological test that measures these factors. The distribution of SSHA scores
among the women at a college has mean 120 and standard deviation 28, and the
distribution of scores among men students has mean 105 and standard deviation 35.
You select a single male student and a single female student at random and give
them the SSHA test.
(a) Explain why it is reasonable to assume that the scores of the two students are
independent.
(b) What are the mean and standard deviation of the difference (female minus male)
of their scores?
(c) From the information given, can you find the probability that the woman chosen
scores higher than the man? If so, find this probability. If not, explain why you
cannot.
4.69 The number of offspring produced by a female Asian stochastic beetle is ran-
dom, with this pattern: 20% of females die without female offspring, 30% have one
female offspring, and 50% have two female offspring. Females of the benign boiler
beetle have this reproductive pattern: 40% die without female offspring, 40% have
one female offspring, and 20% have two female offspring.
(a) Find the mean number of female offspring for each species of beetles.
(b) Use the law of large numbers to explain why the population should grow if the
expected number of female offspring is greater than 1 and die out if this expected
value is less than 1.
4.70 A study of the weights of the brains of Swedish men found that the weight
X was a random variable with mean 1400 grams and standard deviation 20 grams.
Find positive numbers a and b such that Y = a + bX has mean 0 and standard
deviation 1.
Section 4.4 105
4.71 In a process for manufacturing glassware, glass stems are sealed by heating
them in a flame. The temperature of the flame varies a bit. Here is the distribution
of the temperature X measured in degrees Celsius:
Temperature 540◦ 545◦ 550◦ 555◦ 560◦
Probability 0.1 0.25 0.3 0.25 0.1
(a) Find the mean temperature µX and the standard deviation σX .

(b) The target temperature is 550◦C. What are the mean and standard deviation
of the number of degrees off target, X − 550?
(c) A manager asks for results in degrees Fahrenheit. The conversion of X into
degrees Fahrenheit is given by
9
Y = X + 32
5
What are the mean µY and standard deviation σY of the temperature of the flame
in the Fahrenheit scale?
4.72 One consequence of the law of large numbers is that once we have a probability
distribution for a random variable, we can find its mean by simulating many out-
comes and averaging them. The law of large numbers says that if we take enough
outcomes, their average value is sure to approach the mean of the distribution.
I have a little bet to offer you. Toss a coin ten times. If there is no run of three
or more straight heads or tails in the ten outcomes, I’ll pay you $2. If there is a run
of three or more, you pay me just $1. Surely you will want to take advantage of me
and play this game?
Simulate enough plays of this game (the outcomes are +$2 if you win and −$1
if you lose) to estimate the mean outcome. Is it to your advantage to play?
4.73 You have two scales for measuring weights in a chemistry lab. Both scales give
answers that vary a bit in repeated weighings of the same item. If the true weight
of a compound is 2 grams (g), the first scale produces readings X that have mean
2.000 g and standard deviation 0.002 g. The second scale’s readings Y have mean
2.001 g and standard deviation 0.001 g.
(a) What are the mean and standard deviation of the difference Y − X between the
readings? (The readings X and Y are independent.)
(b) You measure once with each scale and average the readings. Your result is
Z = (X + Y )/2. What are µZ and σZ ? Is the average Z more or less variable than
the reading Y of the less variable scale?
4.74 Here is the distribution of grades (A = 4, B = 3, and so on) in Statistics 101

at North Carolina State University:
Value of X 0 1 2 3 4
Probability 0.01 0.05 0.30 0.43 0.21
Find the average (that is, the mean) grade in this course.
4.75 Typographical and spelling errors can be either “nonword errors” or “word
errors.” A nonword error is not a real word, as when “the” is typed as “teh.” A
word error is a real word, but not the right word, as when “lose” is typed as “loose.”
When undergraduates are asked to write a 250-word essay (without spell-checking),
the number of nonword errors has the following distribution:
Errors 0 1 2 3 4
Probability 0.1 0.2 0.3 0.3 0.1
The number of word errors has this distribution:

Errors 0 1 2 3
Probability 0.4 0.3 0.2 0.1
What are the mean numbers of nonword errors and word errors in an essay?
4.76 Find the mean and standard deviation of the total number of errors (nonword
errors plus word errors) in an essay if the error counts have the distributions given
in the previous exercise.
(a) The counts of nonword and word errors are independent.
(b) Students who make many nonword errors also tend to make many word errors,
so that the correlation between the two error counts is 0.5.
4.77 You buy a hot stock for $1000. The stock either gains 30% or loses 25% each
day, each with probability 0.5. Its returns on consecutive days are independent of
each other. You plan to sell the stock after two days.
(a) What are the possible values of the stock after two days, and what is the prob-
ability for each value? What is the probability that the stock is worth more after
two days than the $1000 you paid for it?
(b) What is the mean value of the stock after two days? You see that these two
criteria give different answers to the question, “Should I invest?”
4.78 For each of the following situations, would you expect the random variables X
and Y to be independent? Explain your answers.
(a) X is the rainfall (in inches) on November 6 of this year, and Y is the rainfall at
the same location on November 6 of next year.
(b) X is the amount of rainfall today, and Y is the rainfall at the same location
tomorrow.
(c) X is today’s rainfall at the Orlando, Florida, airport, and Y is today’s rainfall
at Disney World just outside Orlando.
4.79 A time and motion study measures the time required for an assembly-line
worker to perform a repetitive task. The data show that the time required to bring
a part from a bin to its position on an automobile chassis varies from car to car with
mean 11 seconds and standard deviation 2 seconds. The time required to attach the
part to the chassis varies with mean 20 seconds and standard deviation 8 seconds.
(a) What is the mean time required for the entire operation of positioning and
attaching the part?
(b) Industry quality programs strive to reduce variation in processes. A training
program reduces the standard deviation for attaching the part from 8 seconds to 4
seconds. Will this change your result in (a) if the mean times don’t change? Why
is reducing the variation nonetheless worthwhile to the automaker?
Section 4.4 107
(c) The study finds that the times required for the two steps are independent. A
part that takes a long time to position, for example, does not take more or less
time to attach than other parts. How would your answer in (a) change if the two
variables were dependent with correlation 0.3?
4.80 Find the standard deviation of the time required for the two-step assembly
operation studied in the previous exercise, assuming that the study shows the two
times to be independent. Redo the calculation assuming that the two times are
dependent, with correlation 0.3. Explain in nontechnical language why positive
correlation increases the variability of the total time.
4.81 You have two instruments with which to measure the height of a tower. If
the true height is 100 meters, measurements with the first instrument vary with
mean 100 meters and standard deviation 1.2 meters. Measurements with the second
instrument vary with mean 100 meters and standard deviation 0.85 meter. You make
one measurement with each instrument. Your results are X1 for the first and X2
for the second, and are independent.
(a) To combine the two measurements, you might average them, Y = (X1 + X2)/2.
What are the mean and standard deviation of Y ?
(b) It makes sense to give more weight to the less variable measurement because
it is more likely to be close to the truth. Statistical theory says that to make the
standard deviation as small as possible you should weight the two measurements
inversely proportional to their variances. The variance of X2 is very close to half
the variance of X1 , so X2 should get twice the weight of X1. That is, use
1 2
W = X1 + X2
3 3
What are the mean and standard deviation of W ?
4.82 Means of the numbers of rooms in housing units. How do rented

housing units differ from units occupied by their owners? Here are the distributions
of the number of rooms for owner-occupied units and renter-occupied units in San
Jose, California (from the Census Bureau’s 1998 American Housing Survey):
Rooms 1 2 3 4 5 6 7 8 9 10
Owned 0.003 0.002 0.023 0.104 0.210 0.224 0.197 0.149 0.053 0.035
Rented 0.008 0.027 0.287 0.363 0.164 0.093 0.039 0.013 0.003 0.003
Find the mean number of rooms for both types of housing units. Make probability
histograms of these two distributions, using the same scales. How do the means
reflect the differences you see in the histograms?
4.83 Standard deviations of numbers of rooms in housing units. Refer to

the previous exercise. Which of the two distributions of room counts appears more
spread out in the probability histograms? Why? Find the standard deviation for
both distributions. The standard deviation provides a numerical measure of spread.
Portfolio analysis. Here are the means, standard deviations, and correlations for
the annual returns from three Fidelity mutual funds for the 10 years ending in Febru-
ary 2004. (Means and standard deviations from Fidelity Investments, fidelity.com.
Correlations from the Fidelity Insight newsletter, www.fidelityinsight.com. The
correlations concern an unspecified period, and other online sources give different
correlations, so these should be regarded as approximate at best.) Because there
are three random variables, there are three correlations. We use subscripts to show
which pair of random variables a correlation refers to.
W = annual return on 500 Index Fund µW = 11.12% σW = 17.46%

X = annual return on Investment Grade Bond Fund µX = 6.46% σX = 4.18%
Y = annual return on Diversified International Fund µY = 11.10% σY = 15.62%
Correlations
ρW X = −0.22 ρW Y = 0.56 ρXY = −0.12
The following three exercises make use of these historical data.
4.84 Investing in a mix of U.S. stocks and foreign stocks. Many advisers rec-
ommend using roughly 20% foreign stocks to diversify portfolios of U.S. stocks. You
see that the 500 Index (U.S. stocks) and Diversified International (foreign stocks)
Funds had almost the same mean returns. A portfolio of 80% 500 Index and 20%
Diversified International will deliver this mean return with less risk. Verify this by
finding the mean and standard deviation of returns on this portfolio.
4.85 The effect of correlation. Diversification works better when the invest-
ments in a portfolio have small correlations. To demonstrate this, suppose that
returns on 500 Index Fund and Diversified International Fund had the means and
standard deviations we have given but were uncorrelated (ρW Y = 0). Show that the
standard deviation of a portfolio that combines 80% 500 Index with 20% Diversified
International is then smaller than your result from the previous exercise. What
happens to the mean return if the correlation is 0?
4.86 A portfolio with three investments. Portfolios often contain more than
two investments. The rules for means and variances continue to apply, though the
arithmetic gets messier. A portfolio containing proportions a of 500 Index Fund, b
of Investment Grade Bond Fund, and c of Diversified International Fund has return
R = aW + bX + cY . Because a, b, and c are the proportions invested in the three
funds, a + b + c = 1. The mean and variance of the portfolio return R are
µR = aµW + bµX + cµY

2
σR = a2 σW
2
+ b2σX
2
+ c2σY2 + 2abρW X σW σX + 2acρW Y σW σY + 2bcρXY σX σY
A basic well-diversified portfolio has 60% in 500 Index, 20% in Investment Grade
Bond, and 20% in Diversified International. What are the (historical) mean and
standard deviation of the annual returns for this portfolio? What does an investor
gain by choosing this diversified portfolio over 100% U.S. stocks? What does the
investor lose (at least in this time period)?
Section 4.5 109
Section 4.5
4.87 Here is a two-way table of all suicides committed in a recent year by sex of the
victim and method used.
Male Female
Firearms 15,802 2,367
Poison 3,262 2,233
Hanging 3,822 856
Other 1,571 571
Total 24,457 6,027
(a) What is the probability that a randomly selected suicide victim is male?
(b) What is the probability that the suicide victim used a firearm?
(c) What is the conditional probability that a suicide used a firearm, given that it
was a man? Given that it was a woman?
(d) Describe in simple language (don’t use the word “probability”) what your results
in (a) tell you about the difference between men and women with respect to suicide.
4.88 Consolidated Builders has bid on two large construction projects. The company
president believes that the probability of winning the first contract (event A) is 0.6,
that the probability of winning the second (event B) is 0.5, and that the probability
of winning both jobs (event {A and B}) is 0.3. What is the probability of the event
{A or B} that Consolidated will win at least one of the jobs?
4.89 In the setting of the previous exercise, are events A and B independent? Do
a calculation that proves your answer.
4.90 Draw a Venn diagram that illustrates the relation between events A and B in
the previous exercise. Write each of the following events in terms of A, B, Ac , and
B c . Indicate the events on your diagram and use the information in the previous
exercise to calculate the probability of each.
(a) Consolidated wins both jobs.
(b) Consolidated wins the first job but not the second.
(c) Consolidated does not win the first job but does win the second.
(d) Consolidated does not win either job.
4.91 Choose an employed person at random. Let A be the event that the person
chosen is a woman, and B the event that the person holds a managerial or pro-
fessional job. Government data tell us that P (A) = 0.46 and the probability of
managerial and professional jobs among women is P (B | A) = 0.32. Find the prob-
ability that a randomly chosen employed person is a woman holding a managerial
or professional position.
4.92 Common sources of caffeine in the diet are coffee, tea, and cola drinks. Suppose
that
55% of adults drink coffee
25% of adults drink tea
45% of adults drink cola
and also that

15% drink both coffee and tea
5% drink all three beverages
25% drink both coffee and cola
5% drink only tea
Draw a Venn diagram marked with this information. Use it along with the addition
rules to answer the following questions.
(a) What percent of adults drink only cola?
(b) What percent drink none of these beverages?
4.93 Functional Robotics Corporation buys electrical controllers from a Japanese

supplier. The company’s treasurer thinks that there is probability 0.4 that the dollar
will fall in value against the Japanese yen in the next month. The probability that
the supplier will demand that the contract be renegotiated is 0.8 if the dollar falls,
and 0.2 if the dollar does not fall. What is the probability that the supplier will
demand renegotiation? (Use a tree diagram to organize the information given.)
Here are data on the age and marital status of adult American women. Exercises
4.94 and 4.95 use this information.
Age
18 to 29 30 to 64 65 and over Total
Married 7,842 43,808 8,270 59,920
Never married 13,930 7,184 751 21,865
Widowed 36 2,523 8,385 10,944
Divorced 704 9,174 1,263 11,141
Total 22,512 62,689 18,669 103,870
4.94 Choose an adult American woman at random.

(a) What is the probability that the woman chosen is 65 years old or older?
(b) What is the conditional probability that the woman chosen is married, given
that she is 65 or over?
(c) How many women are both married and in the 65 and over age group? What
is the probability that the woman we choose is a married woman at least 65 years
old?
(d) Verify that the three probabilities you found in (a), (b), and (c) satisfy the
multiplication rule.
4.95 Choose an adult American woman at random.

(a) What is the conditional probability that the woman chosen is 18 to 29 years old,
given that she is married?
(b) Verify that P (married | age 18 to 29) = 0.348. Complete this sentence: 0.348
is the proportion of women who are among those women who are
.
(c) In (a), you found P (age 18 to 29 | married). Write a sentence of the form given
in (b) that describes the meaning of this result. The two conditional probabilities
give us very different information.
Section 4.5 111
4.96 A telemarketing company calls telephone numbers chosen at random. It finds

that 70% of calls are not completed (the party does not answer or refuses to talk),
that 20% result in talking to a woman, and that 10% result in talking to a man.
After that point, 30% of the women and 20% of the men actually buy something.
What percent of calls result in a sale?
4.97 An examination consists of multiple-choice questions, each having five possible

answers. Linda estimates that she has probability 0.75 of knowing the answer to
any question that may be asked. If she does not know the answer, she will guess,
with conditional probability 1/5 of being correct. What is the probability that
Linda gives the correct answer to a question? (Draw a tree diagram to guide the
calculation.)
4.98 In the setting of Exercise 4.96, what percent of sales are made to women?
(Write this as a conditional probability.)
4.99 In the setting of Exercise 4.97, find the conditional probability that Linda
knows the answer, given that she supplies the correct answer. (You can use the
result of the previous exercise and the definition of conditional probability, or you
can use Bayes’s rule.)
4.100 Zipdrive, Inc. has developed a new disk drive for small computers. The
demand for the new product is uncertain but can be described as “high” or “low”
in any one year. After 4 years, the product is expected to be obsolete. Management
must decide whether to build a plant or to contract with a factory in Hong Kong to
manufacture the new drive. Building a plant will be profitable if demand remains
high but could lead to a loss if demand drops in future years.
After careful study of the market and of all relevant costs, Zipdrive’s planning
office provides the following information. Let A be the event that the first year’s
demand is high, and B be the event that the following 3 years’ demand is high. The
marketing division’s best estimate of the probabilities is
P (A) = 0.9
P (B | A) = 0.36
P (B | Ac ) = 0
The probability that building a plant is more profitable than contracting the pro-
duction to Hong Kong is 0.95 if demand is high all 4 years, 0.3 if demand is high
only in the first year, and 0.1 if demand is low all 4 years.
Draw a tree diagram that organizes this information. The tree will have three
stages: first year’s demand, next 3 years’ demand, and whether building or con-
tracting is more profitable. Which decision has the higher probability of being more
profitable? (When decision analysis is used for investment decisions like this, firms
in fact compare the mean profits rather than the probability of a profit. We ignore
this complication.)
4.101 John has coronary artery disease. He and his doctor must decide between
medical management of the disease and coronary bypass surgery. Because John has
been quite active, he is concerned about his quality of life as well as the length of
life. He wants to make the decision that will maximize the probability of the event
A that he survives for 5 years and is able to carry on moderate activity during that
time. The doctor makes the following probability estimates for patients of John’s
age and condition (based loosely on M. C. Weinstein, J. S. Pliskin, and W. B. Stason,
“Coronary artery bypass surgery: decision and policy analysis,” in J. P. Bunker, B.
A. Barnes, and F. W. Mosteller (eds.), Costs, Risks and Benefits of Surgery, Oxford
University Press, 1977, pp. 342–371):
• Under medical management, P (A) = 0.7.
• There is probability 0.05 that John will not survive bypass surgery, probability
0.10 that he will survive with serious complications, and probability 0.85 that
he will survive the surgery without complications.
• If he survives with complications, the conditional probability of the desired

outcome A is 0.73. If there are no serious complications, the conditional
probability of A is 0.76.
Draw a tree diagram that summarizes this information. Then calculate P (A) as-
suming that John chooses the surgery. Does surgery or medical management offer
him the better chance of achieving his goal?
4.102 In 2000, the Internal Revenue Service received 129,075,000 individual tax
returns. Of these, 10,855,000 reported an adjusted gross income of at least $100,000,
and 240,000 reported at least $1 million. If you know that a randomly chosen return
shows an income of $100,000 or more, what is the conditional probability that the
income is at least $1 million?
4.103 Julie is graduating from college. She has studied biology, chemistry, and
computing and hopes to work as a forensic scientist applying her science background
to crime investigation. Late one night she thinks about some jobs she has applied
for. Let A, B, and C be the events that Julie is offered a job by
A = the Connecticut Office of the Chief Medical Examiner

B = the New Jersey Division of Criminal Justice
C = the federal Disaster Mortuary Operations Response Team
Julie writes down her personal probabilities for being offered these jobs:
P (A) = 0.6 P (B) = 0.4 P (C) = 0.2

P (A and B) = 0.1 P (A and C) = 0.05 P (B and C) = 0.05
P (A and B and C) = 0
Make a Venn diagram of the events A, B, and C. Mark the probabilities of every
intersection involving these events and their complements. Use this diagram for
Exercises 4.104 to 4.106.
4.104 What is the probability that Julie is offered at least one of the three jobs?
4.105 What is the probability that Julie is offered both the Connecticut and New
Jersey jobs, but not the federal job?
Section 4.5 113
4.106 If Julie is offered the federal job, what is the conditional probability that she
is also offered the New Jersey job? If Julie is offered the New Jersey job, what is
the conditional probability that she is also offered the federal job?
4.107 Here are the counts (in thousands) of earned degrees in the United States
in the 2005–2006 academic year, classified by level and by the sex of the degree
recipient:
Bachelor’s Master’s Professional Doctorate Total
Female ?784 276 39 20 1119
Male ?559 197 44 25 ?825
Total 1343 473 83 45 1944
(a) If you choose a degree recipient at random, what is the probability that the
person you choose is a woman?
(b) What is the conditional probability that you choose a woman, given that the
person chosen received a professional degree?
(c) Are the events “choose a woman” and “choose a professional degree recipient”
independent? How do you know?
4.108 The previous exercise gives the counts (in thousands) of earned degrees in
the United States in the 2005–2006 academic year. Use these data to answer the
following questions.
(a) What is the probability that a randomly chosen degree recipient is a man?
(b) What is the conditional probability that the person chosen received a bachelor’s
degree, given that he is a man?
(c) Use the multiplication rule to find the probability of choosing a male bachelor’s
degree recipient. Check your result by finding this probability directly from the
table of counts.
4.109 The probability that a randomly chosen student at the University of New
Harmony is a woman is 0.6. The probability that the student is studying education
is 0.15. The conditional probability that the student is a woman, given that the
student is studying education, is 0.8. What is the conditional probability that the
student is studying education, given that she is a woman?
4.110 The voters in a large city are 40% white, 40% black, and 20% Hispanic.
(Hispanics may be of any race in official statistics, but in this case we are speaking
of political blocks.) A black mayoral candidate anticipates attracting 30% of the
white vote, 90% of the black vote, and 50% of the Hispanic vote. Draw a tree
diagram with probabilities for the race (white, black, or Hispanic) and vote (for or
against the candidate) of a randomly chosen voter. What percent of the overall vote
does the candidate expect to get?
4.111 In the election described in the previous exercise, what percent of the candi-
date’s votes come from black voters? (Write this as a conditional probability.)
4.112 At a self-service gas station, 40% of the customers pump regular gas, 35%
pump midgrade, and 25% pump premium gas. Of those who pump regular, 30%
pay at least $20. Of those who pump midgrade, 50% pay at least $20. And of those
who pump premium, 60% pay at least $20. What is the probability that the next
customer pays at least $20?
4.113 In the setting of the previous exercise, what percent of customers who pay at
least $20 pump premium? (Write this as a conditional probability.)
4.114 Gender and majors. The probability that a randomly chosen student
at the University of New Harmony is a woman is 0.62. The probability that the
student is studying education is 0.17. The conditional probability that the student
is a woman, given that the student is studying education, is 0.8. What is the
conditional probability that the student is studying education, given that she is a
woman?
4.115 Cystic fibrosis. Cystic fibrosis is a lung disorder that often results in
death. It is inherited but can be inherited only if both parents are carriers of an
abnormal gene. In 1989, the CF gene that is abnormal in carriers of cystic fibrosis
was identified. The probability that a randomly chosen person of European ancestry
carries an abnormal CF gene is 1/25. (The probability is less in other ethnic groups.)
The CF20m test detects most but not all harmful mutations of the CF gene. The
test is positive for 90% of people who are carriers. It is (ignoring human error)
never positive for people who are not carriers. Jason tests positive. What is the
probability that he is a carrier?
4.116 Use Bayes’s rule. Refer to the previous exercise. Jason knows that he
is a carrier of cystic fibrosis. His wife, Julianne, has a brother with cystic fibrosis,
which means the probability is 2/3 that she is a carrier. If Julianne is a carrier,
each child she has with Jason has probability 1/4 of having cystic fibrosis. If she
is not a carrier, her children cannot have the disease. Jason and Julianne have one
child, who does not have cystic fibrosis. This information reduces the probability
that Julianne is a carrier. Use Bayes’s rule to find the conditional probability that
Julianne is a carrier, given that she and Jason have one child who does not have
cystic fibrosis.
4.117 Deal a five-card poker hand from a shuffled deck. The probabilities of several
types of hands are approximately as follows:
Hand Worthless One pair Two pairs Better hands

Probability 0.50 0.42 0.05 ?
What must be the probability of getting a hand better than two pairs? What is the
probability of getting a hand that is not worthless?
4.118 You have torn a tendon and are facing surgery to repair it. The orthopedic
surgeon explains the risks to you. Infection occurs in 3% of such operations, the
repair fails in 14%, and both infection and failure occur together in 1%. What
percent of these operations succeed and are free from infection?
Review Exercises 115
4.119 You are playing a board game in which the severity of a penalty is determined
by rolling three dice and adding the spots on the up-faces. The dice are all balanced
so that each face is equally likely, and the three dice fall independently. If X1,
X2, and X3 are the number of spots on the up-faces of the three dice, then X =
X1 + X2 + X3. Use this fact to find the mean µX and the standard deviation σX
without finding the distribution of X. (Start with the distribution of each of the
Xi .)
4.120 Enzyme immunoassay (EIA) tests are used to screen blood specimens for the
presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the
presence of the virus. The test is quite accurate but is not always correct. Here are
approximate probabilities of positive and negative EIA outcomes when the blood
tested does and does not actually contain antibodies to HIV (J. Richard George,
“Alternative specimen sources: methods for confirming positives,” 1998 Conference
on the Laboratory Science of HIV, found online at the Centers for Disease Control
and Prevention, www.cdc.gov):
Test result
+ −
Antibodies present 0.9985 0.0015
Antibodies absent 0.0060 0.9940
Suppose that 1% of a large population carries antibodies to HIV in their blood.

(a) Draw a tree diagram for selecting a person from this population (outcomes: an-
tibodies present or absent) and for testing his or her blood (outcomes: EIA positive
or negative).
(b) What is the probability that the EIA is positive for a randomly chosen person
from this population?
(c) What is the probability that a person has the antibody, given that the EIA test
is positive?
(This exercise illustrates a fact that is important when considering proposals for
widespread testing for HIV, illegal drugs, or agents of biological warfare: if the
condition being tested is uncommon in the population, many positives will be false
positives.)
4.121 The previous exercise gives data on the results of EIA tests for the presence
of antibodies to HIV. Repeat part (c) of that exercise for two different populations:
(a) Blood donors are prescreened for HIV risk factors, so perhaps only 0.1% (0.001)
of this population carries HIV antibodies.
(b) Clients of a drug rehab clinic are a high-risk group, so perhaps 10% of this
population carries HIV antibodies.
(c) What general lesson do your calculations illustrate?
4.122 Two wine tasters rate each wine they taste on a scale of 1 to 5. From data
on their ratings of a large number of wines, we obtain the following probabilities for
both tasters’ ratings of a randomly chosen wine:
Taster 2
Taster 1 1 2 3 4 5
1 0.03 0.02 0.01 0.00 0.00
2 0.02 0.08 0.05 0.02 0.01
3 0.01 0.05 0.25 0.05 0.01
4 0.00 0.02 0.05 0.20 0.02
5 0.00 0.01 0.01 0.02 0.06
(a) Why is this a legitimate assignment of probabilities to outcomes?
(b) What is the probability that the tasters agree when rating a wine?
(c) What is the probability that Taster 1 rates a wine higher than 3? What is the
probability that Taster 2 rates a wine higher than 3?
4.123 In the setting of the previous exercise, Taster 1’s rating for a wine is 3. What
is the conditional probability that Taster 2’s rating is higher than 3?
4.124 Rotter Partners is planning a major investment. The amount of profit X is

uncertain but a probabilistic estimate gives the following distribution (in millions of
dollars):
Profit 1 1.5 2 4 10
Probability 0.1 0.2 0.4 0.2 0.1
(a) Find the mean profit µX and the standard deviation σX of the profit.
(b) Rotter Partners owes its source of capital a fee of $200,000 plus 10% of the
profits X. So the firm actually retains
Y = 0.9X − 0.2
from the investment. Find the mean and standard deviation of Y .
4.125 A grocery store gives its customers cards that may win them a prize when
matched with other cards. The back of the card announces the following probabili-
ties of winning various amounts if a customer visits the store 10 times:
Amount $1000 $200 $50 $10
Probability 1/10,000 1/1000 1/100 1/20
(a) What is the probability of winning nothing?
(b) What is the mean amount won?
(c) What is the standard deviation of the amount won?
4.126 Profits from an investment. Rotter Partners is planning a major invest-

ment. The amount of profit X is uncertain but a probabilistic estimate gives the
following distribution (in millions of dollars):
Profit 1 1.5 2 4 10
Probability 0.4 0.2 0.2 0.1 0.1
(a) Find the mean profit µX and the standard deviation σX of the profit.
(b) Rotter Partners owes its source of capital a fee of $200,000 plus 10% of the
profits X. So the firm actually retains
Y = 0.9X − 0.2
from the investment. Find the mean and standard deviation of Y .
4.127 Prizes for grocery store customers. A grocery store gives its customers
cards that may win them a prize when matched with other cards. The back of the
card announces the following probabilities of winning various amounts if a customer
visits the store 10 times:
Amount $1000 $250 $100 $10
Probability 1/10,000 1/1000 1/100 1/20
(a) What is the probability of winning nothing?

(b) What is the mean amount won?
(c) What is the standard deviation of the amount won?
CHAPTER 5
Section 5.1
5.1 A company that owns and services a fleet of cars for its sales force has found
that the service lifetime of disc brake pads varies from car to car according to a
Normal distribution with mean µ = 55, 000 miles and standard deviation σ = 4500
miles. The company installs a new brand of brake pads on 8 cars.
(a) If the new brand has the same lifetime distribution as the previous type, what
is the distribution of the sample mean lifetime for the 8 cars?
(b) The average life of the pads on these 8 cars turns out to be x = 51, 800 miles.
What is the probability that the sample mean lifetime is 51,800 miles or less if the
lifetime distribution is unchanged? The company takes this probability as evidence
that the average lifetime of the new brand of pads is less than 55,000 miles.
5.2 Investors remember 1987 as the year stocks lost 22% of their value in a single
day. For 1987 as a whole, the mean return of all common stocks on the New York
Stock Exchange was µ = −3.5%. (That is, these stocks lost an average of 3.5% of
their value in 1987.) The standard deviation of the returns was about σ = 26%.
The distribution of annual returns for stocks is roughly Normal.
(a) What percent of stocks lost money? (That is the same as the probability that a
stock chosen at random has a return less than 0.)
(b) Suppose that you held a portfolio of 5 stocks chosen at random from New York
Stock Exchange stocks. What are the mean and standard deviation of the returns
of randomly chosen portfolios of 5 stocks?
(c) What percent of such portfolios lost money? Explain the difference between this
result and the result of (a).
5.3 Newly manufactured automobile radiators may have small leaks. Most have
no leaks, but some have 1, 2, or more. The number of leaks in radiators made by
one supplier has mean 0.15 and standard deviation 0.4. The distribution of number
of leaks cannot be Normal because only whole-number counts are possible. The
supplier ships 400 radiators per day to an auto assembly plant. Take x to be the
mean number of leaks in these 400 radiators. Over several years of daily shipments,
what range of values will contain the middle 95% of the many x’s?
5.4 Children in kindergarten are sometimes given the Ravin Progressive Matrices
Test (RPMT) to assess their readiness for learning. Experience at Southwark El-
ementary School suggests that the RPMT scores for its kindergarten pupils have
mean 13.6 and standard deviation 3.1. The distribution is close to Normal. Mr.
Lavin has 22 children in his kindergarten class this year. He suspects that their
RPMT scores will be unusually low because the test was interrupted by a fire drill.
To check this suspicion, he wants to find the level L such that there is probability
only 0.05 that the mean score of 22 children falls below L when the usual Southwark
distribution remains true. What is the value of L?
118
5.5 A laboratory weighs filters from a coal mine to measure the amount of dust in
the mine atmosphere. Repeated measurements of the weight of dust on the same
filter vary Normally with standard deviation σ = 0.08 milligram (mg) because the
weighing is not perfectly precise. The dust on a particular filter actually weighs 123
mg. Repeated weighings will then have the Normal distribution with mean 123 mg
and standard deviation 0.08 mg.
(a) The laboratory reports the mean of 3 weighings. What is the distribution of this
mean?
(b) What is the probability that the laboratory reports a weight of 124 mg or higher
for this filter?
5.6 The scores of twelfth-grade students on the National Assessment of Educational

Progress year 2000 mathematics test have a distribution that is approximately Nor-
mal with mean µ = 300 and standard deviation σ = 35.
(a) Choose one twelfth-grader at random. What is the probability that his or her
score is higher than 300? Higher than 335?
(b) Now choose an SRS of four twelfth-graders. What is the probability that their
mean score is higher than 300? Higher than 335?
5.7 The number of accidents per week at a hazardous intersection varies with mean
2.2 and standard deviation 1.4. This distribution takes only whole-number values,
so it is certainly not Normal.
(a) Let x be the mean number of accidents per week at the intersection during a
year (52 weeks). What is the approximate distribution of x according to the central
limit theorem?
(b) What is the approximate probability that x is less than 2?
(c) What is the approximate probability that there are fewer than 100 accidents at
the intersection in a year? (Hint: Restate this event in terms of x.)
5.8 A roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green.
When the wheel is spun, the ball is equally likely to come to rest in any of the slots.
Gamblers can place a number of different bets in roulette. One of the simplest
wagers chooses red or black. A bet of $1 on red will pay off an additional dollar if
the ball lands in a red slot. Otherwise, the player loses his dollar. When gamblers
bet on red or black, the two green slots belong to the house.
(a) A gambler’s winnings on a $1 bet are either $1 or −$1. Give the probabilities of
these outcomes. Find the mean and standard deviation of the gambler’s winnings.
(b) Explain briefly what the law of large numbers tells us about what will happen
if the gambler makes a large number of bets on red.
(c) The central limit theorem tells us the approximate distribution of the gambler’s
mean winnings in 50 bets. What is this distribution? Use the 68–95–99.7 rule to
give the range in which the mean winnings will fall 95% of the time. Multiply by 50
to get the middle 95% of the distribution of the gambler’s winnings on nights when
he places 50 bets.
(d) What is the probability that the gambler will lose money if he makes 50 bets?
(This is the probability that the mean is less than 0.)
(e) The casino takes the other side of these bets. If 100,000 bets on red are placed in
a week at the casino, what is the distribution of the mean winnings of gamblers on
119
these bets? What range covers the middle 95% of mean winnings in 100,000 bets?
Multiply by 100,000 to get the range of gamblers’ losses. (Gamblers’ losses are the
casino’s winnings. Part (c) shows that a gambler gets excitement. Now we see that
the casino has a business.)
5.9 An experiment to compare the nutritive value of normal corn and high-lysine
corn divides 40 chicks at random into two groups of 20. One group is fed a diet
based on normal corn while the other receives high-lysine corn. At the end of
the experiment, inference about which diet is superior is based on the difference
y − x between the mean weight gain y of the 20 chicks in the high-lysine group
and the mean weight gain x of the 20 in the normal-corn group. Because of the
randomization, the two sample means are independent.
(a) Suppose that µX = 360 grams (g) and σX = 55 g in the population of all chicks
fed normal corn, and that µY = 385 g and σY = 50 g in the high-lysine population.
What are the mean and standard deviation of y − x?
(b) The weight gains are Normally distributed in both populations. What is the
distribution of x? Of y? What is the distribution of y − x?
(c) What is the probability that the mean weight gain in the high-lysine group
exceeds the mean weight gain in the normal-corn group by 25 g or more?
5.10 An experiment on the teaching of reading compares two methods, A and B. The
response variable is the Degree of Reading Power (DRP) score. The experimenter
uses Method A in a class of 26 students and Method B in a comparable class of 24
students. The classes are assigned to the teaching methods at random. Suppose
that in the population of all children of this age the DRP score has the N (34, 12)
distribution if Method A is used and the N (37, 11) distribution if Method B is used.
(a) What is the distribution of the mean DRP score x for the 26 students in the A
group? (Assume that this group can be regarded as an SRS from the population of
all children of this age.)
(b) What is the distribution of the mean score y for the 24 students in the B group?
(c) Use the results of (a) and (b), keeping in mind that x and y are independent,
to find the distribution of the difference y − x between the mean scores in the two
groups.
(d) What is the probability that the mean score for the B group will be at least 4
points higher than the mean score for the A group?
5.11 Leona and Fred are friendly competitors in high school. Both are about to
take the ACT college entrance examination. They agree that if one of them scores 5
or more points better than the other, the loser will buy the winner a pizza. Suppose
that in fact Fred and Leona have equal ability, so that each score varies Normally
with mean 24 and standard deviation 2. (The variation is due to luck in guessing
and the accident of the specific questions being familiar to the student.) The two
scores are independent. What is the probability that the scores differ by 5 or more
points in either direction?
5.12 The design of an electronic circuit calls for a 100-ohm resistor and a 250-ohm
resistor connected in series so that their resistances add. The components used are
not perfectly uniform, so that the actual resistances vary independently according to
120
Normal distributions. The resistance of 100-ohm resistors has mean 100 ohms and
standard deviation 2.5 ohms, while that of 250-ohm resistors has mean 250 ohms
and standard deviation 2.8 ohms.
(a) What is the distribution of the total resistance of the two components in series?
(b) What is the probability that the total resistance lies between 345 and 355 ohms?
5.13 ACT, Inc., the producer of the ACT test of readiness for college work, also
produces tests for eighth- and ninth-grade students designed to help them plan for
the future. Two of these tests measure reading and mathematics achievement. Each
has scores that range from 1 to 25. For students tested in the fall of their eighth-
grade year, the reading test has mean 13.9 and standard deviation 3.63. The mean
score on the math test is 14.4 and the standard deviation is 3.46. Scores roughly
follow a Normal distribution. (Based on data from over 7000 students, reported at
www.act.org/explore/newscale/summary.html.)
(a) If a student’s reading score X and mathematics score Y were independent, what
would be the distribution of total X + Y ?
(b) Using the distribution you found in (a), what percent of eighth-graders have a
total score of 30 or higher?
(c) In fact, the X and Y scores are strongly correlated. In this case, does the mean
of X + Y still have the value you found in (a)? Does the standard deviation still
have the value you found in (a)?
5.14 Antonio measures the alcohol content of whiskey for his Chemistry 101 lab.
He actually measures the mass of 5 milliliters of whiskey—a chemical calculation
then finds the percent alcohol from the mass. The standard deviation of students’
measurements of mass is σ = 10 milligrams (mg). Antonio repeats the measurement
3 times and records the mean x of his 3 measurements.
(a) What is the standard deviation σx of Antonio’s mean result?
(b) How many times must Antonio repeat the measurement to reduce the standard
deviation of x to 5 mg? Explain to someone who knows no statistics the advantage
of reporting the average of several measurements rather than the result of a single
measurement.
5.15 An automatic grinding machine in an auto parts plant prepares axles with a
target diameter µ = 40.125 millimeters (mm). The machine has some variability,
so the standard deviation of the diameters is σ = 0.002 mm. A sample of 4 axles is
inspected each hour for process control purposes, and records are kept of the sample
mean diameter. If the process mean is exactly equal to the target value, what will
be the mean and standard deviation of the numbers recorded?
5.16 Averages are less variable than individual observations. Suppose that the axle
diameters in the previous exercise vary according to a Normal distribution. In that
case, the mean x of an SRS of axles also has a Normal distribution.
(a) Make a sketch of the Normal curve for a single axle. Add the Normal curve for
the mean of an SRS of 4 axles on the same sketch.
(b) What is the probability that the diameter of a single randomly chosen axle differs
from the target value by 0.004 mm or more?
121
(c) What is the probability that the mean diameter of an SRS of 4 axles differs from
the target value by 0.004 mm or more?
5.17 Averages of several measurements are less variable than individual measure-
ments. The true mass of the whiskey sample in Exercise 5.14 is 4.6 grams, or 4600
milligrams (mg). Antonio’s measurements have the Normal distribution with mean
4600 mg and standard deviation 10 mg. In this case, the mean of his 3 measure-
ments also has a Normal distribution.
(a) Sketch on the same graph the two Normal curves, for individual measurements
and for means of 3 measurements.
(b) What is the probability that Antonio misses the true mass by more than 10 mg
in either direction if he makes one measurement?
(c) What is the probability that the mean of three independent measurements misses
the true mass by more than 10 mg?
5.18 North Carolina State University posts the grade distributions for its courses
online. You can find that the distribution of grades in Statistics 101 in the fall 2003
semester was
Grade A B C D F
Probability 0.21 0.43 0.30 0.05 0.01
(a) Using the common scale A = 4, B = 3, C = 2, D = 1, F = 0, take X to be the

grade of a randomly chosen Statistics 101 student. Use the definitions of the mean
and standard deviation for discrete random variables to find the mean µ and the
standard deviation σ of grades in this course.
(b) Statistics 101 is a large course. We can take the grades of an SRS of 50 students
to be independent of each other. If x is the average of these 50 grades, what are the
mean and standard deviation of x?
(c) What is the probability P (X ≥ 3) that a randomly chosen Statistics 101 student
gets a B or better? What is the approximate probability P (x ≥ 3) that the grade
point average for 50 randomly chosen Statistics 101 students is B or better?
5.19 A $1 bet in a state lottery’s Pick 3 game pays $500 if the three-digit number
you choose exactly matches the winning number, which is drawn at random. Here
is the distribution of the payoff X:
Payoff X $0 $500
Probability 0.999 0.001
Each day’s drawing is independent of other drawings.

(a) What are the mean and standard deviation of X?
(b) Joe buys a Pick 3 ticket every day. What does the law of large numbers say
about the average payoff Joe receives from his bets?
(c) What does the central limit theorem say about the distribution of Joe’s average
payoff after 365 bets in a year?
(d) Joe comes out ahead for the year if his average payoff is greater than $1 (the
amount he spent each day on a ticket). What is the probability that Joe ends the
year ahead?
122
5.20 In response to the increasing weight of airline passengers, the Federal Aviation
Administration in 2003 told airlines to assume that passengers average 190 pounds
in the summer, including clothing and carry-on baggage. But passengers vary: the
FAA gave a mean but not a standard deviation. A reasonable standard deviation is
35 pounds. Weights are not Normally distributed, especially when the population
includes both men and women, but they are not very non-Normal. A commuter
plane carries 19 passengers. What is the approximate probability that the total
weight of the passengers exceeds 4000 pounds? (Hint: To apply the central limit
theorem, restate the problem in terms of the mean weight.)
5.21 A hole in an engine block is 2.5 centimeters (cm) in diameter. Shafts manufac-
tured to go through this hole must have 0.025 cm clearance for unforced fit. That
is, shaft diameter cannot exceed 2.475 cm. The shafts vary in diameter according
to the Normal distribution with mean 2.45 cm and standard deviation 0.01 cm.
(a) What percent of shafts will fit into the hole?
(b) Redo the problem assuming that the hole diameter also varies, independently
of the shaft diameter, following the Normal distribution with mean 2.5 cm and
standard deviation 0.01 cm. You must find the probability that the hole diameter
exceeds the shaft diameter by at least 0.025 cm.
5.22 “Durable press” cotton fabrics are treated to improve their recovery from
wrinkles after washing. Unfortunately, the treatment also reduces the strength of
the fabric. The breaking strength of untreated fabric is Normally distributed with
mean 58 pounds and standard deviation 2.3 pounds. The same type of fabric after
treatment has Normally distributed breaking strength with mean 30 pounds and
standard deviation 1.6 pounds. A clothing manufacturer tests 5 specimens of each
fabric. All 10 strength measurements are independent.
(a) What is the probability that the mean breaking strength of the 5 untreated
specimens exceeds 50 pounds?
(b) What is the probability that the mean breaking strength of the 5 untreated
specimens is at least 25 pounds greater than the mean strength of the 5 treated
specimens?
5.23 Many companies place advertisements to improve the image of their brand
rather than to promote specific products. In a randomized comparative experiment,
business students read ads that cited either the Wall Street Journal or the National
Enquirer for important facts about a fictitious company. The students then rated
the trustworthiness of the source on a 7-point scale. Suppose that in the population
of all students scores for the Journal have mean 4.8 and standard deviation 1.5,
while scores for the Enquirer have mean 2.4 and standard deviation 1.6.
(a) There are 30 students in each group. Although individual scores are discrete,
the mean score for a group of 30 will be close to Normal. Why?
(b) What are the means and standard deviations of the sample mean scores y for
the Journal group and x for the Enquirer group?
(c) We can take all 60 scores to be independent because students are not told each
other’s scores. What is the distribution of the difference y − x between the mean
scores in the two groups?
(d) Find P (y − x ≥ 1).
123
5.24 Linda invests her money in a portfolio that consists of 80% Fidelity 500 Index
Fund and 20% Fidelity Diversified International Fund. Suppose that in the long
run the annual real return X on the 500 Index Fund has mean 9% and standard
deviation 20%, the annual real return Y on the Diversified International Fund has
mean 10% and standard deviation 17%, and the correlation between X and Y is
0.6.
(a) The return on Linda’s portfolio is R = 0.8X + 0.2Y . What are the mean and
standard deviation of R?
(b) The distribution of returns is typically roughly symmetric but with more extreme
high and low observations than a Normal distribution. The average return over a
number of years, however, is close to Normal. If Linda holds her portfolio for 20
years, what is the approximate probability that her average return is less than 5%?
(c) The calculation you just made is not overly helpful, because Linda isn’t really
concerned about the mean return R. To see why, suppose that her portfolio returns
12% this year and 6% next year. The mean return for the two years is 9%. If Linda
starts with $1000, how much does she have at the end of the first year? At the end
of the second year? How does this amount compare with what she would have if
both years had the mean return, 9%? Over 20 years, there may be a large difference
between the ordinary mean R and the geometric mean, which reflects the fact that
returns in successive years multiply rather than add.
5.25 The amount that households pay service providers for access to the Internet
varies quite a bit, but the mean monthly fee is $28 and the standard deviation is $10.
The distribution is not Normal: many households pay about $10 for limited dial-up
access or about $25 for unlimited dial-up access, but many pay more for broadband
connections. A sample survey asks an SRS of 500 households with Internet access
how much they pay. What is the probability that the average fee paid by the sample
households exceeds $29?
5.26 A particle moves along the line in a random walk. That is, the particle starts at
the origin (position 0) and moves either right or left in independent steps of length
1. If the particle moves to the right with probability 3/4, its movement at the ith
step is a random variable Xi with distribution
P (Xi = 1) = 0.75
P (Xi = −1) = 0.25
The position of the particle after k steps is the sum of these random movements,
Y = X 1 + X2 + · · · + X k
Use the central limit theorem to find the approximate probability that the position
of the particle after 500 steps is at least 200 to the right.
5.27 An automatic grinding machine in an auto parts plant prepares axles with a
target diameter µ = 40.135 millimeters (mm). The machine has some variability,
so the standard deviation of the diameters is σ = 0.003 mm. A sample of 4 axles is
inspected each hour for process control purposes, and records are kept of the sample
mean diameter. If the process mean is exactly equal to the target value, what will
be the mean and standard deviation of the numbers recorded?
124
5.28 Averages are less variable than individual observations. Suppose that the axle
diameters in the previous exercise vary according to a Normal distribution. In that
case, the mean x̄ of an SRS of axles also has a Normal distribution.
(a) Make a sketch of the Normal curve for a single axle. Add the Normal curve for
the mean of an SRS of 4 axles on the same sketch.
(b) What is the probability that the diameter of a single randomly chosen axle differs
from the target value by 0.006 mm or more?
(c) What is the probability that the mean diameter of an SRS of 4 axles differs from
the target value by 0.006 mm or more?
5.29 The number of lightning strikes on a square kilometer of open ground in a year
has mean 6 and standard deviation 2.4. (These values are typical of much of the
United States.) Counts of strikes on separate areas are independent. The National
Lightning Detection Network uses automatic sensors to watch for lightning in an
area of 10 square kilometers.
(a) What are the mean and standard deviation of the total number of lightning
strikes observed?
(b) What are the mean and standard deviation of the mean number of strikes per
square kilometer?
5.30 The distribution of annual returns on common stocks is roughly symmetric,

but extreme observations are more frequent than in a Normal distribution. Because
the distribution is not strongly non-Normal, the mean return over even a moderate
number of years is close to Normal. Annual real returns on the Standard & Poor’s
500 stock Index over the period 1871 to 2004 have varied with mean 9.2% and
standard deviation 20.6%. Andrew plans to retire in 45 years and is considering
investing in stocks. What is the probability (assuming that the past pattern of
variation continues) that the mean annual return on common stocks over the next
45 years will exceed 15%? What is the probability that the mean return will be less
than 5%?
5.31 A hole in an engine block is 2.5 centimeters (cm) in diameter. Shafts manufac-
tured to go through this hole must have 0.024 cm clearance for unforced fit. That
is, shaft diameter cannot exceed 2.474 cm. The shafts vary in diameter according
to the Normal distribution with mean 2.45 cm and standard deviation 0.01 cm.
(a) What percent of shafts will fit into the hole?
(b) Redo the problem assuming that the hole diameter also varies, independently
of the shaft diameter, following the Normal distribution with mean 2.5 cm and
standard deviation 0.01 cm. You must find the probability that the hole diameter
exceeds the shaft diameter by at least 0.025 cm.
Section 5.2
5.32 For each of the following situations, indicate whether a binomial distribution
is a reasonable probability model for the random variable X. Give your reasons in
each case.
(a) You observe the sex of the next 50 children born at a local hospital; X is the
125
number of girls among them.

(b) A couple decides to continue to have children until their first girl is born; X is
the total number of children the couple has.
(c) You want to know what percent of married people believe that mothers of young
children should not be employed outside the home. You plan to interview 50 people,
and for the sake of convenience you decide to interview both the husband and the
wife in 25 married couples. The random variable X is the number among the 50
persons interviewed who think mothers should not be employed.
5.33 In each of the following cases, decide whether or not a binomial distribution is
an appropriate model, and give your reasons.
(a) Fifty students are taught about binomial distributions by a television program.
After completing their study, all students take the same examination. The number
of students who pass is counted.
(b) A student studies binomial distributions using computer-assisted instruction.
After the initial instruction is completed, the computer presents 10 problems. The
student solves each problem and enters the answer; the computer gives additional
instruction between problems if the student’s answer is wrong. The number of
problems that the student solves correctly is counted.
(c) A chemist repeats a solubility test 10 times on the same substance. Each test
is conducted at a temperature 10◦ higher than the previous test. She counts the
number of times that the substance dissolves completely.
5.34 A factory employs several thousand workers, of whom 30% are Hispanic. If
the 15 members of the union executive committee were chosen from the workers
at random, the number of Hispanics on the committee would have the binomial
distribution with n = 15 and p = 0.3.
(a) What is the probability that exactly 3 members of the committee are Hispanic?
(b) What is the probability that 3 or fewer members of the committee are Hispanic?
5.35 A university that is better known for its basketball program than for its aca-
demic strength claims that 80% of its basketball players get degrees. An investi-
gation examines the fate of all 20 players who entered the program over a period
of several years that ended 6 years ago. Of these players, 11 graduated and the
remaining 9 are no longer in school. If the university’s claim is true, the number of
players who graduate among the 20 studied should have the B(20, 0.8) distribution.
(a) Find the probability that exactly 11 of the 20 players graduate.
(b) Find the probability that 11 or fewer players graduate. This probability is so
small that it casts doubt on the university’s claim.
5.36 (a) What is the mean number of Hispanics on randomly chosen committees of
15 workers in Exercise 5.34?
(b) What is the standard deviation σ of the count X of Hispanic members?
(c) Suppose that 10% of the factory workers were Hispanic. Then p = 0.1. What
is σ in this case? What is σ if p = 0.01? What does your work show about the
behavior of the standard deviation of a binomial distribution as the probability of
a success gets closer to 0?
Section 5.1 127
5.37 (a) Find the mean number of graduates out of 20 players in the setting of
Exercise 5.35 if the university’s claim is true.
(b) Find the standard deviation σ of the count X.
(c) Suppose that the 20 players came from a population of which p = 0.9 graduated.
What is the standard deviation σ of the count of graduates? If p = 0.99, what is
σ? What does your work show about the behavior of the standard deviation of a
binomial distribution as the probability p of success gets closer to 1?
5.38 You are planning a sample survey of small businesses in your area. You will
choose an SRS of businesses listed in the telephone book’s Yellow Pages. Experience
shows that only about half the businesses you contact will respond.
(a) If you contact 150 businesses, it is reasonable to use the B(150, 0.5) distribution
for the number X who respond. Explain why.
(b) What is the expected number (the mean) who will respond?
(c) What is the probability that 70 or fewer will respond? (Use software or the
Normal approximation.)
(d) How large a sample must you take to increase the mean number of respondents
to 100?
5.39 Your mail-order company advertises that it ships 90% of its orders within three
working days. You select an SRS of 100 of the 5000 orders received in the past week
for an audit. The audit reveals that 86 of these orders were shipped on time.
(a) What is the sample proportion of orders shipped on time?
(b) If the company really ships 90% of its orders on time, what is the probability
that the proportion in an SRS of 100 orders is as small as the proportion in your
sample or smaller? (Use software or the Normal approximation.)
(c) A critic says, “Aha! You claim 90%, but in your sample the on-time percentage
is lower than that. So the 90% claim is wrong.” Explain in simple language why
your probability calculation in (b) shows that the result of the sample does not
refute the 90% claim.
5.40 You operate a restaurant. You read that a sample survey by the National
Restaurant Association shows that 40% of adults are committed to eating nutritious
food when eating away from home. To help plan your menu, you decide to conduct
a sample survey in your own area. You will use random digit dialing to contact an
SRS of 200 households by telephone.
(a) If the national result holds in your area, it is reasonable to use the B(200, 0.4)
distribution to describe the count X of respondents who seek nutritious food when
eating out. Explain why.
(b) What is the mean number of nutrition-conscious people in your sample if p = 0.4
is true? What is the probability that X lies between 75 and 85?
(c) You find 100 of your 200 respondents concerned about nutrition. Is this reason
to believe that the percent in your area is higher than the national 40%? To answer
this question, find the probability that X is 100 or larger if p = 0.4 is true. If this
probability is very small, that is reason to think that p is actually greater than 0.4.
5.41 “How would you describe your own physical health at this time? Would you
say your physical health is—excellent, good, only fair, or poor?” The Gallup Poll
asked this question of 1005 randomly selected adults, of whom 29% said “excellent.”
(David W. Moore and Joseph Carroll, “Most Americans call their physical and
mental health ‘good’ or ‘excellent,’ ” www.gallup.com/poll/releases/, November
28, 2001.) Suppose that in fact the proportion of the adult population who say their
health is excellent is p = 0.29.
(a) What is the probability that the sample proportion p̂ of an SRS of size n = 1000
who say their health is excellent lies between 26% and 32%? (That is, within ±3%
of the truth about the population.)
(b) Repeat the probability calculation of (a) for SRSs of sizes n = 250 and n = 4000.
What general conclusion can you draw from your calculations?
5.42 “How would you describe your own personal weight situation right now—
very overweight, somewhat overweight, about right, somewhat underweight, or very
underweight?” When the Gallup Poll asked an SRS of 1005 adults this question, 51%
answered “about right.” (Lydia Saad, “Hold the gravy! Six in ten Americans want
to lose weight,” www.gallup.com/poll/releases/, November 21, 2001.) Suppose
that in fact 51% of the entire adult population think their weight is about right.
(a) Many opinion polls announce a “margin of error” of about ±3%. What is the
probability that an SRS of size 1005 has a sample proportion p̂ that is within ±3%
(±0.03) of the population proportion p = 0.51?
(b) Answer the same question if the population proportion is p = 0.06. (This is
the proportion who told Gallup that they were “very overweight.”) How does the
probability change as p moves from near 0.5 to near zero?
5.43 A student organization is planning to ask a sample of 50 students if they

have noticed alcohol abuse brochures on campus. The sample percentage who say
“Yes” will be reported. The organization’s statistical adviser says that the standard
deviation of this percentage will be about 7%.
(a) What would the standard deviation be if the sample contained 100 students
rather than 50?
(b) How large a sample is required to reduce the standard deviation of the percentage
who say “Yes” from 7% to 3.5%? Explain to someone who knows no statistics the
advantage of taking a larger sample in a survey of opinion.
5.44 A national opinion poll found that 44% of all American adults agree that par-
ents should be given vouchers good for education at any public or private school of
their choice. Suppose that in fact the population proportion who feel this way is
p = 0.44.
(a) Many opinion polls have a “margin of error” of about ±3%. What is the proba-
bility that an SRS of size 300 has a sample proportion p̂ that is within ±3% of the
population proportion p = 0.44? (Use software or the Normal approximation.)
(b) Answer the same question for SRSs of sizes 600 and 1200. What is the effect of
increasing the size of the sample?
5.45 A sociology professor asks her class to observe cars having a man and a woman
in the front seat and record which of the two is the driver.
(a) Explain why it is reasonable to use the binomial distribution for the number of
male drivers in n cars if all observations are made in the same location at the same
Section 5.1 129
time of day.
(b) Explain why the binomial model may not apply if half the observations are made
outside a church on Sunday morning and half are made on campus after a dance.
(c) The professor requires students to observe 10 cars during business hours in a
retail district close to campus. Past observations have shown that the man is driv-
ing in about 85% of cars in this location. What is the probability that the man is
driving in 8 or fewer of the 10 cars?
(d) The class has 10 students, who will observe 100 cars in all. What is the proba-
bility that the man is driving in 80 or fewer of these?
5.46 A study by a federal agency concludes that polygraph (lie detector) tests
given to truthful persons have probability about 0.2 of suggesting that the person
is deceptive. (Office of Technology Assessment, Scientific Validity of Polygraph
Testing: A Research Review and Evaluation, Government Printing Office, 1983.)
(a) A firm asks 12 job applicants about thefts from previous employers, using a
polygraph to assess their truthfulness. Suppose that all 12 answer truthfully. What
is the probability that the polygraph says at least 1 is deceptive?
(b) What is the mean number among 12 truthful persons who will be classified as
deceptive? What is the standard deviation of this number?
(c) What is the probability that the number classified as deceptive is less than the
mean?
5.47 In each situation below, is it reasonable to use a binomial distribution for

the random variable X? Give reasons for your answer in each case. If a binomial
distribution applies, give the values of n and p.
(a) Most calls made at random by sample surveys don’t succeed in talking with
a live person. Of calls to New York City, only 1/12 succeed. A survey calls 500
randomly selected numbers in New York City. X is the number that reach a live
person.
(b) At peak periods, 25% of attempted log-ins to an Internet service provider fail.
Log-in attempts are independent and each has the same probability of failing. Darci
logs in repeatedly until she succeeds. X is the number of the login attempt that
finally succeeds.
(c) On a bright October day, Canada geese arrive to foul the pond at an apartment
complex at the average rate of 12 geese per hour; X is the number of geese that
arrive in the next three hours.
5.48 In each situation below, is it reasonable to use a binomial distribution for the
random variable X? Give reasons for your answer in each case.
(a) An auto manufacturer chooses one car from each hour’s production for a detailed
quality inspection. One variable recorded is the count X of finish defects (dimples,
ripples, etc.) in the car’s paint.
(b) The pool of potential jurors for a murder case contains 100 persons chosen at
random from the adult residents of a large city. Each person in the pool is asked
whether he or she opposes the death penalty; X is the number who say “Yes.”
(c) Joe buys a ticket in his state’s Pick 3 lottery game every week; X is the number
of times in a year that he wins a prize.
5.49 Typographic errors in a text are either nonword errors (as when “the” is typed
as “teh”) or word errors that result in a real but incorrect word. Spell-checking
software will catch nonword errors but not word errors. Human proofreaders catch
70% of word errors. You ask a fellow student to proofread an essay in which you
have deliberately made 20 word errors.
(a) If the student matches the usual 70% rate, what is the distribution of the number
of errors caught? What is the distribution of the number of errors missed?
(b) Missing 9 or more out of 20 errors seems a poor performance. What is the
probability that a proofreader who catches 70% of word errors misses 9 or more out
of 20?
5.50 What kinds of Web sites do males aged 18 to 34 visit most often? Pornographic
sites take first place, but about 50% of male Internet users in this age group visit
an auction site such as eBay at least once a month. Interview a random sample of
12 male Internet users aged 18 to 34.
(a) What is the distribution of the number who have visited an online auction site
in the past month?
(b) What is the probability that at least 8 of the 12 have visited an auction site in
the past month?
5.51 Return to the proofreading setting of Exercise 5.49.

(a) What is the mean number of errors caught? What is the mean number of errors
missed? You see that these two means must add to 20, the total number of errors.
(b) What is the standard deviation σ of the number of errors caught?
(c) Suppose that a proofreader catches 90% of word errors, so that p = 0.9. What
is σ in this case? What is σ if p = 0.99? What happens to the standard deviation
of a binomial distribution as the probability of a success gets close to 1?
5.52 Suppose that 50% of male Internet users aged 18 to 34 have visited an auction
site at least once in the past month.
(a) If you interview 12 at random, what is the mean of the count X who have visited
an auction site? What is the mean of the proportion p̂ in your sample who have
visited an auction site?
(b) Repeat the calculations in (a) for samples of size 120 and 1200. What happens
to the mean count of successes as the sample size increases? What happens to the
mean proportion of successes?
5.53 In the proofreading setting of Exercise 5.49, what is the smallest number of
misses m with P (X ≥ m) no larger than 0.05? You might consider m or more misses
as evidence that a proofreader actually catches fewer than 70% of word errors.
5.54 Some of the methods in this section are approximations rather than exact
probability results. We have given rules of thumb for safe use of these approxima-
tions.
(a) You are interested in attitudes toward drinking among the 75 members of a
fraternity. You choose 25 members at random to interview. One question is “Have
you had five or more drinks at one time during the last week?” Suppose that in fact
20% of the 75 members would say “Yes.” Explain why you cannot safely use the
Section 5.1 131
B(25, 0.2) distribution for the count X in your sample who say “Yes.”
(b) The National AIDS Behavioral Surveys found that 0.2% (that’s 0.002 as a deci-
mal fraction) of adult heterosexuals had both received a blood transfusion and had
a sexual partner from a group at high risk of AIDS. Suppose that this national
proportion holds for your region. Explain why you cannot safely use the Normal
approximation for the sample proportion who fall in this group when you interview
an SRS of 500 adults.
5.55 Each entry in a table of random digits like Table B has probability 0.1 of being
a 0, and digits are independent of each other.
(a) What is the probability that a group of five digits from the table will contain at
least one 0?
(b) What is the mean number of 0s in lines 40 digits long?
5.56 The Probability applet on the text CD and Web site simulates tosses of a coin.
You can choose the number of tosses n and the probability p of a head. You can
therefore use the applet to simulate binomial random variables.
In an audit of sales records, the count of misclassified sales records has the
binomial distribution with n = 15 and p = 0.08. Set these values for the number of
tosses and probability of heads in the applet. Table C shows that the probability
of getting a sample with exactly 1 misclassified record is 0.3734. This is the long-
run proportion of samples with exactly 1 bad record. Click “Toss” and “Reset”
repeatedly to simulate 20 samples. Record the number of bad records (the count of
heads) in each of the 20 samples. What proportion of the 20 samples had exactly 1
bad record? Remember that probability tells us only what happens in the long run.
5.57 In 1998, Mark McGwire of the St. Louis Cardinals hit 70 home runs, a new
major league record. Was this feat as surprising as most of us thought? In the three
seasons before 1998, McGwire hit a home run in 11.6% of his times at bat. He went
to bat 509 times in 1998. If he continues his past performance, McGwire’s home run
count in 509 times at bat has approximately the binomial distribution with n = 509
and p = 0.116.
(a) What is the mean number of home runs McGwire will hit in 509 times at bat?
(b) What is the probability that he hits 70 or more home runs?
(c) In 2001, Barry Bonds of the San Francisco Giants hit 73 home runs, breaking
McGwire’s record. This was surprising. In the three previous seasons, Bonds hit a
home run in 8.65% of his times at bat. He batted 476 times in 2001. Considering
his home run count as a binomial random variable with n = 476 and p = 0.0865,
what is the probability of 73 or more home runs?
5.58 A believer in the “random walk” theory of the behavior of stock prices thinks
that an index of stock prices has probability 0.65 of increasing in any year. Moreover,
the change in the index in any given year is not influenced by whether it rose or fell
in earlier years. Let X be the number of years among the next 6 years in which the
index rises.
(a) What are n and p in the binomial distribution of X?
(b) Give the possible values that X can take and the probability of each value. Draw
a probability histogram for the distribution of X.
(c) Find the mean of the number X of years in which the stock price index rises and
mark the mean on your probability histogram.
(d) Find the standard deviation of X. What is the probability that X takes a value
within one standard deviation of its mean?
5.59 “What do you think is the ideal number of children for a family to have?”
A Gallup Poll asked this question of 1006 randomly chosen adults. Almost half
(49%) thought two children was ideal. Suppose that p = 0.49 is exactly true for
the population of all adults. Gallup announced a margin of error of ±3 percentage
points for this poll. What is the probability that the sample proportion p̂ for an
SRS of size n = 1006 falls between 0.46 and 0.52? You see that it is likely, but not
certain, that polls like this give results that are correct within their margin of error.
5.60 Return to the Gallup Poll setting of the previous exercise. We are supposing
that the proportion of all adults who think that two children is ideal is p = 0.49.
What is the probability that a sample proportion p̂ falls between 0.46 and 0.52 (that
is, within ±3 percentage points of the true p) if the sample is an SRS of size n = 250?
Of size n = 4000? Combine these results with your work in Exercise 5.59 to make
a general statement about the effect of larger samples in a sample survey.
5.61 The changing probabilities you found in the previous two exercises are due to
the fact that the standard deviation of the sample proportion p̂ gets smaller as the
sample size n increases. If the population proportion is p = 0.49, how large a sample
is needed to reduce the standard deviation of p̂ to σp̂ = 0.005? (The 68–95–99.7 rule
then says that about 95% of all samples will have p̂ within 0.01 of the true p.)
5.62 A Gallup Poll finds that 30% of adults visited a casino in the past 12 months,
and that 6% bet on college sports. These results come from a random sample of
1011 adults. For an SRS of size n = 1011:
(a) What is the probability that the sample proportion p̂ is between 0.29 and 0.31
if the population proportion is p = 0.30?
(b) What is the probability that the sample proportion p̂ is between 0.05 and 0.07
if the population proportion is p = 0.06?
(c) How does the probability that p̂ falls within ±0.01 of the true p change as p gets
closer to 0?
5.63 The Harvard College Alcohol Study finds that 67% of college students support
efforts to “crack down on underage drinking.” The study took a sample of almost
15,000 students, so the population proportion who support a crackdown is very close
to p = 0.67. The administration of your college surveys an SRS of 100 students and
finds that 62 support a crackdown on underage drinking.
(a) What is the sample proportion who support a crackdown on underage drinking?
(b) If in fact the proportion of all students on your campus who support a crackdown
is the same as the national 67%, what is the probability that the proportion in an
SRS of 100 students is as small or smaller than the result of the administration’s
sample?
(c) A writer in the student paper says that support for a crackdown is lower on
Section 5.1 133
your campus than nationally. Write a short letter to the editor explaining why the
survey does not support this conclusion.
5.64 A selective college would like to have an entering class of 1200 students. Be-
cause not all students who are offered admission accept, the college admits more
than 1200 students. Past experience shows that about 70% of the students admit-
ted will accept. The college decides to admit 1500 students. Assuming that students
make their decisions independently, the number who accept has the B(1500, 0.7) dis-
tribution. If this number is less than 1200, the college will admit students from its
waiting list.
(a) What are the mean and the standard deviation of the number X of students
who accept?
(b) Use the Normal approximation to find the probability that at least 1000 students
accept.
(c) The college does not want more than 1200 students. What is the probability
that more than 1200 will accept?
(d) If the college decides to increase the number of admission offers to 1700, what
is the probability that more than 1200 will accept?
5.65 The mailing list of an agency that markets scuba-diving trips to the Florida
Keys contains 70% males and 30% females. The agency calls 30 people chosen at
random from its list.
(a) What is the probability that 20 of the 30 are men? (Use the binomial probability
formula.)
(b) What is the probability that the first woman is reached on the fourth call? (That
is, the first 4 calls give MMMF.)
5.66 One way of checking the effect of undercoverage, nonresponse, and other
sources of error in a sample survey is to compare the sample with known demo-
graphic facts about the population. The 2000 census found that 23,772,494 of the
209,128,094 adults (age 18 and over) in the United States called themselves “Black
or African American.”
(a) What is the population proportion p of blacks among American adults?
(b) An opinion poll chooses 1500 adults at random. What is the mean number of
blacks in such samples? (Explain the reasoning behind your calculation.)
(c) Use a Normal approximation to find the probability that such a sample will con-
tain 170 or fewer blacks. Be sure to check that you can safely use the approximation.
5.67 Here is a simple probability model for multiple-choice tests. Suppose that
each student has probability p of correctly answering a question chosen at random
from a universe of possible questions. (A strong student has a higher p than a
weak student.) The correctness of an answer to a question is independent of the
correctness of answers to other questions. Jodi is a good student for whom p = 0.75.
(a) Use the Normal approximation to find the probability that Jodi scores 70% or
lower on a 100-question test.
(b) If the test contains 250 questions, what is the probability that Jodi will score
70% or lower?
(c) How many questions must the test contain in order to reduce the standard
deviation of Jodi’s proportion of correct answers to half its value for a 100-item
test?
(d) Laura is a weaker student for whom p = 0.6. Does the answer you gave in (c)
for the standard deviation of Jodi’s score apply to Laura’s standard deviation also?
5.68 What kinds of Web sites do males aged 18 to 34 visit most often? Pornographic
sites take first place, but about 50% of male Internet users in this age group visit an
auction site such as eBay at least once a month (John Schwartz, ”Leisure pursuits
of today’s young men,” New York Times, March 29, 2004). Interview a random
sample of 15 male Internet users aged 18 to 34.
(a) What is the distribution of the number who have visited an online auction site
in the past month?
(b) What is the probability that at least 8 of the 15 have visited an auction site in
the past month?
5.69 Suppose that 50% of male Internet users aged 18 to 34 have visited an auction
site at least once in the past month.
(a) If you interview 15 at random, what is the mean of the count X who have visited
an auction site? What is the mean of the proportion p̂ in your sample who have
visited an auction site?
(b) Repeat the calculations in (a) for samples of size 150 and 1500. What happens
to the mean count of successes as the sample size increases? What happens to the
mean proportion of successes?
5.70 An opinion poll asks a sample of 500 adults whether they favor giving parents
of school-age children vouchers that can be exchanged for education at any public or
private school of their choice. Each school would be paid by the government on the
basis of how many vouchers it collected. Suppose that in fact 45% of the population
favor this idea. What is the probability that at least half of the sample are in favor?
(Assume that the sample is an SRS.)
5.71 A political activist is gathering signatures on a petition by going door-to-door

asking citizens to sign. She wants 100 signatures. Suppose that the probability of
getting a signature at each household is 1/10 and let the random variable X be the
number of households visited to collect exactly 100 signatures. Is it reasonable to
use a binomial distribution for X? If so, give n and p. If not, explain why not.
5.72 High school dropouts make up 12.1% of all Americans aged 18 to 24. A
vocational school that wants to attract dropouts mails an advertising flyer to 25,000
persons between the ages of 18 and 24.
(a) If the mailing list can be considered a random sample of the population, what
is the mean number of high school dropouts who will receive the flyer? What is the
standard deviation of this number?
(b) What is the probability that at least 3500 dropouts will receive the flyer?
5.73 According to a market research firm, 52% of all residential telephone numbers
in Los Angeles are unlisted. A telemarketing company uses random digit dialing
equipment that dials residential numbers at random, regardless of whether they are
listed in the telephone directory. The firm calls 500 numbers in Los Angeles.
(a) What is the exact distribution of the number X of unlisted numbers that are
called?
(b) Use a suitable approximation to calculate the probability that at least half of
the numbers called are unlisted.
5.74 The level of nitrogen oxides (NOX) in the exhaust of a particular car model
varies with mean 0.9 grams per mile (g/mi) and standard deviation 0.15 g/mi. A
company has 125 cars of this model in its fleet.
(a) What is the approximate distribution of the mean NOX emission level x for
these cars?
(b) What is the level L such that the probability that x is greater than L is only
0.01?
5.75 The World Health Organization MONICA Project collected health information
from random samples of adults in many nations. One question asked was, “Have
you had your blood pressure measured in the past year?” Of 186 American males
aged 25 to 34, 153 said “Yes.” For females in the same age group, 235 of the sample
of size 248 said “Yes.” (From the Web site of the Finnish National Public Health
Institute, www.ktl.fi/monica/.) Let us suppose that for the entire population in
this age group, 82% of men and 95% of women have had their blood pressure mea-
sured in the past year.
(a) What is the approximate distribution of the proportion p̂1 of “Yes” responses in
an SRS of 186 men? Of the corresponding proportion p̂2 for an SRS of 248 women?
(b) The samples of women and men are of course independent. What is the approx-
imate distribution of the difference p̂2 − p̂1?
(c) What is the approximate probability that the female proportion exceeds the
male proportion by at least 10 percentage points?
5.76 The distribution of scores for persons over 16 years of age on the Wechsler Adult
Intelligence Scale (WAIS) is approximately Normal with mean 100 and standard
deviation 15. The WAIS is one of the most common “IQ tests” for adults.
(a) What is the probability that a randomly chosen individual has a WAIS score of
105 or higher?
(b) What are the mean and standard deviation of the average WAIS score x for an
SRS of 60 people?
(c) What is the probability that the average WAIS score of an SRS of 60 people is
105 or higher?
(d) Would your answers to any of (a), (b), or (c) be affected if the distribution of
WAIS scores in the adult population were distinctly non-Normal?
5.77 The study habits portion of the Survey of Study Habits and Attitudes (SSHA)
psychological test consists of two sets of questions. One set measures “delay avoid-
ance” and the other measures “work methods.” A subject’s study habits score is
the sum X + Y of the delay avoidance score X and the work methods score Y .
The distribution of X in a broad population of first-year college students is close to

N (25, 10), while the distribution of Y in the same population is close to N (25, 9).
(a) If a subject’s X and Y scores were independent, what would be the distribution
of the study habits score X + Y ?
(b) Using the distribution you found in (a), what percent of the population have a
study habits score of 60 or higher?
(c) In fact, the X and Y scores are strongly correlated. In this case, does the mean
of X + Y still have the value you found in (a)? Does the standard deviation still
have the value you found in (a)?
5.78 The probability that a randomly chosen driver will be involved in an accident
in the next year is about 0.2. This is based on the proportion of millions of drivers
who have accidents. “Accident” includes things like crumpling a fender in your own
driveway, not just highway accidents. Carlos, David, Jermaine, Ramon, Scott, and
Sean are college students who live together in an off-campus apartment. Last year,
3 of the 6 had accidents. What is the probability that 3 or more of 6 randomly
chosen drivers have an accident in the same year? Why does your calculation not
apply to drivers like the 6 students?
5.79 Consider that the total SAT scores of high school seniors in a recent year had
mean µ = 1026 and standard deviation σ = 209. The distribution of SAT scores is
roughly Normal.
(a) Ramon scored 1100. If scores have a Normal distribution, what percentile of the
distribution is this?
(b) Now consider the mean x of the scores of 70 randomly chosen students. If
x = 1100, what percentile of the sampling distribution of x is this?
(c) Which of your calculations, (a) or (b), is less accurate because SAT scores do
not have an exactly Normal distribution? Explain your answer.
5.80 Although cities encourage carpooling to reduce traffic congestion, most vehicles
carry only one person. For example, 70% of vehicles on the roads in the Minneapolis-
St. Paul metropolitan area are occupied by just the driver.
(a) If you choose 10 vehicles at random, what is the probability that more than half
(that is, 6 or more) carry just one person?
(b) If you choose 100 vehicles at random, what is the probability that more than
half (that is, 51 or more) carry just one person?
5.81 The Census Bureau says that the 10 most common last names in the United
States are (in order) Smith, Johnson, Williams, Jones, Brown, Davis, Miller, Wilson,
Moore, and Taylor. These names account for 5.6% of all U.S. residents. Out of
curiosity, you look at the authors of the textbooks for your current courses. There
are 9 authors in all. Would you be surprised if none of the names of these authors
were among the 10 most common? Give a probability to support your answer and
explain the reasoning behind your calculation.
5.82 It is a striking fact that the first digits of numbers in legitimate records often
follow a distribution known as Benford’s law. Here it is:
First digit 1 2 3 4 5 6 7 8 9
Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
Fake records usually have fewer first digits 1, 2, and 3. What is the approximate
probability, if Benford’s law holds, that among 1200 randomly chosen invoices there
are no more than 680 in amounts with first digit 1, 2, or 3?
5.83 According to genetic theory, the blossom color in the second generation of a
certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each
plant has probability 3/4 of having red blossoms, and the blossom colors of separate
plants are independent.
(a) What is the probability that exactly 6 out of 8 of these plants have red blossoms?
(b) What is the mean number of red-blossomed plants when 80 plants of this type
are grown from seeds?
(c) What is the probability of obtaining at least 50 red-blossomed plants when 80
plants are grown from seeds?
5.84 The weight of the eggs produced by a certain breed of hen is Normally dis-
tributed with mean 65 grams (g) and standard deviation 5 g. If cartons of such eggs
can be considered to be SRSs of size 12 from the population of all eggs, what is the
probability that the weight of a carton falls between 750 g and 825 g?
5.85 Suppose (as is roughly true) that 20% of all Internet users have posted photos
online. A sample survey interviews an SRS of 1555 Internet users.
(a) What is the actual distribution of the number X in the sample who have posted
photos online?
(b) What is the probability that 300 or fewer of the people in the sample have posted
photos online? (Use either software or a suitable approximation.)
5.86 A study of rush-hour traffic in San Francisco records the number of people in
each car entering a freeway at a suburban interchange. Suppose that this number
X has mean 1.5 and standard deviation 0.75 in the population of all cars that enter
at this interchange during rush hours.
(a) Does the count X have a binomial distribution? Why or why not?
(b) Could the exact distribution of X be Normal? Why or why not?
(c) Traffic engineers estimate that the capacity of the interchange is 700 cars per
hour. According to the central limit theorem, what is the approximate distribution
of the mean number of persons x in 700 randomly selected cars at this interchange?
(d) The count of people in 700 cars is 700x. Use your result from (c) to give an
approximate distribution for the count. What is the probability that 700 cars will
carry more than 1075 people?
5.87 A sample survey interviews an SRS of 267 college women. Suppose (as is
roughly true) that 70% of all college women have been on a diet within the last 12
months. What is the probability that 75% or more of the women in the sample have
been on a diet?
5.88 A machine fastens plastic screw-on caps onto containers of motor oil. If the
machine applies more torque than the cap can withstand, the cap will break. Both
the torque applied and the strength of the caps vary. The capping machine torque
has the Normal distribution with mean 7 inch-pounds and standard deviation 0.9
inch-pounds. The cap strength (the torque that would break the cap) has the Normal
distribution with mean 10 inch-pounds and standard deviation 1.2 inch-pounds.
(a) Explain why it is reasonable to assume that the cap strength and the torque
applied by the machine are independent.
(b) What is the probability that a cap will break while being fastened by the capping
machine?
5.89 The unique colors of the cashmere sweaters your firm makes result from heating
undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical
for regulating dye uptake and hence the final color. There are 5 kettles, all of which
receive dye liquor from a common source. Past data show that pH varies according
to a Normal distribution with µ = 4.22 and σ = 0.127. You use statistical process
control to check the stability of the process. Twice each day, the pH of the liquor
in each kettle is measured, each time giving a sample of size 5. The mean pH x
is compared with “control limits” given by the 99.7 part of the 68–95–99.7 rule for
Normal distributions, namely µx ± 3σx. What are the numerical values of these
control limits for x?
5.90 Suppose (as is roughly true) that 88% of college men and 82% of college women
were employed last summer. A sample survey interviews SRSs of 500 college men
and 500 college women. The two samples are of course independent.
(a) What is the approximate distribution of the proportion p̂F of women who worked
last summer? What is the approximate distribution of the proportion p̂M of men
who worked?
(b) The survey wants to compare men and women. What is the approximate dis-
tribution of the difference in the proportions who worked, p̂M − p̂F ? Explain the
reasoning behind your answer.
(c) What is the probability that in the sample a higher proportion of women than
men worked last summer?
5.91 The probability that a randomly chosen driver will be involved in an accident
in the next year is about 0.2. This is based on the proportion of millions of drivers
who have accidents. “Accident” includes things like crumpling a fender in your own
driveway, not just highway accidents. Carlos, David, Jermaine, Ramon, and Scott
are college students who live together in an off-campus apartment. Last year, 3 of
the 5 had accidents. What is the probability that 3 or more of 5 randomly chosen
drivers have an accident in the same year? Why does your calculation not apply to
drivers like the 5 students?
5.92 Suppose that the total SAT scores of high school seniors in a recent year had
mean µ = 1026 and standard deviation σ = 209. The distribution of SAT scores is
roughly Normal.
(a) Julie scored 1110. If scores have a Normal distribution, what percentile of the
distribution is this?
(b) Now consider the mean x̄ of the scores of 80 randomly chosen students. If
x̄ = 1110, what percentile of the sampling distribution of x̄ is this?
(c) Which of your calculations, (a) or (b), is less accurate because SAT scores do
not have an exactly Normal distribution? Explain your answer.
5.93 Serving in a bomber crew in World War II was dangerous. The British esti-
mated that the probability of an aircraft loss due to enemy action was 1/20 for each
mission. A tour of duty for British airmen in Bomber Command was 30 missions.
What is the probability that an airman would complete a tour of duty without being
on an aircraft lost from enemy action?
5.94 A sample survey interviews an SRS of 280 college women. Suppose (as is
roughly true) that 70% of all college women have been on a diet within the last 12
months. What is the probability that 75% or more of the women in the sample have
been on a diet?
CHAPTER 6
Section 6.1
6.1 The Acculturation Rating Scale for Mexican Americans (ARSMA) is a psy-
chological test developed to measure the degree of Mexican/Spanish versus An-
glo/English acculturation of Mexican Americans. The distribution of ARSMA scores
in a population used to develop the test was approximately Normal, with mean 3.0
and standard deviation 0.8. A further study gave ARSMA to 50 first-generation
Mexican Americans. The mean of their scores is x = 2.36. If the standard deviation
for the first-generation population is also σ = 0.8, give a 95% confidence interval for
the mean ARSMA score for first-generation Mexican Americans.
6.2 The 2000 census “long form” asked the total 1999 income of the householder,
the person in whose name the dwelling unit was owned or rented. This census
form was sent to a random sample of 17% of the nation’s households. Suppose
that the households that returned the long form are an SRS of the population
of all households in each district. In Middletown, a city of 40,000 persons, 2621
householders reported their income. The mean of the responses was x = $33, 453,
and the standard deviation was s = $8721. The sample standard deviation for so
large a sample will be very close to the population standard deviation σ. Use these
facts to give an approximate 99% confidence interval for the 1999 mean income of
Middletown householders who reported income.
6.3 Refer to the previous problem. Give a 99% confidence interval for the total 1999
income of the households that reported income in Middletown.
6.4 A newspaper headline describing a poll of registered voters taken two weeks
before a recent election read “Ringel leads with 52%.” The accompanying article
describing the poll stated that the margin of error was 2% with 95% confidence.
(a) Explain in plain language to someone who knows no statistics what “95% con-
fidence” means.
(b) The poll shows Ringel leading. But the newspaper article said that the election
was too close to call. Explain why.
6.5 A student reads that a 95% confidence interval for the mean SAT Math score
of California high school seniors is 452 to 470. Asked to explain the meaning of this
interval, the student says, “95% of California high school seniors have SAT Math
scores between 452 and 470.” Is the student right? Justify your answer.
6.6 As we prepare to take a sample and compute a 95% confidence interval, we know
that the probability that the interval we compute will cover the parameter is 0.95.
That’s the meaning of 95% confidence. If we use several such intervals, however,
our confidence that all give correct results is less than 95%.
6.7 In an agricultural field trial a corn variety is planted in seven separate locations,
which may have different mean yields due to differences in soil and climate. At the
end of the experiment, seven independent 95% confidence intervals will be calculated,
140
Section 6.1 141
one for the mean yield at each location.

(a) What is the probability that every one of the seven intervals covers the true
mean yield at its location? This probability (expressed as a percent) is our overall
confidence level for the seven simultaneous statements.
(b) What is the probability that at least six of the seven intervals cover the true
mean yields?
6.8 A newspaper ad for a manager trainee position contained the statement “Our
manager trainees have a first-year earnings average of $20,000 to $24,000.” Do you
think that the ad is describing a confidence interval? Explain your answer.
6.9 A survey of users of the Internet found that males outnumbered females by
nearly 2 to 1. This was a surprise, because earlier surveys had put the ratio of men
to women closer to 9 to 1. Later in the article we find this information:
Detailed surveys were sent to more than 13,000 organizations on the
Internet; 1,468 usable responses were received. According to Mr. Quar-
terman, the margin of error is 2.8 percent, with a confidence level of 95
percent.
(a) What was the response rate for this survey? (The response rate is the percent
of the planned sample that responded.)
(b) Do you think that the small margin of error is a good measure of the accuracy
of the survey’s results? Explain your answer.
6.10 The mean amount µ for all of the invoices for your company last month is not
known. Based on your past experience, you are willing to assume that the standard
deviation of invoice amounts is about $200.
(a) If you take a random sample of 100 invoices, what is the value of the standard
deviation for x?
(b) The 68–95–99.7 rule says that the probability is about 0.95 that x is within
of the population mean µ. Fill in the blank.
(c) About 95% of all samples will capture the true mean of all of the invoices in the
interval x plus or minus . Fill in the blank.
6.11 You measure the weights of 24 male runners. You do not actually choose an
SRS, but you are willing to assume that these runners are a random sample from the
population of male runners in your town or city. Here are their weights in kilograms:
67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9
60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8
66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7
Suppose that the standard deviation of the population is known to be σ = 4.5 kg.
(a) What is σx , the standard deviation of x?
(b) Give a 95% confidence interval for µ, the mean of the population from which
the sample is drawn. Are you quite sure that the average weight of the population
of runners is less than 65 kg?
6.12 Suppose that you had measured the weights of the runners in the previous
exercise in pounds rather than kilograms. Use your answers to the previous exercise
and the fact that 1 kilogram equals 2.2 pounds to answer these questions.
(a) What is the mean weight of these runners?
(b) What is the standard deviation of the mean weight?
(c) Give a 95% confidence interval for the mean weight of the population of runners
that these runners represent.
6.13 Crop researchers plant 15 plots with a new variety of corn. The yields in
bushels per acre are
138.0 139.1 113.0 132.5 140.7 109.7 118.9 134.8

109.6 127.3 115.6 130.4 130.2 111.7 105.5
Assume that σ = 10 bushels per acre.

(a) Find the 90% confidence interval for the mean yield µ for this variety of corn.
(b) Find the 95% confidence interval.
(c) Find the 99% confidence interval.
(d) How do the margins of error in (a), (b), and (c) change as the confidence level
increases?
6.14 Suppose that the crop researchers in the previous exercise obtained the same
value of x from a sample of 50 plots rather than 15.
(a) Compute the 95% confidence interval for the mean yield µ.
(b) Is the margin of error larger or smaller than the margin of error found for the
sample of 15 plots in the previous exercise? Explain in plain language why the
change occurs.
(c) Will the 90% and 99% intervals for a sample of size 50 be wider or narrower
than those for n = 15? (You need not actually calculate these intervals.)
6.15 In the two previous exercises, we compared confidence intervals based on corn
yields from 15 and 50 small plots of ground. How large a sample is required to
estimate the mean yield within ±6 bushels per acre with 90% confidence?
6.16 A test for the level of potassium in the blood is not perfectly precise. Moreover,
the actual level of potassium in a person’s blood varies slightly from day to day.
Suppose that repeated measurements for the same person on different days vary
Normally with σ = 0.2.
(a) Julie’s potassium level is measured once. The result is x = 3.4. Give a 90%
confidence interval for her mean potassium level.
(b) If three measurements were taken on different days and the mean result is
x = 3.4, what is a 90% confidence interval for Julie’s mean blood potassium level?
6.17 How large a sample of Julie’s potassium levels in the previous exercise would
be needed to estimate her mean µ within ±0.06 with 95% confidence?
6.18 A study of the career paths of hotel general managers sent questionnaires to an
SRS of 160 hotels belonging to major U.S. hotel chains. There were 114 responses.
The average time these 114 general managers had spent with their current company
was 11.78 years. Give a 99% confidence interval for the mean number of years
general managers of major-chain hotels have spent with their current company.
Section 6.1 143
(Take it as known that the standard deviation of time with the company for all
general managers is 3.2 years.)
6.19 Researchers planning a study of the reading ability of third-grade children want
to obtain a 95% confidence interval for the population mean score on a reading test,
with margin of error no greater than 3 points. They carry out a small pilot study
to estimate the variability of test scores. The sample standard deviation is s = 12
points in the pilot study, so in preliminary calculations the researchers take the
population standard deviation to be σ = 12.
(a) The study budget will allow as many as 100 students. Calculate the margin of
error of the 95% confidence interval for the population mean based on n = 100.
(b) There are many other demands on the research budget. If all of these demands
were met, there would be funds to measure only 10 children. What is the margin of
error of the confidence interval based on n = 10 measurements?
(c) Find the smallest value of n that would satisfy the goal of a 95% confidence
interval with margin of error 3 or less. Is this sample size within the limits of the
budget?
6.20 The Gallup Poll asked 1571 adults what they considered to be the most serious
problem facing the nation’s public schools; 30% said drugs. This sample percent
is an estimate of the percent of all adults who think that drugs are the schools’
most serious problem. The news article reporting the poll result adds, “The poll
has a margin of error—the measure of its statistical accuracy—of three percentage
points in either direction; aside from this imprecision inherent in using a sample to
represent the whole, such practical factors as the wording of questions can affect
how closely a poll reflects the opinion of the public in general.”
The Gallup Poll uses a complex multistage sample design, but the sample per-
cent has approximately a Normal distribution. Moreover, it is standard practice to
announce the margin of error for a 95% confidence interval unless a different confi-
dence level is stated.
(a) The announced poll result was 30% ± 3%. Can we be certain that the true
population percent falls in this interval?
(b) Explain to someone who knows no statistics what the announced result 30%±3%
means.
(c) This confidence interval has the same form we have met earlier:
estimate ± z ∗ σestimate
(Actually σ is estimated from the data, but we ignore this for now.) What is the
standard deviation σestimate of the estimated percent?
(d) Does the announced margin of error include errors due to practical problems
such as undercoverage and nonresponse?
6.21 When the statistic that estimates an unknown parameter has a Normal distri-
bution, a confidence interval for the parameter has the form
estimate ± z ∗ σestimate
In a complex sample survey design, the appropriate unbiased estimate of the pop-
ulation mean and the standard deviation of this estimate may require elaborate
computations. But when the estimate is known to have a Normal distribution and
its standard deviation is given, we can calculate a confidence interval for µ from
complex sample designs without knowing the formulas that led to the numbers
given.
A report based on the Current Population Survey estimates the 1999 median
annual earnings of households as $40,816 and also estimates that the standard de-
viation of this estimate is $191. The Current Population Survey uses an elaborate
multistage sampling design to select a sample of about 50,000 households. The sam-
pling distribution of the estimated median income is approximately Normal. Give
a 95% confidence interval for the 1999 median annual earnings of households.
6.22 The previous problem reports data on the median household income for the
entire United States. In a detailed report based on the same sample survey, you find
that the estimated median income for four-person families in Michigan is $65,467.
Is the margin of error for this estimate with 95% confidence greater or less than the
margin of error for the national median? Why?
6.23 The Bowl Championship Series (BCS) was designed to select the top two teams
in college football for a final championship game. The teams are selected by a com-
plicated formula. In 2001, the University of Miami Hurricanes and the University
of Nebraska Cornhuskers played for the championship. However, many football fans
thought that Nebraska should not have played in the game because it was rated
only fourth in both major opinion polls. Third-ranked University of Colorado fans
were particularly upset because Colorado soundly beat the Cornhuskers late in the
season. A new CNN/USA Today/Gallup Poll reports that a majority of fans would
prefer a national championship play-off as an alternative to the BCS. The news
media polled a random sample of 1019 adults 18 years of age or older. A summary
of the results states that 54% prefer the play-off, and the margin of error is 3% for
95% confidence.
(a) Give the 95% confidence interval.
(b) Do you think that a newspaper headline stating that a majority of fans prefer a
play-off is justified by the results of this study? Explain your answer.
6.24 An advertisement in the student newspaper asks you to consider working for a
telemarketing company. The ad states, “Earn between $500 and $1000 per week.”
Do you think that the ad is describing a confidence interval? Explain your answer.
6.25 A New York Times poll on women’s issues interviewed 1025 women and 472
men randomly selected from the United States, excluding Alaska and Hawaii. The
poll found that 47% of the women said they do not get enough time for themselves.
(a) The poll announced a margin of error of ±3 percentage points for 95% confidence
in conclusions about women. Explain to someone who knows no statistics why we
can’t just say that 47% of all adult women do not get enough time for themselves.
(b) Then explain clearly what “95% confidence” means.
(c) The margin of error for results concerning men was ±4 percentage points. Why
is this larger than the margin of error for women?
Section 6.1 145
6.26 A radio talk show invites listeners to enter a dispute about a proposed pay
increase for city council members. “What yearly pay do you think council members
should get? Call us with your number.” In all, 958 people call. The mean pay
they suggest is x = $9740 per year, and the standard deviation of the responses is
s = $1125. For a large sample such as this, s is very close to the unknown population
σ. The station calculates the 95% confidence interval for the mean pay µ that all
citizens would propose for council members to be $9669 to $9811. Is this result
trustworthy? Explain your answer.
6.27 A study based on a sample of size 25 reported a mean of 76 with a margin of

error of 12 for 95% confidence. Give the 95% confidence interval.
6.28 Refer to the previous exercise. If you wanted 99% confidence for the same study,
would your margin of error be greater than, equal to, or less than 12? Explain your
answer.
6.29 Suppose that the sample mean is 50 and the standard deviation is assumed to
be 5. Make a diagram that illustrates the effect of sample size on the width of a
95% interval. Use the following sample sizes: 10, 20, 40, and 100. Summarize what
the diagram shows.
6.30 A study with 25 observations gave a mean of 70. Assume that the standard
deviation is 15. Make a diagram that illustrates the effect of the confidence level
on the width of the interval. Use 80%, 90%, 95%, and 99%. Summarize what the
diagram shows.
6.31 Consider the following two scenarios. (A) Take a simple random sample of 100
students from an elementary school with children in grades kindergarten through
fifth grade; (B) Take a simple random sample of 100 third-graders from the same
school. For each of these samples you will measure the height of each child in the
sample. Which sample should have the smaller margin of error for 95% confidence?
Explain your answer.
6.32 You want to rent an unfurnished one-bedroom apartment for next semester.
The mean monthly rent for a random sample of 10 apartments advertised in the local
newspaper is $580. Assume that the standard deviation is $90. Find a 95% confi-
dence interval for the mean monthly rent for unfurnished one-bedroom apartments
available for rent in this community.
6.33 A questionnaire about study habits was given to a random sample of students
taking a large introductory statistics class. The sample of 25 students reported that
they spent an average of 80 minutes per week studying statistics. Assume that the
standard deviation is 35 minutes.
(a) Give a 95% confidence interval for the mean time spent studying statistics by
students in this class.
(b) Is it true that 95% of the students in the class have weekly study times that lie
in the interval you found in part (a)? Explain your answer.
6.34 Refer to the previous exercise.

(a) Give the mean and standard deviation in hours.
(b) Calculate the 95% confidence interval in hours from your answer to part (a).
(c) Explain how you could have directly calculated this interval from the 95% interval
that you calculated in the previous exercise.
6.35 Computers in some vehicles calculate various quantities related to performance.

One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon
(mpg). For one vehicle equipped in this way, the mpg were recorded each time the
gas tank was filled, and the computer was then reset. Here are the mpg values for
a random sample of 20 of these records:
15.8 13.6 15.6 19.1 22.4 15.6 22.5 17.2 19.4 22.6
19.4 18.0 14.6 18.7 21.0 14.8 22.6 21.5 14.3 20.9
Suppose that the standard deviation of the population is known to be σ = 2.9 mpg.
(a) What is σx , the standard deviation of x?
(b) Give a 95% confidence interval for µ, the mean mpg for this vehicle.
6.36 Refer to the previous exercise. Here are the values of the average speed in
miles per hour (mph) for the same sample:
21.0 19.0 18.7 39.2 45.8 19.8 48.4 21.0 29.1 35.7
31.6 49.0 16.0 34.6 36.3 19.0 43.3 37.5 16.5 34.5
Assume that the standard deviation is 10.3 mph. Estimate the mean speed at which
this vehicle was driven with a margin of error for 95% confidence.
6.37 Here are the Degree of Reading Power (DRP) scores for a sample of 44 third-
grade students:
40 26 39 14 42 18 25 43 46 27 19
47 19 26 35 34 15 44 40 38 31 46
52 25 35 35 33 29 34 41 49 28 52
47 35 48 22 33 41 51 27 14 54 45
Suppose that the standard deviation of the population of DRP scores is known to
be σ = 11. Give a 95% confidence interval for the population mean score.
6.38 You are planning a survey of starting salaries for recent liberal arts major
graduates from your college. From a pilot study you estimate that the standard
deviation is about $9000. What sample size do you need to have a margin of error
equal to $400 with 95% confidence?
6.39 Suppose that in the setting of the previous exercise you are willing to settle
for a margin of error of $800. Will the required sample size be larger or smaller?
Verify your answer by performing the calculations.
6.40 How large a sample of one-bedroom apartments in Exercise 6.32 would be

needed to estimate the mean µ within ±$20 with 90% confidence?
Section 6.1 147
6.41 A newspaper invites readers to send email stating whether or not they are in
favor of making full-day kindergarten available to all students in the state. A total
of 320 responses are received and, of these, 80% are in favor of the new program.
In an article describing the results, the authors state that the margin of error is 4%
for 95% confidence. Assume that they have computed this number correctly.
(a) Use the sample proportion and the margin of error to compute the 95% confi-
dence interval.
(b) Do you think that these results are trustworthy? Discuss your answer.
6.42 A recent Gallup Poll conducted telephone interviews with a random sample of
adults aged 18 and older. Data were obtained for 1011 people. Of these, 37% said
that football is their favorite sport to watch on television.
(a) The poll announced a margin of error of ±3 percentage points for 95% confidence.
Explain to someone who knows no statistics why we can’t just say that 37% of
Americans would say that football is their favorite sport to watch on television.
(b) Then explain clearly what “95% confidence” means.
(c) Give the 95% confidence interval.
(d) The poll was taken in December, an exciting part of the football season. Do you
think that a similar poll conducted in June might produce different results? Explain
why or why not.
6.43 A survey of users of the Internet found that males outnumbered females by
nearly 2 to 1 (P. H. Lewis, “Technology” column, New York Times, May 29, 1995).
This was a surprise, because earlier surveys had put the ratio of men to women
closer to 9 to 1. Later in the article we find this information:
Detailed surveys were sent to more than 13,000 organizations on the

Internet; 1,468 usable responses were received. According to Mr. Quar-
terman, the margin of error is 2.8 percent, with a confidence level of 95
percent.
Do you think that the small margin of error is a good measure of the accuracy of
the survey’s results? Explain your answer.
6.44 The mean number of calories consumed by women in the United States who
are 19 to 30 years of age is µ = 1791 calories per day. The standard deviation is
31 calories (Dietary Reference Intakes for Energy, Carbohydrate, Fiber, Fat, Fatty
Acids, Cholesterol, Protein, and Amino Acids (Macronutrients), Food and Nutrition
Board, Institute of Medicine, 2002). You will study a sample of 200 women in this
age range, and one of the variables you will collect is calories consumed per day.
(a) What is the standard deviation of the sample mean x̄?
(b) The 68–95–99.7 rule says that the probability is about 0.95 that x is within
calories of the population mean µ. Fill in the blank.
(c) About 95% of all samples will capture the true mean of calories consumed per
day in the interval x plus or minus calories. Fill in the blank.
6.45 A Gallup Poll asked working adults about their job satisfaction (Chris Cham-
bers, “Americans skeptical about mergers of big companies,” Gallup News Service,
November 1, 2000). One question was “All in all, which best describes how you feel
about your job?” The possible answers were “love job,” “like job,” “dislike job,”
and “hate job.” Fifty-nine percent of the sample responded that they liked their
job. Material provided with the results of the poll noted:
Results are based on telephone interviews with 1,001 national adults,

aged 18 and older, conducted Aug. 8–11, 2005. For results based on the
total sample of national adults, one can say with 95% confidence that the
maximum margin of sampling error is 3 percentage points.
The Gallup Poll uses a complex multistage sample design, but the sample percent
has approximately a Normal sampling distribution.
(a) The announced poll result was 59% ± 3%. Can we be certain that the true
population percent falls in this interval?
(b) Explain to someone who knows no statistics what the announced result 59%±3%
means.
(c) This confidence interval has the same form we have met earlier:
estimate ± z ∗σestimate
What is the standard deviation σestimate of the estimated percent?

(d) Does the announced margin of error include errors due to practical problems
such as undercoverage and nonresponse?
Section 6.2
6.46 Each of the following situations requires a significance test about a population
mean µ. State the appropriate null hypothesis H0 and alternative hypothesis Ha in
each case.
(a) The mean area of the several thousand apartments in a new development is
advertised to be 1250 square feet. A tenant group thinks that the apartments are
smaller than advertised. They hire an engineer to measure a sample of apartments
to test their suspicion.
(b) Larry’s car averages 32 miles per gallon on the highway. He now switches to
a new motor oil that is advertised as increasing gas mileage. After driving 3000
highway miles with the new oil, he wants to determine if his gas mileage actually
has increased.
(c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If
the spindle is either too small or too large, the motor will not perform properly. The
manufacturer measures the diameter in a sample of motors to determine whether
the mean diameter has moved away from the target.
6.47 In each of the following situations, a significance test for a population mean
µ is called for. State the null hypothesis H0 and the alternative hypothesis Ha in
each case.
(a) Experiments on learning in animals sometimes measure how long it takes a mouse
to find its way through a maze. The mean time is 18 seconds for one particular maze.
A researcher thinks that a loud noise will cause the mice to complete the maze faster.
She measures how long each of 10 mice takes with a noise as stimulus.
Section 6.2 149
(b) The examinations in a large accounting class are scaled after grading so that
the mean score is 50. A self-confident teaching assistant thinks that his students
have a higher mean score than the class as a whole. His students this semester can
be considered a sample from the population of all students he might teach, so he
compares their mean score with 50.
(c) A university gives credit in French language courses to students who pass a
placement test. The language department wants to know if students who get credit
in this way differ in their understanding of spoken French from students who actually
take the French courses. Some faculty think the students who test out of the courses
are better, but others argue that they are weaker in oral comprehension. Experience
has shown that the mean score of students in the courses on a standard listening
test is 24. The language department gives the same listening test to a sample of 40
students who passed the credit examination to see if their performance is different.
6.48 You have performed a two-sided test of significance and obtained a value of
z = 3.3. Use Table D to find the approximate P -value for this test.
6.49 You have performed a one-sided test of significance and obtained a value of
6.50 An understanding of cockroach biology may lead to an effective control strategy

for these annoying insects. Researchers studying the absorption of sugar by insects
feed cockroaches a diet containing measured amounts of a particular sugar. After 10
hours, the cockroaches are killed and the concentration of the sugar in various body
parts is determined by a chemical analysis. The paper that reports the research
states that a 95% confidence interval for the mean amount (in milligrams) of the
sugar in the hindguts of the cockroaches is 4.2 ± 2.3. (From D. L. Shankland et al.,
“The effect of 5-thio-D-glucose on insect development and its absorption by insects,”
Journal of Insect Physiology, 14 (1968), pp. 63–72.)
(a) Does this paper give evidence that the mean amount of sugar in the hindguts
under these conditions is not equal to 7 mg? State H0 and Ha and base a test on
the confidence interval.
(b) Would the hypothesis that µ = 5 mg be rejected at the 5% level in favor of a
two-sided alternative?
6.51 Market pioneers, companies that are among the first to develop a new product
or service, tend to have higher market shares than latecomers to the market. What
accounts for this advantage? Here is an excerpt from the conclusions of a study of
a sample of 1209 manufacturers of industrial goods:
Can patent protection explain pioneer share advantages? Only 21% of

the pioneers claim a significant benefit from either a product patent or a
trade secret. Though their average share is two points higher than that of
pioneers without this benefit, the increase is not statistically significant
(z = 1.13). Thus, at least in mature industrial markets, product patents
and trade secrets have little connection to pioneer share advantages.
Find the P -value for the given z. Then explain to someone who knows no statistics
what “not statistically significant” in the study’s conclusion means. Why does the
author conclude that patents and trade secrets don’t help, even though they con-
tributed 2 percentage points to average market share? (From William T. Robinson,
“Sources of market pioneer advantages: the case of industrial goods industries,”
Journal of Marketing Research, 25 (1988), pp. 87–94.)
6.52 Each of the following situations requires a significance test about a population
mean µ. State the appropriate null hypothesis H0 and alternative hypothesis Ha in
each case.
(a) A dual X-ray absorptiometry (DXA) scanner is used to measure bone mineral
density for people who may be at risk for osteoporosis. To be sure that the measure-
ments are accurate, an object called a “phantom” that has known mineral density
µ = 1.4 grams per square centimeter is measured. This phantom is scanned 10
times.
(b) Feedback from your customers shows that many think it takes too long to fill
out the online order form for your products. You redesign the form and survey a
random sample of customers to determine whether or not they think that the new
form is actually an improvement. The response uses a five-point scale: −2 if the
new form takes much less time than the old form; −1 if the new form takes a little
less time; 0 if the new form takes about the same time; +1 if the new form takes a
little more time; and +2 if the new form takes much more time.
(c) You purchase a shipment of 60-watt light bulbs to be used in a variety of your
products. If the wattage is too low or two high, your product will not look good.
You measure the wattage of a random sample of 20 bulbs.
6.53 In each of the following situations, a significance test for a population mean
µ is called for. State the null hypothesis H0 and the alternative hypothesis Ha in
each case.
(a) Experiments on learning in animals sometimes measure how long it takes a mouse
to find its way through a maze. The mean time is 18 seconds for one particular maze.
A researcher thinks that a loud noise will cause the mice to complete the maze faster.
She measures how long each of 10 mice takes with a noise as stimulus.
(b) The examinations in a large history class are scaled after grading so that the
mean score is 50. A self-confident teaching assistant thinks that his students have
a higher mean score than the class as a whole. His students this semester can
(c) The Census Bureau reports that households spend an average of 31% of their
total spending on housing. A homebuilders association in Cleveland wonders if the
national finding applies in their area. They interview a sample of 40 households in
the Cleveland metropolitan area to learn what percent of their spending goes toward
housing.
6.54 In each of the following situations, state an appropriate null hypothesis H0

and alternative hypothesis Ha . Be sure to identify the parameters that you use to
state the hypotheses. (We have not yet learned how to test these hypotheses.)
(a) A sociologist asks a large sample of high school students which academic subject
they like best. She suspects that a higher percent of males than of females will name
mathematics as their favorite subject.
Section 6.2 151
(b) An education researcher randomly divides sixth-grade students into two groups
for physical education class. He teaches both groups basketball skills, using the same
methods of instruction in both classes. He encourages Group A with compliments
and other positive behavior but acts cool and neutral toward Group B. He hopes
to show that positive teacher attitudes result in a higher mean score on a test of
basketball skills than do neutral attitudes.
(c) An economist believes that among employed young adults there is a positive
correlation between income and the percent of disposable income that is saved. To
test this, she gathers income and savings data from a sample of employed persons
in her city aged 25 to 34.
6.55 A test of the null hypothesis H0: µ = µ0 gives test statistic z = 1.8.
(a) What is the P -value if the alternative is Ha: µ > µ0 ?
(b) What is the P -value if the alternative is Ha: µ < µ0 ?
(c) What is the P -value if the alternative is Ha: µ 6= µ0 ?
6.56 The P -value for a two-sided test of the null hypothesis H0: µ = 10 is 0.06.
(a) Does the 95% confidence interval include the value 10? Why?
(b) Does the 90% confidence interval include the value 10? Why?
6.57 A 95% confidence interval for a population mean is (28, 35).

(a) Can you reject the null hypothesis that µ = 34 at the 5% significance level?
Why?
(b) Can you reject the null hypothesis that µ = 36 at the 5% significance level?
Why?
6.58 A new supplier offers a good price on a catalyst used in your production process.
You compare the purity of this catalyst with that from your current supplier. The
P -value for a test of “no difference” is 0.15. Can you be confident that the purity of
the new product is the same as the purity of the product that you have been using?
Discuss.
6.59 We often see televised reports of brushfires threatening homes in California.

Some people argue that the modern practice of quickly putting out small fires allows
fuel to accumulate and so increases the damage done by large fires. A detailed study
of historical data suggests that this is wrong—the damage has risen simply because
there are more houses in risky areas. (Jon E. Keeley, C. J. Fotheringham, and Marco
Morais, “Reexamining fire suppression impacts on brushland fire regimes,” Science,
284 (1999), pp. 1829–1831.) As usual, the study report gives statistical information
tersely. Here is the summary of a regression of number of fires on decade (9 data
points, for the 1910s to the 1990s):
Collectively, since 1910, there has been a highly significant increase (r2 =
0.61, P < 0.01) in the number of fires per decade.
How would you explain this statement to someone who knows no statistics? Include
an explanation of both the description given by r2 and the statistical significance.
6.60 A randomized comparative experiment examined whether a calcium supple-

ment in the diet reduces the blood pressure of healthy men. The subjects received
either a calcium supplement or a placebo for 12 weeks. The statistical analysis was
quite complex, but one conclusion was that “the calcium group had lower seated
systolic blood pressure (P = 0.008) compared with the placebo group.” (R. M.
Lyle et al., “Blood pressure and metabolic effects of calcium supplementation in
normotensive white and black men,” Journal of the American Medical Association,
257 (1987), pp. 1772–1776.) Explain this conclusion, especially the P -value, as if
you were speaking to a doctor who knows no statistics.
6.61 A social psychologist reports that “ethnocentrism was significantly higher (P <
0.05) among church attenders than among nonattenders.” Explain what this means
in language understandable to someone who knows no statistics. Do not use the
word “significance” in your answer.
6.62 A study examined the effect of exercise on how students perform on their final
exam in statistics. The P -value was given as 0.87.
(a) State null and alternative hypotheses that could have been used for this study.
(Note that there is more than one correct answer.)
(b) Do you reject the null hypothesis?
(c) What is your conclusion?
(d) What other facts about the study would you like to know for a proper interpre-
tation of the results?
6.63 The financial aid office of a university asks a sample of students about their
employment and earnings. The report says that “for academic year earnings, a
significant difference (P = 0.038) was found between the sexes, with men earning
more on the average. No difference (P = 0.476) was found between the earnings
of black and white students.” (From a study by M. R. Schlatter et al., Division of
Financial Aid, Purdue University.) Explain both of these conclusions, for the effects
of sex and of race on mean earnings, in language understandable to someone who
knows no statistics.
6.64 The mean yield of corn in the United States is about 120 bushels per acre. A
survey of 40 farmers this year gives a sample mean yield of x = 123.8 bushels per
acre. We want to know whether this is good evidence that the national mean this
year is not 120 bushels per acre. Assume that the farmers surveyed are an SRS from
the population of all commercial corn growers and that the standard deviation of
the yield in this population is σ = 10 bushels per acre. Give the P -value for the
test of
H0: µ = 120
Ha: µ 6= 120
Are you convinced that the population mean is not 120 bushels per acre? Is your
conclusion correct if the distribution of corn yields is somewhat non-Normal? Why?
6.65 In the past, the mean score of the seniors at South High on the American
College Testing (ACT) college entrance examination has been 20. This year a special
preparation course is offered, and all 53 seniors planning to take the ACT test enroll
in the course. The mean of their 53 ACT scores is 22.1. The principal believes that
the new course has improved the students’ ACT scores.
Section 6.2 153
(a) Assume that ACT scores vary Normally with standard deviation 6. Is the
outcome x = 22.1 good evidence that the population mean score is greater than
20? State H0 and Ha , compute the test statistic and the P -value, and answer the
question by interpreting your result.
(b) The results are in any case inconclusive because of the design of the study. The
effects of the new course are confounded with any change from past years, such as
other new courses or higher standards. Briefly outline the design of a better study
of the effect of the new course on ACT scores.
6.66 There are other z statistics that we have not yet studied. We can use Table D to
assess the significance of any z statistic. A study compares the habits of students who
are on academic probation with students whose grades are satisfactory. One variable
measured is the hours spent watching television last week. The null hypothesis is “no
difference” between the means for the two populations. The alternative hypothesis
is two-sided. The value of the test statistic is z = −1.37.
(a) Is the result significant at the 5% level?
(b) Is the result significant at the 1% level?
6.67 You measure the weights of 24 male runners. These runners are not a random
sample from a population, but you are willing to assume that their weights represent
the weights of similar runners. Here are their weights in kilograms:
67.8 61.9 63.0 53.1 62.3 59.7 55.4 58.9

60.9 69.2 63.7 68.3 64.7 65.6 56.0 57.8
66.0 62.9 53.6 65.0 55.8 60.4 69.3 61.7
Exercise 6.11 asks you to find a 95% confidence interval for the mean weight of the
population of all such runners, assuming that the population standard deviation is
σ = 4.5 kg.
(a) Give the confidence interval from that exercise, or calculate the interval if you
did not do the exercise.
(b) Based on this confidence interval, does a test of
H0: µ = 61.3 kg
Ha: µ 6= 61.3 kg
reject H0 at the 5% significance level?
(c) Would H0: µ = 63 be rejected at the 5% level if tested against a two-sided
alternative?
6.68 An old farmer claims to be able to detect the presence of water with a forked
stick. In a test of this claim, he is presented with 5 identical barrels, some containing
water and some not. He is right in 4 of the 5 cases.
(a) Suppose the farmer has probability p of being correct. If he is just guessing,
p = 0.5. State an appropriate H0 and Ha in terms of p for a test of whether he does
better than guessing.
(b) If the farmer is simply guessing, what is the distribution of the number X of
correct answers in 5 tries?
(c) The observed outcome is X = 4. What is the P -value of the test that takes large
values of X to be evidence against H0?
6.69 Here are several situations where there is an incorrect application of the ideas
presented in this section. Write a short paragraph explaining what is wrong in each
situation and why it is wrong.
(a) A climatologist wants to test the null hypothesis that it will rain tomorrow.
(b) A random sample of size 20 is taken from a population that is assumed to have
a standard deviation of 15. The standard deviation of the sample mean is 15/20.
(c) A researcher tests the following null hypothesis: H0: x̄ = 10.
6.70 Here are several situations where there is an incorrect application of the ideas
presented in this section. Write a short paragraph explaining what is wrong in each
situation and why it is wrong.
(a) A change is made that should improve student satisfaction with the way grades
are processed at your college. The null hypothesis, that there is an improvement, is
tested versus the alternative, that there is no change.
(b) A significance test rejected the null hypothesis that the sample mean is 25.
(c) A report on a study says that the results are statistically significant and the
P -value is 0.95.
6.71 State the appropriate null hypothesis H0 and alternative hypothesis Ha in each
of the following cases.
(a) An experiment is designed to examine the effect of a diet high in soy products
on the bone density of adult rats.
(b) The student newspaper at your college recently changed the format for their
news stories. You take a random sample of students and select those who regularly
read the newspaper. These are asked to indicate their opinions on the changes using
a five-point scale: −2 if the new format is much worse than the old, −1 if the new
format is somewhat worse than the old, 0 if the new format is the same as the old,
+1 if the new format is somewhat better than the old, and +2 if the new format is
much better than the old.
(c) The examinations in a large history class are scaled after grading so that the
mean score is 75. A self-confident teaching assistant thinks that his students have
a higher mean score than the class as a whole. His students this semester can
6.72 State the null hypothesis H0 and the alternative hypothesis Ha in each case.
(a) A national study reports that households spend an average of 30% of their food
expenditures in restaurants. A restaurant association in your area wonders if the
national finding applies locally. They interview a sample of 40 households and ask
about their total food budget and the amount spent in restaurants.
(b) Experiments on learning in animals sometimes measure how long it takes a
mouse to find its way through a maze. The mean time is 20 seconds for one par-
ticular maze. A researcher thinks that playing rap music will cause the mice to
complete the maze faster. She measures how long each of 12 mice takes with the
rap music as a stimulus.
Section 6.2 155
(c) A dual X-ray absorptiometry (DXA) scanner is used to measure bone mineral
density for people who may be at risk for osteoporosis. To be sure that the measure-
ments are accurate, an object called a “phantom” that has known mineral density
µ = 1.3 grams per square centimeter is measured. This phantom is scanned 8 times.
6.73 In each of the following situations, state an appropriate null hypothesis H0

and alternative hypothesis Ha . Be sure to identify the parameters that you use to
state the hypotheses. (We have not yet learned how to test these hypotheses.)
(a) An education researcher randomly divides sixth-grade students into two groups
for physical education class. He teaches both groups volleyball skills, using the same
methods of instruction in both classes. He encourages Group A with compliments
and other positive behavior but acts cool and neutral toward Group B. He hopes
to show that positive teacher attitudes result in a higher mean score on a test of
volleyball skills than do neutral attitudes.
(b) An education researcher believes that among college students there is a positive
correlation between grade point average and self-esteem. To test this, she gathers
grade point average and self-esteem data from a sample of students at your college.
(c) A sociologist asks a large sample of high school students which academic subject
they like best. She suspects that a higher percent of females than of males will name
English as their favorite subject.
6.74 Translate each of the following research questions into appropriate H0 and Ha.
(a) Census Bureau data show that the mean household income in the area served by
a shopping mall is $72,500 per year. A market research firm questions shoppers at
the mall to find out whether the mean household income of mall shoppers is higher
than that of the general population.
(b) Last year, your company’s service technicians took an average of 1.8 hours
to respond to trouble calls from business customers who had purchased service
contracts. Do this year’s data show a different average response time?
6.75 A test statistic for a two-sided significance test for a population mean is z = 2.3.
Sketch a standard Normal curve and mark this value of z on it. Find the P -value
and shade the appropriate areas under the curve to illustrate your calculations.
6.76 A test statistic for a two-sided significance test for a population mean is z =
−1.4. Sketch a standard Normal curve and mark this value of z on it. Find the P -
value and shade the appropriate areas under the curve to illustrate your calculations.
6.77 The P -value for a significance test is 0.082.

(a) Do you reject the null hypothesis at level α = 0.05?
(b) Do you reject the null hypothesis at level α = 0.01?
(c) Explain your answers.
6.78 The P -value for a significance test is 0.032.

(a) Do you reject the null hypothesis at level α = 0.05?
(b) Do you reject the null hypothesis at level α = 0.01?
(c) Explain your answers.
6.79 A test of the null hypothesis H0: µ = µ0 gives test statistic z = −1.6.
(a) What is the P -value if the alternative is Ha: µ > µ0 ?
(b) What is the P -value if the alternative is Ha: µ < µ0 ?
(c) What is the P -value if the alternative is Ha: µ 6= µ0 ?

Why?
Why?

Why?
Why?
6.84 A report based on the National Assessment of Educational Progress (NAEP)

states that the average score on their mathematics test for eighth-grade students in
the District of Columbia was 243 in 2003, which was 235. The report then says that
this value is higher than the average in 2000. A footnote states that comparisons
(higher/lower/different) are determined by statistical tests with 0.05 as the level of
significance. Explain what this means in language understandable to someone who
knows no statistics. Do not use the word “significance” in your answer.
6.85 An NAEP report similar to the one described in the previous exercise states
that the average score on their mathematics test for eighth-grade students in Boston
was 262. The report then says that this value was not significantly different from 287,
the average score for eighth-grade students in U.S. public schools that are located in
large central cities. A footnote states that comparisons (higher/lower/different) are
determined by statistical tests with 0.05 as the level of significance. Explain what
this means in language understandable to someone who knows no statistics. Do not
use the word “significance” in your answer.
6.86 Here are the Degree of Reading Power (DRP) scores for a sample of 44 third-
grade students:
40 26 39 14 42 18 25 43 46 27 19
47 19 26 35 34 15 44 40 38 31 46
52 25 35 35 33 29 34 41 49 28 52
47 35 48 22 33 41 51 27 14 54 45
Section 6.2 157
These students can be considered to be an SRS of the third-graders in a suburban

school district. DRP scores are approximately Normal. Suppose that the standard
deviation of scores in this school district is known to be σ = 11. The researcher
believes that the mean score µ of all third-graders in this district is higher than the
national mean, which is 32.
(a) State the appropriate H0 and Ha to test this suspicion.
(b) Carry out the test. Give the P -value, and then interpret the result in plain
language.
6.87 To determine whether the mean nicotine content of a brand of cigarettes is

greater than the advertised value of 1.4 milligrams, a health advocacy group tests
H0: µ = 1.4
Ha: µ > 1.4

The calculated value of the test statistic is z = 1.75.
(a) Is the result significant at the 5% level?
(b) Is the result significant at the 1% level?
6.88 A computer has a random number generator designed to produce random

numbers that are uniformly distributed on the interval from 0 to 1. If this is true,
the numbers generated come from a population with µ = 0.5 and σ = 0.2887. A
command to generate 100 random numbers gives outcomes with mean x = 0.4365.
Assume that the population σ remains fixed. We want to test
H0: µ = 0.5
Ha: µ 6= 0.5
(a) Calculate the value z of the z test statistic.
(b) Is the result significant at the 5% level (α = 0.05)?
(c) Is the result significant at the 1% level (α = 0.01)?
6.89 Consider a significance test for a null hypothesis versus a two-sided alternative
with a z test statistic. Give a value of z that will give a result significant at the 1%
level but not at the 0.5% level.
6.90 You have performed a two-sided test of significance and obtained a value of
z = −4.3. Use Table D to find the approximate P -value for this test.
6.91 You have performed a one-sided test of significance and obtained a value of
6.92 You will perform a significance test of
H0: µ = 0
versus
Ha: µ > 0
(a) What values of z would lead you to reject H0 at the 5% level?

(b) If the alternative hypothesis was
Ha: µ 6= 0
what values of z would lead you to reject H0 at the 5% level?

(c) Explain why your answers to parts (a) and (b) are different.
6.93 Consider a significance test for a null hypothesis versus a two-sided alternative.
Between what values from Table D does the P -value for an outcome z = 1.34 lie?
Calculate the P -value using Table A, and verify that it lies between the values you
found from Table D.
6.94 Refer to the previous exercise. Find the P -value for z = −1.34.
6.95 Radon is a colorless, odorless gas that is naturally released by rocks and soils
and may concentrate in tightly closed houses. Because radon is slightly radioactive,
there is some concern that it may be a health hazard. Radon detectors are sold to
homeowners worried about this risk, but the detectors may be inaccurate. Univer-
sity researchers placed 12 detectors in a chamber where they were exposed to 105
picocuries per liter (pCi/l) of radon over 3 days. Here are the readings given by the
detectors:
91.9 97.8 111.4 122.3 105.4 95.0
103.8 99.6 96.6 119.3 104.8 101.7
Assume (unrealistically) that you know that the standard deviation of readings for
all detectors of this type is σ = 9.
(a) Give a 95% confidence interval for the mean reading µ for this type of detector.
(b) Is there significant evidence at the 5% level that the mean reading differs from
the true value 105? State hypotheses and base a test on your confidence interval
from (a).
6.96 A new supplier offers a good price on a catalyst used in your production
process. You compare the purity of this catalyst with that of the catalyst offered
by your current supplier. The P -value for a test of “no difference” is 0.27. Can you
be confident that the purity of the new product is the same as the purity of the
product that you have been using? Discuss.
6.97 The level of calcium in the blood in healthy young adults varies with mean
about 9.5 milligrams per deciliter and standard deviation about σ = 0.4. A clinic in
rural Guatemala measures the blood calcium level of 160 healthy pregnant women
at their first visit for prenatal care. The mean is x = 9.57. Is this an indication
that the mean calcium level in the population from which these women come differs
from 9.5?
(a) State H0 and Ha .
(b) Carry out the test and give the P -value, assuming that σ = 0.4 in this population.
Report your conclusion.
(c) Give a 95% confidence interval for the mean calcium level µ in this population.
We are confident that µ lies quite close to 9.5. This illustrates the fact that a test
Section 6.3 159
based on a large sample (n = 160 here) will often declare even a small deviation
from H0 to be statistically significant.
Section 6.3
6.98 Give an example of a set of data for which statistical inference is not valid.
6.99 More than 200,000 people worldwide take the GMAT examination each year as
they apply for MBA programs. Their scores vary Normally with mean about µ = 525
and standard deviation about σ = 100. One hundred students go through a rigorous
training program designed to raise their GMAT scores. Test the hypotheses
H0: µ = 525
Ha: µ > 525
in each of the following situations:

(a) The students’ average score is x = 541.4. Is this result significant at the 5%
level?
(b) The average score is x = 541.5. Is this result significant at the 5% level?
The difference between the two outcomes in (a) and (b) is of no importance. Beware
attempts to treat α = 0.05 as sacred.
6.100 How much education children get is strongly associated with the wealth and
social status of their parents. In social science jargon, this is “socioeconomic status,”
or SES. But the SES of parents has little influence on whether children who have
graduated from college go on to yet more education. One study looked at whether
college graduates took the graduate admissions tests for business, law, and other
graduate programs. The effects of the parents’ SES on taking the LSAT test for law
school were “both statistically insignificant and small.”
(a) What does “statistically insignificant” mean?
(b) Why is it important that the effects were small in size as well as insignificant?
6.101 A local television station announces a question for a call-in opinion poll on the
six o’clock news and then gives the response on the eleven o’clock news. Today’s
question concerns a proposed increase in funds for student loans. Of the 2372
calls received, 1921 oppose the increase. The station, following standard statistical
practice, makes a confidence statement: “81% of the Channel 13 Pulse Poll sample
oppose the increase. We can be 95% confident that the proportion of all viewers who
oppose the increase is within 1.6% of the sample result.” Is the station’s conclusion
justified? Explain your answer.
6.102 A researcher looking for evidence of extrasensory perception (ESP) tests 500
subjects. Four of these subjects do significantly better (P < 0.01) than random
guessing.
(a) Is it proper to conclude that these four people have ESP? Explain your answer.
(b) What should the researcher now do to test whether any of these four subjects
have ESP?
6.103 The text cites an example in which researchers carried out 77 separate signif-
icance tests, of which 2 were significant at the 5% level. Suppose that these tests are
independent of each other. (In fact they were not independent, because all involved
the same subjects.) If all of the null hypotheses are true, each test has probability
0.05 of being significant at the 5% level.
(a) What is the distribution of the number X of tests that are significant?
(b) Find the probability that 2 or more of the tests are significant.
6.104 You are the statistical expert on a team that is planning a study. After you
have made a careful presentation of the mechanics of significance testing, one of the
team members suggests using α = 0.50 for the study because you would be more
likely to obtain statistically significant results with this choice. Explain in simple
terms why this would not be a good use of statistical methods.
6.105 A study with 5000 subjects reported a result that was statistically significant
at the 5% level. Explain why this result might not be particularly large or important.
6.106 A study with 12 subjects reported a result that failed to achieve statistical
significance at the 5% level. The P -value was 0.052. Write a short summary of how
you would interpret these findings.
6.107 A P -value of 0.95 is reported for a significance test for a population mean.
Interpret this result.
6.108 Every user of statistics should understand the distinction between statistical
significance and practical importance. A sufficiently large sample will declare very
small effects statistically significant. Let us suppose that SAT Mathematics (SATM)
scores in the absence of coaching vary Normally with mean µ = 480 and σ = 100.
Suppose further that coaching may change µ but does not change σ. An increase
in the SATM score from 480 to 483 is of no importance in seeking admission to
college, but this unimportant change can be statistically very significant. To see
this, calculate the P -value for the test of
H0: µ = 480
Ha: µ > 480

(a) A coaching service coaches 100 students; their SATM scores average x = 483.
(b) By the next year, the service has coached 1000 students; their SATM scores
average x = 483.
(c) An advertising campaign brings the number of students coached to 10,000; their
average score is still x = 483.
6.109 Give a 99% confidence interval for the mean SATM score µ after coaching in
each part of the previous exercise. For large samples, the confidence interval says,
“Yes, the mean score is higher after coaching, but only by a small amount.”
6.110 As in the previous exercises, suppose that SATM scores vary Normally with
σ = 100. One hundred students go through a rigorous training program designed to
Section 6.4 161
raise their SATM scores by improving their mathematics skills. Carry out a test of
H0: µ = 480
Ha: µ > 480

(a) The students’ average score is x = 496.4. Is this result significant at the 5%
level?
(b) The average score is x = 496.5. Is this result significant at the 5% level?
The difference between the two outcomes in (a) and (b) is of no importance. Beware
attempts to treat α = 0.05 as sacred.
6.111 Refer to the previous problem. A researcher has performed 10 tests of signif-
icance and wants to apply the Bonferroni procedure with α = 0.05. The calculated
P -values are 0.045, 0.888, 0.050, 0.004, 0.001, 0.041, 0.888, 0.010, 0.002, 0.223.
Which of the null hypotheses are rejected with this procedure?
6.112 The table in Exercise 1.44 gives average property damage per year due to
tornadoes for each of the states. Is it appropriate to use the statistical methods we
discussed in this chapter for these data? Explain why or why not.
6.113 Give an example of an interesting set of data for which statistical inference
is valid. Explain your answer.
6.114 A P -value of 0.90 is reported for a significance test for a population mean.
Interpret this result.
Section 6.4
6.115 A previous example gives a test of a hypothesis about the SAT scores of
California high school students based on an SRS of 500 students. The hypotheses
are
H0: µ = 450
Ha: µ > 450

Assume that the population standard deviation is σ = 100. The test rejects H0 at
the 1% level of significance when z ≥ 2.326, where
x − 450
z= √
100/ 500
Is this test sufficiently sensitive to usually detect an increase of 10 points in the

population mean SAT score? Answer this question by calculating the power of the
test against the alternative µ = 460.
6.116 Use the result of the previous exercise to give the probability of a Type I
error and the probability of a Type II error for the test in that exercise when the
alternative is µ = 462.
6.117 A previous example discusses a test about the mean contents of cola bottles.
The hypotheses are
H0: µ = 300
Ha: µ < 300
The sample size is n = 6, and the population is assumed to have a Normal distri-
bution with σ = 3. A 5% significance test rejects H0 if z ≤ −1.645, where the test
statistic z is
x − 300
z= √
3/ 6
Power calculations help us see how large a shortfall in the bottle contents the test
can be expected to detect.
(a) Find the power of this test against the alternative µ = 299.
(b) Find the power against the alternative µ = 295.
(c) Is the power against µ = 290 higher or lower than the value you found in (b)?
Explain why this result makes sense.
6.118 Use the result of the previous exercise to give the probabilities of Type I and
Type II errors for the test discussed there. Take the alternative hypothesis to be
µ = 295.
6.119 You have an SRS of size n = 9 from a Normal distribution with σ = 1. You
wish to test
H0: µ = 0
Ha: µ > 0
You decide to reject H0 if x > 0 and to accept H0 otherwise.
(a) Find the probability of a Type I error, that is, the probability that your test
rejects H0 when in fact µ = 0.
(b) Find the probability of a Type II error when µ = 0.3. This is the probability
that your test accepts H0 when in fact µ = 0.3.
(c) Find the probability of a Type II error when µ = 1.
6.120 (Optional) An acceptance sampling test has probability 0.05 of rejecting a

good lot of bearings and probability 0.08 of accepting a bad lot. The consumer of
the bearings may imagine that acceptance sampling guarantees that most accepted
lots are good. Alas, it is not so. Suppose that 90% of all lots shipped by the pro-
ducer are bad.
(a) Draw a tree diagram for shipping a lot (the branches are “bad” and “good”)
and then inspecting it (the branches at this stage are “accept” and “reject”).
(b) Write the appropriate probabilities on the branches, and find the probability
that a lot shipped is accepted.
(c) Use the definition of conditional probability or Bayes’s formula to find the prob-
ability that a lot is bad, given that the lot is accepted. This is the proportion of
bad lots among the lots that the sampling plan accepts.
6.121 You want to see if a redesign of the cover of a mail-order catalog will increase
sales. A very large number of customers will receive the original catalog, and a
Section 6.4 163
random sample of customers will receive the one with the new cover. For planning
purposes, you are willing to assume that the sales from the new catalog will be
approximately Normal with σ = 60 dollars and that the mean for the original
catalog will be µ = 40 dollars. You decide to use a sample size of n = 1000. You
wish to test
H0: µ = 40
Ha: µ > 40
You decide to reject H0 if x > 43.12 and to accept H0 otherwise.
rejects H0 when in fact µ = 40 dollars.
(b) Find the probability of a Type II error when µ = 45 dollars. This is the
probability that your test accepts H0 when in fact µ = 45.
(d) The distribution of sales is not Normal, because many customers buy nothing.
Why is it nonetheless reasonable in this circumstance to assume that the mean will
be approximately Normal?
6.122 Consider a test of a hypothesis about the SAT scores of California high school
students based on an SRS of 500 students. The hypotheses are
H0: µ = 450
Ha: µ > 450

Assume that the population standard deviation is σ = 100. The test rejects H0 at
the 1% level of significance when z ≥ 2.326, where
x − 450
z= √
100/ 500
Is this test sufficiently sensitive to usually detect an increase of 12 points in the
population mean SAT score? Answer this question by calculating the power of the
test against the alternative µ = 462.
6.123 You want to see if a redesign of the cover of a mail-order catalog will increase
sales. A very large number of customers will receive the original catalog, and a
random sample of customers will receive the one with the new cover. For planning
purposes, you are willing to assume that the sales from the new catalog will be
approximately Normal with σ = 50 dollars and that the mean for the original
catalog will be µ = 25 dollars. You decide to use a sample size of n = 900. You
wish to test
H0: µ = 25
Ha: µ > 25
You decide to reject H0 if x > 26.
rejects H0 when in fact µ = 25 dollars.
(b) Find the probability of a Type II error when µ = 28 dollars. This is the
probability that your test accepts H0 when in fact µ = 28.

(d) The distribution of sales is not Normal, because many customers buy nothing.
Why is it nonetheless reasonable in this circumstance to assume that the mean will
be approximately Normal?
6.124 You are in charge of marketing for a Web site that offers automated medical
diagnoses. The program will scan the results of routine medical tests (pulse rate,
blood pressure, urinalysis, etc.) and either clear the patient or refer the case to a
doctor. You are marketing the program for use as part of a preventive-medicine
system to screen many thousands of persons who do not have specific medical com-
plaints. The program makes a decision about each patient.
(a) What are the two hypotheses and the two types of errors that the program can
make? Describe the two types of errors in terms of “false positive” and “false neg-
ative” test results.
(b) The program can be adjusted to decrease one error probability at the cost of an
increase in the other error probability. Which error probability would you choose to
make smaller, and why? (This is a matter of judgment. There is no single correct
answer.)
6.125 A study compares two groups of mothers with young children who were on
welfare two years ago. One group attended a voluntary training program offered
free of charge at a local vocational school and advertised in the local news media.
The other group did not choose to attend the training program. The study finds a
significant difference (P < 0.01) between the proportions of the mothers in the two
groups who are still on welfare. The difference is not only significant but quite large.
The report says with 95% confidence the percent of the nonattending group still on
welfare is 21% ± 4% higher than that of the group who attended the program. You
are on the staff of a member of Congress who is interested in the plight of welfare
mothers and who asks you about the report.
(a) Explain briefly and in nontechnical language what “a significant difference (P <
0.01)” means.
(b) Explain clearly and briefly what “95% confidence” means.
(c) Is this study good evidence that requiring job training of all welfare mothers
would greatly reduce the percent who remain on welfare for several years?
6.126 Use a computer to generate n = 5 observations from a Normal distribution

with mean 20 and standard deviation 5–N (20, 5). Find the 95% confidence interval
for µ. Repeat this process 100 times and then count the number of times that the
confidence interval includes the value µ = 20. Explain your results.

with mean 20 and standard deviation 5–N (20, 5). Test the null hypothesis that
µ = 20 using a two-sided significance test. Repeat this process 100 times and then
count the number of times that you reject H0. Explain your results.
6.128 Use the same procedure for generating data as in the previous exercise. Now
test the null hypothesis that µ = 22.5. Explain your results.
6.129 Figure 6.2 demonstrates the behavior of a confidence interval in repeated

sampling by showing the results of 25 samples from the same population. Now you
will do a similar demonstration. Suppose that (unknown to the researcher) the mean
SATM score of all California high school seniors is µ = 460, and that the standard
deviation is known to be σ = 100. The scores vary Normally.
(a) Simulate the drawing of 25 SRSs of size n = 100 from this population.
(b) The 95% confidence interval for the population mean µ has the form x ± m.
What is the margin of error m? (Remember that we know σ = 100.)
(c) Use your software to calculate the 95% confidence interval for µ when σ = 100
for each of your 25 samples. Verify the computer’s calculations by checking the
interval given for the first sample against your result in (b). Use the x reported by
the software.
(d) How many of the 25 confidence intervals contain the true mean µ = 460? If
you repeated the simulation, would you expect exactly the same number of intervals
to contain µ? In a very large number of samples, what percent of the confidence
intervals would contain µ?
6.130 In the previous exercise you simulated the SATM scores of 25 SRSs of 100
California seniors. Now use these samples to demonstrate the behavior of a sig-
nificance test. We know that the population of all SATM scores is Normal with
standard deviation σ = 100.
(a) Use your software to carry out a test of
H0: µ = 460
Ha: µ 6= 460
for each of the 25 samples.
(b) Verify the computer’s calculations by using Table A to find the P -value of the
test for the first of your samples. Use the x reported by your software.
(c) How many of your 25 tests reject the null hypothesis at the α = 0.05 significance
level? (That is, how many have P -value 0.05 or smaller?) Because the simulation
was done with µ = 460, samples that lead to rejecting H0 produce the wrong
conclusion. In a very large number of samples, what percent would falsely reject the
hypothesis?
6.131 Suppose that in fact the mean SATM score of California high school seniors
is µ = 480. Would the test in the previous exercise usually detect a mean this far
from the hypothesized value? This is a question about the power of the test.
(a) Simulate the drawing of 25 SRSs from a Normal population with mean µ = 480
and σ = 100. These represent the results of sampling when in fact the alternative
µ = 480 is true.
(b) Repeat on these new data the test of
H0: µ = 460
Ha: µ 6= 460
that you did in the previous exercise. How many of the 25 tests have P -values 0.05
or smaller? These tests reject the null hypothesis at the α = 0.05 significance level,
which is the correct conclusion.
(c) The power of the test against the alternative µ = 480 is the probability that
the test will reject H0: µ = 460 when in fact µ = 480. Calculate this power. In a
very large number of samples from a population with mean 480, what percent would
reject H0 ?
6.132 In a study of possible iron deficiency in infants, researchers compared sev-

eral groups of infants who were following different feeding patterns. One group of
26 infants was being breast-fed. At 6 months of age, these children had a mean
hemoglobin level of x = 12.9 grams per 100 milliliters of blood and a standard devi-
ation of 1.6. Taking the standard deviation to be the population value σ, give a 95%
confidence interval for the mean hemoglobin level of breast-fed infants. What as-
sumptions are required for the validity of the method you used to get the confidence
interval?
6.133 Statisticians prefer large samples. Describe briefly the effect of increasing
the size of a sample (or the number of subjects in an experiment) on each of the
following:
(a) The width of a level C confidence interval.
(b) The P -value of a test, when H0 is false and all facts about the population remain
unchanged as n increases.
(c) The power of a fixed level α test when α, the alternative hypothesis, and all
facts about the population remain unchanged.
6.134 A roulette wheel has 18 red slots among its 38 slots. You observe many
spins and record the number of times that red occurs. Now you want to use these
data to test whether the probability of a red has the value that is correct for a fair
roulette wheel. State the hypotheses H0 and Ha that you will test. (The test for
this situation is discussed in Chapter 8.)
6.135 The text demonstrates the behavior of a confidence interval in repeated sam-
pling by showing the results of 25 samples from the same population. Now you will
do a similar demonstration. Suppose that (unknown to the researcher) the mean
SATM score of all California high school seniors is µ = 475, and that the standard
deviation is known to be σ = 100. The scores vary Normally.
(a) Simulate the drawing of 50 SRSs of size n = 100 from this population.
(b) The 95% confidence interval for the population mean µ has the form x ± m.
What is the margin of error m? (Remember that we know σ = 100.)
(c) Use your software to calculate the 95% confidence interval for µ when σ = 100
for each of your 50 samples. Verify the computer’s calculations by checking the
interval given for the first sample against your result in (b). Use the x reported by
the software.
(d) How many of the 50 confidence intervals contain the true mean µ = 475? If
you repeated the simulation, would you expect exactly the same number of intervals
to contain µ? In a very large number of samples, what percent of the confidence
intervals would contain µ?
6.136 In the previous exercise you simulated the SATM scores of 50 SRSs of 100
California seniors. Now use these samples to demonstrate the behavior of a sig-
nificance test. We know that the population of all SATM scores is Normal with
standard deviation σ = 100.
(a) Use your software to carry out a test of
H0: µ = 475
Ha: µ 6= 475
for each of the 50 samples.
(b) Verify the computer’s calculations by using Table A to find the P -value of the
test for the first of your samples. Use the x reported by your software.
(c) How many of your 50 tests reject the null hypothesis at the α = 0.05 significance
level? (That is, how many have P -value 0.05 or smaller?) Because the simulation
was done with µ = 475, samples that lead to rejecting H0 produce the wrong
conclusion. In a very large number of samples, what percent would falsely reject the
hypothesis?
6.137 Suppose that in fact the mean SATM score of California high school seniors
is µ = 500. Would the test in the previous exercise usually detect a mean this far
from the hypothesized value? This is a question about the power of the test.
(a) Simulate the drawing of 50 SRSs from a Normal population with mean µ = 500
and σ = 100. These represent the results of sampling when in fact the alternative
µ = 500 is true.
(b) Repeat on these new data the test of
H0: µ = 475
Ha: µ 6= 475
that you did in the previous exercise. How many of the 50 tests have P -values 0.05
or smaller? These tests reject the null hypothesis at the α = 0.05 significance level,
which is the correct conclusion.
(c) The power of the test against the alternative µ = 500 is the probability that
the test will reject H0: µ = 475 when in fact µ = 500. Calculate this power. In a
very large number of samples from a population with mean 500, what percent would
reject H0 ?
6.138 You are testing the null hypothesis that µ = 0 versus the alternative µ > 0
using α = 0.05. Assume σ = 17. Suppose x̄ = 5 and n = 10. Calculate the test
statistic and its P -value. Repeat assuming the same value of x̄ but with n = 20. Do
the same for sample sizes of 30, 40, and 50. Plot the values of the test statistic versus
the sample size. Do the same for the P -values. Summarize what this demonstration
shows about the effect of the sample size on significance testing.
6.139 An agronomist examines the cellulose content of a variety of alfalfa hay.

Suppose that the cellulose content in the population has standard deviation σ = 8
milligrams per gram (mg/g). A sample of 16 cuttings has mean cellulose content
x = 140 mg/g.
(a) Give a 95% confidence interval for the mean cellulose content in the population.
(b) A previous study claimed that the mean cellulose content was µ = 135 mg/g,
but the agronomist believes that the mean is higher than that figure. State H0 and
Ha and carry out a significance test to see if the new data support this belief.
(c) The statistical procedures used in (a) and (b) are valid when several assumptions
are met. What are these assumptions?
6.140 Because sulfur compounds cause “off-odors” in wine, oenologists (wine ex-
perts) have determined the odor threshold, the lowest concentration of a compound
that the human nose can detect. For example, the odor threshold for dimethyl sul-
fide (DMS) is given in the oenology literature as 25 micrograms per liter of wine
(µg/l). Untrained noses may be less sensitive, however. Here are the DMS odor
thresholds for 10 beginning students of oenology:
32 33 40 35 24 36 31 30 20 25
Assume (this is not realistic) that the standard deviation of the odor threshold for
untrained noses is known to be σ = 7 µg/l.
(a) Make a stemplot to verify that the distribution is roughly symmetric with no
outliers. (A Normal quantile plot confirms that there are no systematic departures
from Normality.)
(b) Give a 95% confidence interval for the mean DMS odor threshold among all
beginning oenology students.
(c) Are you convinced that the mean odor threshold for beginning students is higher
than the published threshold, 25 µg/l? Carry out a significance test to justify your
answer.
6.141 A study of the pay of corporate chief executive officers (CEOs) examined the
increase in cash compensation of the CEOs of 104 companies, adjusted for inflation,
in a recent year. The mean increase in real compensation was x = 6.8%, and the
standard deviation of the increases was s = 53%. Is this good evidence that the
mean real compensation µ of all CEOs increased that year? The hypotheses are
H0: µ = 0 (no increase)
Ha: µ > 0 (an increase)

Because the sample size is large, the sample s is close to the population σ, so take
σ = 53%.
(a) Sketch the Normal curve for the sampling distribution of x when H0 is true.
Shade the area that represents the P -value for the observed outcome x = 6.8%.
(b) Calculate the P -value.
(c) Is the result significant at the α = 0.05 level? Do you think the study gives
strong evidence that the mean compensation of all CEOs went up?
6.142 When asked to explain the meaning of “statistically significant at the α =

0.05 level,” a student says, “This means there is only probability 0.05 that the
null hypothesis is true.” Is this an essentially correct explanation of statistical
significance? Explain your answer.

with mean 20 and standard deviation 5: N (20, 5). Find the 95% confidence interval
for µ. Repeat this process 100 times and then count the number of times that the
confidence interval includes the value µ = 20. Explain your results.

with mean 20 and standard deviation 5: N (20, 5). Test the null hypothesis that
µ = 20 using a two-sided significance test. Repeat this process 100 times and then
count the number of times that you reject H0. Explain your results.
6.145 Use the same procedure for generating data as in the previous exercise. Now
test the null hypothesis that µ = 18. Explain your results.
6.146 A study of late adolescents and early adults reported average months of
full-time employment for individuals aged 18 to 26 (Sabrina Oesterle et al., “Volun-
teerism during the transition to adulthood: a life course perspective,” Social Forces,
83 (2004), pp. 1123–1149). Here are the means:
Age 18 19 20 21 22 23 24 25 26
Months employed 2.9 4.2 5.0 5.3 6.4 7.4 8.5 8.9 9.3
Assume that the standard deviation for each of these means is 4.5 months and that
each sample size is 750.
(a) Calculate the 95% confidence interval for each mean.
(b) Plot the means versus age. Draw a vertical line through the first mean extending
up to the upper confidence limit and down to the lower limit. At the ends of the
line, draw a short dash. Do the same for each of the other means.
(c) Write a summary of what the data show. Note that in circumstances such as
this, it is common practice not to make any adjustments for the fact that several
confidence intervals are being reported. Be sure to include comments about this in
your summary.
6.147 Persons aged 55 and over represented 21.3% of the U.S. population in the
year 2000. This group is expected to increase to 30.5% by 2025. In terms of actual
numbers of people, the increase is from 58.6 million to 101.4 million. Restaurateurs
have found this market to be important and would like to make their businesses
attractive to older customers. One study used a questionnaire to collect data from
people aged 50 and over (Barbara A. Almanza, Richard Ghiselli, and William Jaffe,
“Foodservice design and aging baby boomers: importance and perception of phys-
ical amenities in restaurants,” Foodservice Research International, 12 (2000), pp.
25–40). For one part of the analysis, individuals were classified into two age groups:
50 to 64 and 65 to 79. There were 267 people in the first group and 263 in the
second. One set of items concerned ambiance, menu design, and service. A series
of statements were rated on a 1 to 5 scale with 1 representing “strongly disagree”
and 5 representing “strongly agree.” In some cases the wording has been shortened
in the following table. Here are the means:
Statement 50–64 65–79

Ambiance:
Most restaurants are too dark 2.75 2.93
Most restaurants are too noisy 3.33 3.43
Background music is often too loud 3.27 3.55
Restaurants are too smoky 3.17 3.12
Tables are too small 3.00 3.19
Tables are too close together 3.79 3.81
Menu design:
Print size is not large enough 3.68 3.77
Glare makes menus difficult to read 2.81 3.01
Colors of menus make them difficult to read 2.53 2.72
Service:
It is difficult to hear the service staff 2.65 3.00
I would rather be served than serve myself 4.23 4.14
I would rather pay the server than a cashier 3.88 3.48
Service is too slow 3.13 3.10
First, examine the means of the people who are 50 to 64. Order the statements
according to the means and describe the results. Then do the same for the older
group. For each statement compute the z statistic and the associated P -value for
the comparison between the two groups. For these calculations you can assume
that the standard deviation of the difference is 0.08, so z is simply the difference
in the means divided by 0.08. Note that you are performing 13 significance tests
in this exercise. Keep this in mind when you interpret your results. Write a report
summarizing your work.
CHAPTER 7
Section 7.1
7.1 What critical value t∗ from Table D should be used for a confidence interval for
the mean of the population in each of the following situations?
(a) A 90% confidence interval based on n = 12 observations.
(b) A 95% confidence interval from an SRS of 30 observations.
(c) An 80% confidence interval from a sample of size 18.
7.2 Use software to find the critical values t∗ that you would use for each of the
following confidence intervals for the mean.
7.3 The one-sample t statistic for testing
H0: µ = 0
Ha: µ > 0
from a sample of n = 15 observations has the value t = 1.97.
(a) What are the degrees of freedom for this statistic?
(b) Give the two critical values t∗ from Table D that bracket t.
(c) What are the right-tail probabilities p for these two entries?
(d) Between what two values does the P -value of the test fall?
(e) Is the value t = 1.97 significant at the 5% level? Is it significant at the 1% level?
(f) If you have software available, find the exact P -value.
7.4 The one-sample t statistic from a sample of n = 30 observations for the two-sided
test of
H0: µ = 64
Ha: µ 6= 64
has the value t = 1.12.
(a) What are the degrees of freedom for t?
(b) Locate the two critical values t∗ from Table D that bracket t. What are the
right-tail probabilities p for these two values?
(c) How would you report the P -value for this test?
(d) Is the value t = 1.12 statistically significant at the 10% level? At the 5% level?
(e) If you have software available, find the exact P -value.
7.5 The one-sample t statistic for a test of
H0: µ = 20
Ha: µ < 20
171
based on n = 12 observations has the value t = −2.45.

(b) How would you report the P -value based on Table D?
(c) If you have software available, find the exact P -value.
7.6 A bank wonders whether omitting the annual credit card fee for customers who
charge at least $2400 in a year would increase the amount charged on its credit card.
The bank makes this offer to an SRS of 250 of its existing credit card customers.
It then compares how much these customers charge this year with the amount that
they charged last year. The mean increase is $342, and the standard deviation is
$108.
(a) Is there significant evidence at the 1% level that the mean amount charged
increases under the no-fee offer? State H0 and Ha and carry out a t test.
(b) Give a 95% confidence interval for the mean amount of the increase.
(c) The distribution of the amount charged is skewed to the right, but outliers are
prevented by the credit limit that the bank enforces on each card. Use of the t
procedures is justified in this case even though the population distribution is not
Normal. Explain why.
(d) A critic points out that the customers would probably have charged more this
year than last even without the new offer because the economy is more prosperous
and interest rates are lower. Briefly describe the design of an experiment to study
the effect of the no-fee offer that would avoid this criticism.
7.7 The bank in the previous exercise tested a new idea on a sample of 250 customers.
Suppose that the bank wanted to be quite certain of detecting a mean increase of
µ = $100 in the amount charged, at the α = 0.01 significance level. Perhaps a
sample of only n = 50 customers would accomplish this. Find the approximate
power of the test with n = 50 against the alternative µ = $100 as follows:
(a) What is the t critical value for the one-sided test with α = 0.01 and n = 50?
(b) Write the criterion for rejecting H0 : µ = 0 in terms of the t statistic. Then
take s = 108 (an estimate based on the data in the previous exercise) and state the
rejection criterion in terms of x.
(c) Assume that µ = 100 (the given alternative) and that σ = 108 (an estimate from
the data in the previous exercise). The approximate power is the probability of the
event you found in (b), calculated under these assumptions. Find the power. Would
you recommend that the bank do a test on 50 customers, or should more customers
be included?
7.8 In an experiment on the metabolism of insects, American cockroaches were fed

measured amounts of a sugar solution after being deprived of food for a week and
of water for 3 days. After 2, 5, and 10 hours, the researchers dissected some of the
cockroaches and measured the amount of sugar in various tissues. Five cockroaches
fed the sugar D-glucose and dissected after 10 hours had the following amounts (in
micrograms) of D-glucose in their hindguts:
55.95 68.24 52.73 21.50 23.78
Find a 95% confidence interval for the mean amount of D-glucose in cockroach
hindguts under these conditions. (Based on D. L. Shankland et al., “The effect of
Section 7.1 173
5-thio-D-glucose on insect development and its absorption by insects,” Journal of

Insect Physiology, 14 (1968), pp. 63–72.)
7.9 Poisoning by the pesticide DDT causes tremors and convulsions. In a study of
DDT poisoning, researchers fed several rats a measured amount of DDT. They then
measured electrical characteristics of the rats’ nervous systems that might explain
how DDT poisoning causes tremors. One important variable was the “absolutely
refractory period,” the time required for a nerve to recover after a stimulus. This
period varies Normally. Measurements on four rats gave the data below (in mil-
liseconds). (Data from D. L. Shankland, “Involvement of spinal cord and peripheral
nerves in DDT-poisoning syndrome in albino rats,” Toxicology and Applied Phar-
macology, 6 (1964), pp. 97–213.)
1.6 1.7 1.8 1.9
(a) Find the mean refractory period x and the standard error of the mean.
(b) Give a 90% confidence interval for the mean “absolutely refractory period” for
all rats of this strain when subjected to the same treatment.
7.10 Suppose that the mean “absolutely refractory period” for unpoisoned rats is
known to be 1.3 milliseconds. DDT poisoning should slow nerve recovery and so
increase this period. Do the data in the previous exercise give good evidence for this
supposition? State H0 and Ha and do a t test. Between what levels from Table D
does the P -value lie? What do you conclude from the test?
7.11 The Acculturation Rating Scale for Mexican Americans (ARSMA) measures
the extent to which Mexican Americans have adopted Anglo/English culture. Dur-
ing the development of ARSMA, the test was given to a group of 17 Mexicans. Their
scores, from a possible range of 1.00 to 5.00, had x = 1.67 and s = 0.25. Because
low scores should indicate a Mexican cultural orientation, these results helped to
establish the validity of the test. (Based on I. Cuellar, L. C. Harris, and R. Jasso,
“An acculturation scale for Mexican American normal and clinical populations,”
Hispanic Journal of Behavioral Sciences, 2 (1980), pp. 199–217.)
(a) Give a 95% confidence interval for the mean ARSMA score of Mexicans.
(b) What assumptions does your confidence interval require? Which of these as-
sumptions is most important in this case?
7.12 The ARSMA test discussed in the previous exercise was compared with a
similar test, the Bicultural Inventory (BI), by administering both tests to 22 Mexican
Americans. Both tests have the same range of scores (1.00 to 5.00) and are scaled
to have similar means for the groups used to develop them. There was a high
correlation between the two scores, giving evidence that both are measuring the
same characteristics. The researchers wanted to know whether the population mean
scores for the two tests were the same. The differences in scores (ARSMA − BI) for
the 22 subjects had x = 0.2519 and s = 0.2767.
(a) Describe briefly how the administration of the two tests to the subjects should
be conducted, including randomization.
(b) Carry out a significance test for the hypothesis that the two tests have the same
population mean. Give the P -value and state your conclusion.
(c) Give a 95% confidence interval for the difference between the two population
mean scores.
7.13 The paper reporting the results on ARSMA used in Exercise 7.11 does not
give the raw data or any discussion of Normality. You would like to replace the t
procedure used in Exercise 7.12 by a sign test. Can you do this from the available
information? Carry out the sign test and state your conclusion, or explain why you
are unable to carry out the test.
7.14 Exercise 7.12 reports a small study comparing ARSMA and BI, two tests of the
acculturation of Mexican Americans. Would this study usually detect a difference
in mean scores of 0.2? To answer this question, calculate the approximate power of
the test (with n = 22 subjects and α = 0.05) of
H0: µ = 0
Ha: µ 6= 0
against the alternative µ = 0.2. Note that this is a two-sided test.
(a) From Table D, what is the critical value for α = 0.05?
(b) Write the criterion for rejecting H0 at the α = 0.05 level. Then take s = 0.3,
the approximate value observed in Exercise 7.12, and restate the rejection criterion
in terms of x.
(c) Find the probability of this event when µ = 0.2 (the alternative given) and σ =
0.3 (estimated from the data in Exercise 7.12) by a Normal probability calculation.
This is the approximate power.
7.15 Gas chromatography is a sensitive technique used by chemists to measure

small amounts of compounds. The response of a gas chromatograph is calibrated
by repeatedly testing specimens containing a known amount of the compound to be
measured. A calibration study for a specimen containing 1 nanogram (ng) (that’s
10−9 gram) of a compound gave the following response readings:
21.6 20.0 25.0 21.9
The response is known from experience to vary according to a Normal distribution

unless an outlier indicates an error in the analysis. Estimate the mean response to 1
ng of this substance, and give the margin of error for your choice of confidence level.
Then explain to a chemist who knows no statistics what your margin of error means.
(Data from the appendix of D. A. Kurtz (ed.), Trace Residue Analysis, American
Chemical Society Symposium Series, no. 284, 1985.)
7.16 Your local newspaper contains a large number of advertisements for unfur-
nished one-bedroom apartments. You choose 10 at random and calculate that their
mean monthly rent is $540 and that the standard deviation of their rents is $80.
(a) What is the standard error of the mean?
(b) What are the degrees of freedom for a one-sample t statistic?
7.17 You want to rent an unfurnished one-bedroom apartment for next semester.
You take a random sample of 10 apartments advertised in the local newspaper and
Section 7.1 175
record the rental rates. Here are the rents (in dollars per month):
500, 650, 600, 505, 450, 550, 515, 495, 650, 395
Find a 95% confidence interval for the mean monthly rent for unfurnished one-
bedroom apartments available for rent in this community.
7.18 If you chose 99% rather than 95% confidence, would your margin of error in
the previous exercise be larger or smaller? Explain your answer and verify it by
doing the calculations.
7.19 A random sample of 10 one-bedroom apartments from your local newspaper

has these monthly rents (dollars):
500, 650, 600, 505, 450, 550, 515, 495, 650, 395
Do these data give good reason to believe that the mean rent of all advertised
apartments is greater than $500 per month? State hypotheses, find the t statistic
and its P -value, and state your conclusion.
7.20 National Fuelsaver Corporation manufactures the Platinum Gasaver, a device

they claim “may increase gas mileage by 22%.” Here are the percent changes in
gas mileage for 15 identical vehicles, as presented in one of the company’s advertise-
ments:
48.3 46.9 46.8 44.6 40.2 38.5 34.6 33.7
28.7 28.7 24.8 10.8 10.4 6.9 −12.4
Would you recommend use of a t confidence interval to estimate the mean fuel
savings in the population of all such vehicles? Explain your answer.
7.21 A manufacturer of small appliances employs a market research firm to estimate

retail sales of its products. Here are last month’s sales of electric can openers from
an SRS of 50 stores in the Midwest sales region:
19 19 16 19 25 26 24 63 22 16
13 26 34 10 48 16 20 14 13 24
34 14 25 16 26 25 25 26 11 79
17 25 18 15 13 35 17 15 21 12
19 20 32 19 24 19 17 41 24 27
(a) Make a stemplot of the data. The distribution is skewed to the right and has
several high outliers. The bootstrap is a modern computer-intensive tool for getting
accurate confidence intervals without the Normality condition. Three bootstrap
simulations, each with 10,000 repetitions, give these 95% confidence intervals for
mean sales in the entire region: (20.42, 27.26), (20.40, 27.18), and (20.48, 27.28).
(b) Find the 95% t confidence interval for the mean. It is essentially the same as
the bootstrap intervals. The lesson is that for sample sizes as large as n = 50, t
procedures are very robust.
7.22 Refer to the previous exercise. Each electric can opener sold generates a profit
of $2.50 for the manufacturer.
(a) What is the mean profit per store in the Midwest sales region?
(b) Transform the confidence interval you found in the previous exercise into an
interval for the mean profit for stores in the Midwest region.
7.23 Refer to the previous two exercises. There are 4325 stores that sell can openers
manufactured by this company.
(a) Estimate the total profit for sales last month in the Midwest region.
(b) Give a 95% confidence interval for the total profit for sales last month in the
Midwest region.
7.24 The scores of four roommates on the Law School Admission Test (LSAT) are
628, 593, 455, 503
Find the mean, the standard deviation, and the standard error of the mean. Is it
appropriate to calculate a confidence interval for these data? Explain why or why
not.
7.25 Here are estimates of the daily intakes of calcium (in milligrams) for 38 women
between the ages of 18 and 24 years who participated in a study of women’s bone
health:
808 882 1062 970 909 802 374 416 784 997
651 716 438 1420 1425 948 1050 976 572 403
626 774 1253 549 1325 446 465 1269 671 696
1156 684 1933 748 1203 2433 1255 1100
(a) Display the data using a stemplot and make a Normal quantile plot. Describe
the distribution of calcium intakes for these women.
(b) Calculate the mean, the standard deviation, and the standard error.
(c) Find a 95% confidence interval for the mean.
7.26 Refer to the previous exercise. Eliminate the two largest values and answer
parts (a), (b), and (c).
7.27 Refer to Exercise 7.25. Suppose that the recommended daily allowance (RDA)
of calcium for women in this age range is 1300 milligrams (this value is changed
from time to time on the basis of the statistical analysis of new data). We want to
express the results in terms of percent of the RDA.
(a) Divide each intake by the RDA, multiply by 100, and compute the 95% confi-
dence interval from the transformed data.
(b) Verify that you can obtain the same result by similarly transforming the interval
you calculated in Exercise 7.25.
7.28 Refer to Exercises 7.25 and 7.27. You want to compare the average calcium
intake of these women with the RDA using a significance test.
(a) State appropriate null and alternative hypotheses.
(b) Give the test statistic, the degrees of freedom, and the P -value.
(c) State your conclusion.
(d) Repeat the calculations without the two largest values. Does your conclusion
depend on whether or not these observations are included in the analysis?
Section 7.1 177
7.29 The calcium intake data used in Exercise 7.25 contain two large observations
and we have some concern about the use of the t procedures because of this. In
Exercise 7.27 we compared the mean of the data with 1300 milligrams, the RDA.
We can use a version of the sign test to compare the median intake with the RDA.
First subtract 1300 from each intake. If the population median is 1300, we expect
approximately half of the observations to be above the median and half to be below
it. The number of observations that will be above the median is binomial with
n = 38 and p = 0.5. Carry out the sign test and summarize your results.
7.30 How much do users pay for Internet service? Here are the monthly fees (in
dollars) paid by a random sample of 50 users of commercial Internet service providers
in August 2000: (Data from the August 2000 supplement to the Current Population
Survey, from the Census Bureau Web site, www.census.gov.)
20 40 22 22 21 21 20 10 20 20
20 13 18 50 20 18 15 8 22 25
22 10 20 22 22 21 15 23 30 12
9 20 40 22 29 19 15 20 20 20
20 15 19 21 14 22 21 35 20 22
(a) Make a stemplot of the data. Also make a Normal quantile plot if your software
permits. The data are not Normal: there are stacks of observations taking the same
values, and the distribution is more spread out in both directions and somewhat
skewed to the right. The t procedures are nonetheless approximately correct because
n = 50 and there are no extreme outliers.
(b) Give a 95% confidence interval for the mean monthly cost of Internet access in
August 2000.
7.31 The data in the previous exercise show that many people paid $20 per month
for Internet access, presumably because major providers such as AOL charged this
amount. Do the data give good reason to think that the mean cost for all Internet
users differs from $20 per month?
7.32 Refer to the two previous exercises concerning fees paid for Internet access
by a national random sample of clients of Internet service providers in 2000. The
Census Bureau estimates that 44 million households had Internet access in 2000.
Use the confidence interval that you found to give a 95% confidence interval for the
total amount these households paid in Internet access fees. This is one aspect of the
national economic impact of the Internet.
7.33 Refer to the three previous exercises. Suppose you are interested in the cost
per year rather than the cost per month. Find a 95% confidence interval for the
mean yearly cost of Internet access. How does this interval relate to the one that
you found in Exercise 7.30?
7.34 The cost of health care is the subject of many studies that use statistical
methods. One such study estimated that the average length of service for home
health care among people over the age of 65 who use this type of service is 96.0
days with a standard error of 5.1 days. Assuming that the degrees of freedom are
large, calculate a 90% confidence interval for the true mean length of service. (A. N.
Dey, “Characteristics of elderly home health care users,” National Center for Health
Statistics, 1996.)
7.35 The embryos of brine shrimp can enter a dormant phase in which metabolic
activity drops to a low level. Researchers studying this dormant phase measured
the level of several compounds important to normal metabolism. The results were
reported in a table, with the note, “Values are means ± SEM for three independent
samples.” The table entry for the compound ATP was 0.84 ± 0.01. Biologists
reading the article are presumed to be able to decipher this. (S. C. Hand and E.
Gnaiger, “Anaerobic dormancy quantified in Artemia embryos,” Science, 239 (1988),
pp. 1425–1427.)
(a) What does the abbreviation “SEM” stand for?
(b) The researchers made three measurements of ATP, which had x = 0.84. What
was the sample standard deviation s for these measurements?
(c) Give a 90% confidence interval for the mean ATP level in dormant brine shrimp
embryos.
7.36 The design of controls and instruments has a large effect on how easily people
can use them. A student project investigated this effect by asking 25 right-handed
students to turn a knob (with their right hands) that moved an indicator by screw
action. There were two identical instruments, one with a right-hand thread (the
knob turns clockwise) and the other with a left-hand thread (the knob turns coun-
terclockwise). The following table gives the times required (in seconds) to move the
indicator a fixed distance (data provided by Timothy Sturm, Purdue University):
Subject Right thread Left thread Subject Right thread Left thread
1 113 137 14 107 87
2 105 105 15 118 166
3 130 133 16 103 146
4 101 108 17 111 123
5 138 115 18 104 135
6 118 170 19 111 112
7 87 103 20 89 93
8 116 145 21 78 76
9 75 78 22 100 116
10 96 107 23 89 78
11 122 84 24 85 101
12 103 148 25 88 123
13 116 147
(a) Each of the 25 students used both instruments. Discuss briefly how the experi-
ment should be arranged and how randomization should be used.
(b) The project hoped to show that right-handed people find right-hand threads
easier to use. State the appropriate H0 and Ha about the mean time required to
complete the task.
(c) Carry out a test of your hypotheses. Give the P -value and report your conclu-
sions.
Section 7.1 179
7.37 Refer to the previous exercise. Give a 90% confidence interval for the mean
time advantage of right-hand over left-hand threads in the setting of the previous
exercise. Do you think that the time saved would be of practical importance if the
task were performed many times—for example, by an assembly-line worker? To help
answer this question, find the mean time for right-hand threads as a percent of the
mean time for left-hand threads.
7.38 An agricultural field trial compares the yield of two varieties of tomatoes for
commercial use. The researchers divide in half each of 10 small plots of land in
different locations and plant each tomato variety on one half of each plot. After
harvest, they compare the yields in pounds per plant at each location. The 10
differences (Variety A − Variety B) give the following statistics: x = 0.46 and
s = 0.92. Is there convincing evidence that Variety A has the higher mean yield?
State H0 and Ha, and give a P -value to answer this question.
7.39 The tomato experts who carried out the field trial described in the previous
exercise suspect that the relative lack of significance there is due to low power. They
would like to be able to detect a mean difference in yields of 0.6 pound per plant at
the 0.05 significance level. Based on the previous study, use 0.92 as an estimate of
both the population σ and the value of s in future samples.
(a) What is the power of the test from Exercise 7.38 with n = 12 against the
alternative µ = 0.6?
(b) If the sample size is increased to n = 30 plots of land, what will be the power
against the same alternative?
7.40 The following situations all require inference about a mean or means. Identify
each as (1) a single sample, (2) matched pairs, or (3) two independent samples. The
procedures of this section apply to cases (1) and (2). We will learn procedures for
(3) in the next section.
(a) An education researcher wants to learn whether inserting questions before or
after introducing a new concept in an elementary school mathematics text is more
effective. He prepares two text segments that teach the concept, one with motivat-
ing questions before and the other with review questions after. Each text segment
is used to teach a different group of children, and their scores on a test over the
material are compared.
(b) Another researcher approaches the same problem differently. She prepares text
segments on two unrelated topics. Each segment comes in two versions, one with
questions before and the other with questions after. Each of a group of children
is taught both topics, one topic (chosen at random) with questions before and the
other with questions after. Each child’s test scores on the two topics are compared
to see which topic he or she learned better.
(c) To evaluate a new analytical method, a chemist obtains a reference specimen of
known concentration from the National Institute of Standards and Technology. She
then makes 20 measurements of the concentration of this specimen with the new
method and checks for bias by comparing the mean result with the known concen-
tration.
(d) Another chemist is evaluating the same new method. He has no reference spec-
imen, but a familiar analytic method is available. He wants to know if the new and
old methods agree. He takes a specimen of unknown concentration and measures

the concentration 10 times with the new method and 10 times with the old method.
7.41 A table gives the number of medical doctors per 100,000 people for each of the
50 states. It does not make sense to use the t procedures (or any other statistical
procedures) to give a 95% confidence interval for the mean number of medical doctors
per 100,000 people in the population of the American states. Explain why not.
7.42 Computers in some vehicles calculate various quantities related to the perfor-
mance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles
per gallon (mpg). For one vehicle equipped in this way, the mpg was recorded each
time the gas tank was filled and the computer was then reset. Here are the mpg
values for a random sample of 20 of these records:
15.8 13.6 15.6 19.1 22.4 15.6 22.5 17.2 19.4 22.6
19.4 18.0 14.6 18.7 21.0 14.8 22.6 21.5 14.3 20.9
(a) Describe the distribution using graphical methods and summarize the results.
(b) Is it appropriate to use methods based on Normal distributions to analyze these
data? Explain why or why not.
(c) Find the mean, the standard deviation, the standard error, and the margin of
error for 95% confidence. Report the 95% confidence interval for µ, the mean mpg
for this vehicle based on these data.
(d) Do you think that this interval would apply to other similar vehicles? Give
reasons why and why not.
7.43 Refer to the previous exercise. Here are the values of the average speed in
miles per hour (mph) for the same sample:
21.0 19.0 18.7 39.2 45.8 19.8 48.4 21.0 29.1 35.7
31.6 49.0 16.0 34.6 36.3 19.0 43.3 37.5 16.5 34.5
Answer the questions given in the previous exercise.
7.44 You will have complete sales information for last month in a week, but right
now you have data from a random sample of 40 stores. The mean change in sales in
the sample is +3.8% and the standard deviation of the changes is 12%. Are average
sales for all stores different from last month?
(a) State appropriate null and alternative hypotheses. Explain how you decided
between the one- and two-sided alternatives.
(b) Find the t statistic and its P -value. State your conclusion.
(c) If the test gives strong evidence against the null hypothesis, would you conclude
that sales are up in every one of your stores? Explain your answer.
7.45 For a sample of size 5, a test of a null hypothesis versus a two-sided alternative
gives t = 2.45.
(a) Is the test result significant at the 5% level? Draw a sketch of the appropriate t
distribution and illustrate your calculation with this sketch.
(b) Now assume that the same statistic was obtained for a sample size of n = 10.
Assess the statistical significance of the result and illustrate the calculation with a
Section 7.1 181
sketch. How did the statistical significance change with the sample size? Explain
your answer.
7.46 Assume a sample size of n = 2000. Draw a picture of the distribution of the t
statistic under the null hypothesis. Use your picture to illustrate the values of the
test statistic that would lead to rejection of the null hypothesis at the 1% level for
a two-sided alternative.
7.47 Repeat the previous exercise for the two situations where the alternative is
one-sided.
7.48 Computer software reports x̄ = 12.3 and P = 0.08 for a t test of H0 : µ = 0

versus Ha: µ 6= 0. Based on prior knowledge, you can justify testing the alternative
Ha: µ > 0. What is the P -value for your significance test?
7.49 Suppose that x̄ = −12.3 in the setting of the previous exercise. Would this
change your answer? Use a sketch of the distribution of the test statistic under the
null hypothesis to illustrate and explain your answer.
7.50 Use Table D to find the critical value t∗ to be used for a confidence interval
for the mean of the population in each of the following situations.
7.51 Use software to find the critical values t∗ that you would use for 95% confidence
intervals for sample sizes of 10, 20, 30, 40, 50, 100, 200, and 500. Plot the values of
t∗ versus the sample size and describe the relationship.
7.52 A sample of size n = 15 is used to perform a significance test for H0 : µ = 0

versus Ha: µ > 0. The test statistic is t = 2.15.
(b) Give the two critical values t∗ from Table D that bracket t.
(c) What are the right-tail probabilities p for these two entries?
(d) Between what two values does the P -value of the test fall?
(e) Sketch the t distribution for this exercise and illustrate your answers to parts
(c) and (d) with the sketch.
(f) Is the value t = 2.15 significant at the 5% level? Is it significant at the 1% level?
(g) If you have software available, find the exact P -value.
7.53 The hypotheses H0 : µ = 50 and Ha : µ 6= 50 are examined using a sample of

size n = 30. The one-sample t statistic has the value t = 1.35.
(a) Give the degrees of freedom for the test statistic.
(b) Locate the two critical values t∗ from Table D that bracket t. What are the
right-tail probabilities p for these two values?
(d) Is the value t = 1.35 statistically significant at the 10% level? At the 5% level?
(e) Illustrate your answers to the previous parts of this exercise with a sketch of the
t distribution.
(f) If you have software available, find the exact P -value.
7.54 The one-sample t statistic for a test of H0: µ = 30 versus Ha: µ < 30 based on
n = 12 observations has the value t = −3.21.
(a) Find the degrees of freedom for this statistic.
(b) Use Table D to find an approximate P -value. Use a sketch of the t distribution
to illustrate your work.
(c) Find the exact P -value if you have software available.
7.55 Here are estimates of the daily intakes of calcium (in milligrams) for 40 women
between the ages of 18 and 24 years who participated in a study of women’s bone
health:
725 764 853 559 1225 456 555 1168 682 676
808 882 1062 970 909 802 574 576 784 997
651 716 438 1220 1475 948 1050 976 572 483
1256 685 1144 848 1213 3223 1270 1108 350 421
(a) Use a stemplot or other graphical summary to describe the distribution of in-
takes. If you have software available, make a Normal quantile plot. Write a short
paragraph describing the distribution. Be sure to refer to your graphics in your
summary.
(b) Find the mean and the standard deviation. Sketch a Normal curve with this
mean and standard deviation.
(c) Write a sentence in which you give the 95% confidence interval for the mean and
an explanation of how it should be interpreted.
(d) There is an outlier. Eliminate it and answer parts (a), (b), and (c) again. How
do the results change?
(e) Take one side of the following issue and present reasons for your views. “These
results do not apply to women aged 18 to 24 years from the same community who
did not volunteer to participate in the study.”
7.56 Refer to the previous exercise. For some nutrients including calcium, recom-
mendations are expressed as adequate intakes (AI). The AI for women in this age
range is 1000 milligrams. Let’s compare the intakes of the women in this sample
with the AI using a significance test. Use the data for all 40 women.
(a) What null and alternative hypotheses would you use for this problem?
(b) Give the test statistic, the degrees of freedom, the P -value, and a sketch illus-
trating the P -value.
(c) Write a short paragraph giving the descriptive statistics and the significance test
results for this problem.
(d) (Optional) Use the sign test for this problem. Compare the two approaches to
the analysis of these data.
7.57 Many organizations are doing surveys to determine the satisfaction of their
customers. Attitudes toward various aspects of campus life were the subject of one
such study conducted at Purdue University. Each item was rated on a 1 to 5 scale,
with 5 being the highest rating. The average response of 1406 first-year students to
Section 7.1 183
“Feeling welcomed at Purdue” was 3.9 with a standard deviation of 0.98. Assuming
that the respondents are an SRS, give a 99% confidence interval for the mean of all
first-year students.
7.58 Do piano lessons improve the spatial-temporal reasoning of preschool children?

Neurobiological arguments suggest that this may be true. A study designed to test
this hypothesis measured the spatial-temporal reasoning of 34 preschool children
before and after six months of piano lessons. (The study also included children who
took computer lessons and a control group, but we are not concerned with those
here.) The changes in the reasoning scores are
2 5 7 –2 2 7 4 1 0 7 3 4 3 4 9 4 5
2 9 6 0 3 6 –1 3 4 6 7 –2 7 –3 3 4 4
(a) Display the data and summarize the distribution.

(b) Find the mean, the standard deviation, and the standard error of the mean.
(c) Give a 95% confidence interval for the mean improvement in reasoning scores.
7.59 Refer to the previous exercise. Test the null hypothesis that there is no im-
provement versus the alternative suggested by the neurobiological arguments. State
the hypotheses, and give the test statistic with degrees of freedom and the P -value.
What do you conclude? From your answer to part (c) of the previous exercise what
can be concluded from this significance test?
7.60 The researchers studying vitamin C in CSB were also interested in a similar
commodity called wheat-soy blend (WSB). Both of these commodities are mixed
with other ingredients and cooked. Loss of vitamin C as a result of this process
was another concern of the researchers. One preparation used in Haiti called gruel
(or “bouillie” in Creole) can be made from WSB, salt, sugar, milk, banana, and
other optional items to improve the taste. Samples of gruel prepared in Haitian
households were collected. The vitamin C content (in milligrams per 100 grams of
blend, dry basis) was measured before and after cooking. Here are the results:
Sample 1 2 3 4 5
Before 73 79 86 88 78
After 20 27 29 36 17
Set up appropriate hypotheses and carry out a significance test for these data. (It
is not possible for cooking to increase the amount of vitamin C.)
7.61 Refer to the previous exercise. The fact that vitamin C is destroyed by cooking
is neither new nor surprising, so the significance test you performed in the previous
exercise simply confirms that this fact is evident in the small sample of data collected.
The real question here concerns how much of the vitamin C is lost.
(a) Give a 95% confidence interval for the amount of vitamin C lost by preparing
and cooking gruel in Haiti.
(b) The units for your interval are mg/100 g of blend. To meaningfully interpret
the amount lost, it is necessary to know something about the amount that we
started with. For example, a loss of 50 mg/100 g would probably not be of much
concern if we started with 5000 mg/100 g. The specifications call for the blend to
contain 98 mg/100 g (dry basis). The difference between this specification and the
“before” values above is due to sample variation in the manufacturing process and
the handling of the product from the time it was manufactured until it was used
to prepare gruel in these Haitian homes. Express the “after” data as percent of
specification and give a 95% confidence interval for the mean percent.
7.62 A bank wonders whether omitting the annual credit card fee for customers who
charge at least $3000 in a year would increase the amount charged on its credit card.
The bank makes this offer to an SRS of 500 of its existing credit card customers.
It then compares how much these customers charge this year with the amount that
they charged last year. The mean increase is $565, and the standard deviation is
$267.
(a) Is there significant evidence at the 1% level that the mean amount charged
increases under the no-fee offer? State H0 and Ha and carry out a t test.
(b) Give a 95% confidence interval for the mean amount of the increase.
(c) The distribution of the amount charged is skewed to the right, but outliers are
prevented by the credit limit that the bank enforces on each card. Use of the t
procedures is justified in this case even though the population distribution is not
Normal. Explain why.
(d) A critic points out that the customers would probably have charged more this
year than last even without the new offer because the economy is more prosperous
and interest rates are lower. Briefly describe the design of an experiment to study
the effect of the no-fee offer that would avoid this criticism.
7.63 In a randomized comparative experiment on the effect of dietary calcium on

blood pressure, 54 healthy white males were divided at random into two groups.
One group received calcium; the other, a placebo. At the beginning of the study,
the researchers measured many variables on the subjects. The paper reporting the
study gives x = 114.9 and s = 9.3 for the seated systolic blood pressure of the 27
members of the placebo group.
(a) Give a 95% confidence interval for the mean blood pressure of the population
from which the subjects were recruited.
(b) What assumptions about the population and the study design are required by
the procedure you used in (a)? Which of these assumptions are important for the
validity of the procedure in this case?
7.64 How accurate are radon detectors of a type sold to homeowners? To answer
this question, university researchers placed 12 detectors in a chamber that exposed
them to 105 picocuries per liter (pCi/l) of radon. The detector readings were as
follows:
91.9 97.8 111.4 122.3 105.4 95.0
103.8 99.6 96.6 119.3 104.8 101.7
(a) Make a stemplot of the data. The distribution is somewhat skewed to the right,
but not strongly enough to forbid use of the t procedures.
(b) Is there convincing evidence that the mean reading of all detectors of this type
differs from the true value of 105? Carry out a test in detail and write a brief
conclusion.
Section 7.1 185
7.65 The researchers studying vitamin C in CSB were also interested in a similar
commodity called wheat-soy blend (WSB). A major concern was the possibility
that some of the vitamin C content would be destroyed as a result of storage and
shipment of the commodity to its final destination. The researchers specially marked
a collection of bags at the factory and took a sample from each of these to determine
the vitamin C content. Five months later in Haiti they found the specially marked
bags and took samples. The data consist of two vitamin C measures for each bag,
one at the time of production in the factory and the other five months later in Haiti.
The units are mg/100 g. Here are the data:
Factory Haiti Factory Haiti Factory Haiti
44 40 45 38 39 43
50 37 32 40 52 38
48 39 47 35 45 38
44 35 40 38 37 38
42 35 38 34 38 41
47 41 41 35 44 40
49 37 43 37 43 35
50 37 40 34 39 38
39 34 37 40 44 36
(a) Describe the data graphically and numerically. Summarize your results.
(b) Set up hypotheses to examine the question of interest to these researchers.
(c) Perform the significance test and summarize your results.
(d) Find 95% confidence intervals for the mean at the factory, the mean five months
later in Haiti, and for the change.
7.66 The following table gives the pretest and posttest scores on the MLA listening
test in Spanish for 20 high school Spanish teachers who attended an intensive sum-
mer course in Spanish. The setting is identical to the one described in the previous
exercise.
Teacher Pretest Posttest Teacher Pretest Posttest
1 30 29 11 30 32
2 28 30 12 29 28
3 31 32 13 31 34
4 26 30 14 29 32
5 20 16 15 34 32
6 30 25 16 20 27
7 34 31 17 26 28
8 15 18 18 25 29
9 28 33 19 31 32
10 20 25 20 29 32
Summarize the data graphically and numerically. Then analyze the data using a
significance test and a confidence interval. Write a short report summarizing your
results.
7.67 Exercise 7.60 gives data on the amount of vitamin C in gruel made from wheat-
soy blend in 5 Haitian households before and after cooking. Is there evidence that
the median amount of vitamin C is less after cooking? State hypotheses, carry out
a sign test, and report your conclusion.
7.68 Apply the sign test to the data in Exercise 7.68 to assess the effects of piano
lessons on spatial-temporal reasoning.
(a) State the hypotheses two ways: in terms of a population median and in terms
of the probability of an improvement in the test score.
(b) Carry out the sign test. Find the approximate P -value using the Normal ap-
proximation to the binomial distributions, and report your conclusion.
7.69 Use the sign test to assess whether the summer institute in Exercise 7.66
improves Spanish listening skills. State the hypotheses, give the P -value using the
binomial table (Table C), and report your conclusion.
7.70 C-reactive protein (CRP) is a substance that can be measured in the blood.
Values increase substantially within 6 hours of an infection and reach a peak within
24 to 48 hours after. In adults, chronically high values have been linked to an
increased risk of cardiovascular disease. In a study of apparently healthy children
aged 6 to 60 months in Papua New Guinea, CRP was measured in 90 children (data
provided by Francisco Rosales of the Department of Nutritional Sciences, Penn State
University). The units are milligrams per liter (mg/l). Here are the data from a
random sample of 40 of these children.
0.00 3.90 5.64 8.22 0.00 5.62 3.92 6.81 30.61 0.00
73.20 0.00 46.70 0.00 0.00 26.41 22.82 0.00 0.00 3.49
0.00 0.00 4.81 9.57 5.36 0.00 5.66 0.00 59.76 12.38
15.74 0.00 0.00 0.00 0.00 9.37 20.78 7.10 7.89 5.53
(a) Look carefully at the data above. Do you think that there are outliers or is this a
skewed distribution? Now use a histogram or stemplot to examine the distribution.
Write a short summary describing the distribution.
(b) Do you think that the mean is a good characterization of the center of this
distribution? Explain why or why not.
(c) Find a 95% confidence interval for the mean CRP. Discuss the appropriateness
of using this methodology for these data.
7.71 Refer to the previous exercise. With strongly skewed distributions such as
this, we frequently reduce the skewness by taking a log transformation. We have a
bit of a problem here, however, because some of the data are recorded as 0.00 and
the logarithm of zero is not defined. For this variable, the value 0.00 is recorded
whenever the amount of CRP in the blood is below the level that the measuring
instrument is capable of detecting. The usual procedure in this circumstance is to
add a small number to each observation before taking the logs. Transform these data
by adding 1 to each observation and then taking the logarithm. Use the questions in
the previous exercise as a guide to your analysis, and prepare a summary contrasting
this analysis with the one that you performed in the previous exercise.
7.72 In the Papua New Guinea study that provided the data for the previous two
exercises, the researchers also measured serum retinol. A low value of this variable
Section 7.2 187
can be an indicator of vitamin A deficiency. Below are the data on the same sample
of 40 children from this study. The units are micromoles per liter (µmol/l).
1.15 1.36 0.38 0.34 0.35 0.37 1.17 0.97 0.97 0.67
0.31 0.99 0.52 0.70 0.88 0.36 0.24 1.00 1.13 0.31
1.44 0.35 0.34 1.90 1.19 0.94 0.34 0.35 0.33 0.69
0.69 1.04 0.83 1.11 1.02 0.56 0.82 1.20 0.87 0.41
Analyze these data. Use the questions in the previous two exercises as a guide.
7.73 The level of various substances in the blood of kidney dialysis patients is
of concern because kidney failure and dialysis can lead to nutritional problems.
A researcher performed blood tests on several dialysis patients on 6 consecutive
clinic visits (data from Joan M. Susic, “Dietary phosphorus intakes, urinary and
peritoneal phosphate excretion and clearance in continuous ambulatory peritoneal
dialysis patients,” MS thesis, Purdue University, 1985). One variable measured was
the level of phosphate in the blood. Phosphate levels for an individual tend to
vary Normally over time. The data on one patient, in milligrams of phosphate per
deciliter (mg/dl) of blood, are given below:
5.6 5.1 4.6 4.8 5.7 6.4
(a) Calculate the sample mean x and its standard error.

(b) Use the t procedures to give a 90% confidence interval for this patient’s mean
phosphate level.
7.74 The normal range of values for blood phosphate levels is 2.6 to 4.8 mg/dl. The
sample mean for the patient in the previous exercise falls above this range. Is this
good evidence that the patient’s mean level in fact falls above 4.8? State H0 and Ha
and use the data in the previous exercise to carry out a t test. Between which levels
from Table D does the P -value lie? Are you convinced that the patient’s phosphate
level is higher than normal?
Section 7.2
7.75 In a study of cereal leaf beetle damage on oats, researchers measured the
number of beetle larvae per stem in small plots of oats after randomly applying one
of two treatments: no pesticide or Malathion at the rate of 0.25 pound per acre.
Here are the data:
Control: 2 4 3 4 2 3 3 5 3 2 6 3 4
Treatment: 0 1 1 2 1 2 1 1 2 1 1 1
(Based on M. C. Wilson et al., “Impact of cereal leaf beetle larvae on yields of

oats,” Journal of Economic Entomology, 62 (1969), pp. 699–702.) Is there significant
evidence at the 1% level that the mean number of larvae per stem is reduced by
Malathion? Be sure to state H0 and Ha.
7.76 A bank compares two proposals to increase the amount that its credit card
customers charge on their cards. (The bank earns a percentage of the amount
charged, paid by the stores that accept the card.) Proposal A offers to eliminate
the annual fee for customers who charge $2400 or more during the year. Proposal
B offers a small percent of the total amount charged as a cash rebate at the end of
the year. The bank offers each proposal to an SRS of 150 of its existing credit card
customers. At the end of the year, the total amount charged by each customer is
recorded. Here are the summary statistics:
Group n x s
A 150 $1987 $392
B 150 $2056 $413
(a) Do the data show a significant difference between the mean amounts charged
by customers offered the two plans? Give the null and alternative hypotheses, and
calculate the two-sample t statistic. Obtain the P -value (either approximately from
Table D or more accurately from software). State your practical conclusions.
(b) The distributions of amounts charged are skewed to the right, but outliers are
prevented by the limits that the bank imposes on credit balances. Do you think
that skewness threatens the validity of the test that you used in (a)? Explain your
answer.
7.77 What aspects of rowing technique distinguish between novice and skilled com-
petitive rowers? Researchers compared two groups of female competitive rowers:
a group of skilled rowers and a group of novices. The researchers measured many
mechanical aspects of rowing style as the subjects rowed on a Stanford Rowing Er-
gometer. One important variable is the angular velocity of the knee (roughly, the
rate at which the knee joint opens as the legs push the body back on the sliding
seat). This variable was measured when the oar was at right angles to the machine.
(Based on W. N. Nelson and C. J. Widule, “Kinematic analysis and efficiency es-
timate of intercollegiate female rowers,” unpublished manuscript, 1983.) The data
show no outliers or strong skewness. Here is the SAS computer output:
TTEST PROCEDURE
Variable: KNEE
GROUP N Mean Std Dev Std Error

----------------------------------------------------------------------
SKILLED 10 4.18283335 0.47905935 0.15149187
NOVICE 8 3.01000000 0.95894830 0.33903942
Variances T DF Prob>|T|
---------------------------------------
Unequal 3.1583 9.8 0.0104
Equal 3.3918 16.0 0.0037
(a) The researchers believed that the knee velocity would be higher for skilled rowers.
State H0 and Ha.
(b) Give the value of the two-sample t statistic and its P -value (note that SAS
provides two-sided P -values). What do you conclude?
Section 7.2 189
(c) Give a 90% confidence interval for the mean difference between the knee velocities
of skilled and novice female rowers.
7.78 The novice and skilled rowers in the previous exercise were also compared with
respect to several physical variables. Here is the SAS computer output for weight
in kilograms:
TTEST PROCEDURE
Variable: WEIGHT

----------------------------------------------------------------------
SKILLED 10 70.3700000 6.10034898 1.92909973
NOVICE 8 68.4500000 9.03999930 3.19612240
---------------------------------------
Unequal 0.5143 11.8 0.6165
Equal 0.5376 16.0 0.5982
Is there significant evidence of a difference in the mean weights of skilled and novice
rowers? State H0 and Ha , report the two-sample t statistic and its P -value, and
state your conclusion.
7.79 The Johns Hopkins Regional Talent Searches give the SAT (intended for high
school juniors and seniors) to 13-year-olds. In all, 19,883 males and 19,937 females
took the tests between 1980 and 1982. The mean scores of males and females on
the verbal test are nearly equal, but there is a clear difference between the sexes on
the mathematics test. The reason for this difference is not understood. Here are
the data (from a news article in Science, 224 (1983), pp. 1029–1031):
Group x s
Males 416 87
Females 386 74
Give a 99% confidence interval for the difference between the mean score for males
and the mean score for females in the population that Johns Hopkins searches.
7.80 Plant scientists have developed varieties of corn that have increased amounts
of the essential amino acid lysine. In a test of the protein quality of this corn, an
experimental group of 20 one-day-old male chicks was fed a ration containing the
new corn. A control group of another 20 chicks received a ration that was identical
except that it contained normal corn. Here are the weight gains (in grams) after
21 days. (Based on G. L. Cromwell et al., “A comparison of the nutritive value of
opaque-2, floury-2 and normal corn for the chick,” Poultry Science, 47 (1968), pp.
840–847.)
Control Experimental
380 321 366 356 361 447 401 375
283 349 402 462 434 403 393 426
356 410 329 399 406 318 467 407
350 384 316 272 427 420 477 392
345 455 360 431 430 339 410 326
(a) Present the data graphically. Are there outliers or strong skewness that might
prevent the use of t procedures?
(b) State the hypotheses for a statistical test of the claim that chicks fed high-lysine
corn gain weight faster. Carry out the test. Is the result significant at the 10%
level? At the 5% level? At the 1% level?
(c) Give a 95% confidence interval for the mean extra weight gain in chicks fed
high-lysine corn.
7.81 The data on weights of skilled and novice rowers in Exercise 7.78 can be
analyzed by the pooled t procedures, which assume equal population variances.
Report the value of the t statistic, its degrees of freedom, and its P -value, and then
state your conclusion. (The pooled procedures should not be used for the comparison
of knee velocities in Exercise 7.77, because the sample standard deviations in the
two groups are different enough to cast doubt on the assumption of a common
population standard deviation.)
7.82 Pat wants to compare the cost of one- and two-bedroom apartments in the
area of your campus. She collects data for a random sample of 10 advertisements
of each type. Here are the rents for the two-bedroom apartments (in dollars per
month):
595, 500, 580, 650, 675, 675, 750, 500, 495, 670
Here are the rents for the one-bedroom apartments:
500, 650, 600, 505, 450, 550, 515, 495, 650; 395
Find a 95% confidence interval for the additional cost of a second bedroom.
7.83 Pat wonders if two-bedroom apartments rent for significantly more than one-
bedroom apartments. Use the data in the previous exercise to find out.
(a) State appropriate null and alternative hypotheses.
(b) Report the test statistic, its degrees of freedom, and the P -value. What do you
conclude?
(c) Can you conclude that every one-bedroom apartment costs less than every two-
bedroom apartment?
(d) In the previous exercise you found a confidence interval. In this exercise you per-
formed a significance test. Which do you think is more useful to someone planning
to rent an apartment? Why?
7.84 Physical fitness is related to personality characteristics. In one study of this

relationship, middle-aged college faculty who had volunteered for a fitness program
Section 7.2 191
were divided into low-fitness and high-fitness groups based on a physical examina-
tion. The subjects then took the Cattell Sixteen Personality Factor Questionnaire.
(A. H. Ismail and R. J. Young,“The effect of chronic exercise on the personality of
middle-aged men,”Journal of Human Ergology, 2 (1973), pp. 47–57.) Here are the
data for the “ego strength” personality factor:
Low fitness High fitness

4.99 5.53 3.12 6.68 5.93 5.71
4.24 4.12 3.77 6.42 7.08 6.20
4.74 5.10 5.09 7.32 6.37 6.04
4.93 4.47 5.40 6.38 6.53 6.51
4.16 5.30 6.16 6.68
(a) Is the difference in mean ego strength significant at the 5% level? At the 1%
level? Be sure to state H0 and Ha.
(b) You should be hesitant to generalize these results to the population of all middle-
aged men. Explain why.
7.85 The U.S. Department of Agriculture (USDA) uses many types of surveys to ob-
tain important economic estimates. In one pilot study they estimated wheat prices
in July and in September using independent samples. Here is a brief summary from
the report:
√
Month n x̄ s/ n
July 90 $2.95 $0.023
September 45 $3.61 $0.029
(a) Note that the report gave standard errors. Find the standard deviation for each
of the samples.
(b) Use a significance test to examine whether or not the price of wheat was the same
in July and September. Be sure to give details and carefully state your conclusion.
7.86 Refer to the previous exercise. Give a 95% confidence interval for the increase
in price between July and September.
7.87 A market research firm supplies manufacturers with estimates of the retail
sales of their products from samples of retail stores. Marketing managers are prone
to look at the estimate and ignore sampling error. Suppose that an SRS of 75 stores
this month shows mean sales of 52 units of a small appliance, with standard devi-
ation 13 units. During the same month last year, an SRS of 53 stores gave mean
sales of 49 units, with standard deviation 11 units. An increase from 49 to 52 is a
rise of 6%. The marketing manager is happy, because sales are up 6%.
(a) Use the two-sample t procedure to give a 95% confidence interval for the differ-
ence in mean number of units sold at all retail stores.
(b) Explain in language that the manager can understand why he cannot be certain
that sales rose by 6%, and that in fact sales may even have dropped.
7.88 In a study of heart surgery, one issue was the effect of drugs called beta-blockers
on the pulse rate of patients during surgery. The available subjects were divided
at random into two groups of 30 patients each. One group received a beta-blocker;
the other, a placebo. The pulse rate of each patient at a critical point during the
operation was recorded. The treatment group had mean 65.2 and standard deviation
7.8. For the control group, the mean was 70.3 and the standard deviation was 8.3.
(a) Do beta-blockers reduce the pulse rate? State the hypotheses and do a t test.
Is the result significant at the 5% level? At the 1% level?
(b) Give a 99% confidence interval for the difference in mean pulse rates.
7.89 The following table shows Consumer Reports magazine’s laboratory test results
for calories and milligrams of sodium (mostly due to salt) in a number of major
brands of hot dogs. There are three types: all beef, “meat” (mainly pork and beef,
but government regulations allow up to 15% poultry meat), and poultry. (Consumer
Reports, June 1986, pp. 366–367.)

186 495 173 458 129 430
181 477 191 506 132 375
176 425 182 473 102 396
149 322 190 545 106 383
184 482 172 496 94 387
190 587 147 360 102 542
158 370 146 387 87 359
139 322 139 386 99 357
175 479 175 507 170 528
148 375 136 393 113 513
152 330 179 405 135 426
111 300 153 372 142 513
141 386 107 144 86 358
153 401 195 511 143 581
190 645 135 405 152 588
157 440 140 428 146 522
131 317 138 339 144 545
149 319
135 298
132 253
(a) Give a 95% confidence interval for the difference in mean calorie content between
beef and poultry hot dogs.
(b) Based on your confidence interval, can the hypothesis that the population means
are equal be rejected at the 5% significance level? Explain your answer.
(c) What assumptions does your statistical procedure in (a) require? Which of these
assumptions are justified or not important in this case? Are any of the assumptions
doubtful in this case?
7.90 The following table gives data on the blood pressure before and after treatment
for two groups of black males.
Section 7.2 193
Calcium group Placebo group

Begin End Decrease Begin End Decrease
107 100 7 123 124 –1
110 114 –4 109 97 12
123 105 18 112 113 –1
129 112 17 102 105 –3
112 115 –3 98 95 3
111 116 –5 114 119 –5
107 106 1 119 114 5
112 102 10 114 112 2
136 125 11 110 121 –11
102 104 –2 117 118 –1
130 133 –3
One group took a calcium supplement, and the other group received a placebo.
(a) Perform the significance test using a two-sample t test that does not require equal
population standard deviations. Compare your P -value with the result P = 0.059
for the pooled t test.
(b) Give a 90% confidence interval for the difference in means, again using a proce-
dure that does not require equal standard deviations. How does the margin of error
of your interval compare with 5.6 mm, the margin of error for the pooled t test?
7.91 Researchers studying the learning of speech often compare measurements made
on the recorded speech of adults and children. One variable of interest is called the
voice onset time (VOT). Here are the results for 6-year-old children and adults asked
to pronounce the word “bees.” The VOT is measured in milliseconds and can be
either positive or negative. (M. A. Zlatin and R. A. Koenigsknecht, “Development
of the voicing contrast: a comparison of voice onset time in stop perception and
production,” Journal of Speech and Hearing Research, 19 (1976), pp. 93–111.)
Group n x s
Children 10 −3.67 33.89
Adults 20 −23.17 50.74
(a) What is the standard error of the sample mean VOT for the 20 adult subjects?
What is the standard error of the difference x1 − x2 between the mean VOT for
children and adults?
(b) The researchers were investigating whether VOT distinguishes adults from chil-
dren. State H0 and Ha and carry out a two-sample t test. Give a P -value and
report your conclusions.
(c) Give a 95% confidence interval for the difference in mean VOTs when pronounc-
ing the word “bees.” Explain why you knew from your result in (b) that this interval
would contain 0 (no difference).
7.92 The researchers in the study discussed in the previous exercise looked at VOTs
for adults and children pronouncing several different words. Explain why they should
not perform a separate two-sample t test for each word and conclude that the words
with a significant difference (say, P < 0.05) distinguish children from adults. (The
researchers did not make this mistake.)
7.93 Repeat the comparison of mean VOTs for children and adults in Exercise 7.91
using a pooled t procedure. (In practice, we would not pool in this case, because
the data suggest some difference in the population standard deviations.)
(a) Carry out the significance test, and give a P -value.
(b) Give a 95% confidence interval for the difference in population means.
(c) How similar are your results to those you obtained in Exercise 7.91 from the
two-sample t procedures?
7.94 College financial aid offices expect students to use summer earnings to help
pay for college. But how large are these earnings? One college studied this question
by asking a sample of students how much they earned. Omitting students who
were not employed, 1296 responses were received. (Based on studies conducted by
Marvin Schlatter, Division of Financial Aid, Purdue University.) Here are the data
in summary form:
Group n x s
Males 675 $3297.91 $2394.65
Females 621 $2380.68 $1815.55
(a) Use the two-sample t procedures to give a 90% confidence interval for the dif-
ference between the mean summer earnings of male and female students.
(b) The distribution of earnings is strongly skewed to the right. Nevertheless, use
of t procedures is justified. Why?
(c) Once the sample size was decided, the sample was chosen by taking every kth
name from an alphabetical list of undergraduates. Is it reasonable to consider the
sample as two SRSs chosen from the male and female undergraduate populations?
(d) What other information about the study would you request before accepting the
results as describing all undergraduates?
7.95 The pesticide DDT causes tremors and convulsions if it is ingested by humans
or other mammals. Researchers seek to understand how the convulsions are caused.
In a randomized comparative experiment, 6 white rats poisoned with DDT were
compared with a control group of 6 unpoisoned rats. Electrical measurements of
nerve activity are the main clue to the nature of DDT poisoning. When a nerve is
stimulated, its electrical response shows a sharp spike followed by a much smaller
second spike. Researchers found that the second spike is larger in rats fed DDT
than in normal rats. This observation helps biologists understand how DDT causes
tremors. (This example is loosely based on D. L. Shankland, “Involvement of spinal
cord and peripheral nerves in DDT-poisoning syndrome in albino rats,” Toxicology
and Applied Pharmacology, 6 (1964), pp. 197–213.)
The researchers measured the amplitude of the second spike as a percentage of
the first spike when a nerve in the rat’s leg was stimulated. For the poisoned rats
the results were
12.207 16.869 25.050 22.429 8.456 20.589
The control group data were
11.074 9.686 12.064 9.351 8.182 6.642

Section 7.2 195
Normal quantile plots show no evidence of outliers or strong skewness. Both pop-
ulations are reasonably Normal, as far as can be judged from 6 observations. The
difference in means is quite large, but in such small samples the sample mean is
highly variable. A significance test can help confirm that we are seeing a real effect.
Because the researchers did not conjecture in advance that the size of the second
spike would increase in rats fed DDT, we test
H0: µ1 = µ2
Ha: µ1 6= µ2
Here is the output from a statistical software system for these data:
TTEST PROCEDURE
Variable: SPIKE

----------------------------------------------------------------
DDT 6 17.60000000 6.34014839 2.58835474
CONTROL 6 9.49983333 1.95005932 0.79610839
---------------------------------------
Unequal 2.9912 5.9 0.0247
Equal 2.9912 10.0 0.0135
(a) Interpret the output.

(b) Starting from the computer’s results for xi and si , verify the values given for
the test statistic t = 2.99 and the degrees of freedom df = 5.9.
7.96 The Chapin Social Insight Test is a psychological test designed to measure how
accurately the subject appraises other people. The possible scores on the test range
from 0 to 41. During the development of the Chapin test, it was given to several
different groups of people. Here are the results for male and female college students
majoring in the liberal arts:
Group Sex n x s
1 Male 133 25.34 5.05
2 Female 162 24.94 5.44
Do these data support the contention that female and male students differ in average
social insight? Use the pooled two-sample procedure and the procedure that does
not assume that the standard deviations are the same. Compare the results.
7.97 In each of the following situations explain what is wrong and why.
(a) A researcher wants to test H0: x̄1 = x̄2 versus the two-sided alternative H1: x̄1 6=
x̄2 .
(b) A study recorded the scores of 20 children who were similar in age. The scores
of the 10 boys in the study were compared with the scores of all 20 children using
the two-sample methods of this section.

(c) A two-sample t statistic gave a P -value of 0.96. From this you can reject the
null hypothesis with 95% confidence.
7.98 For each of the following, answer the question and give a short explanation of
your reasoning.
(a) A 95% confidence interval for the difference between two means is reported as
(1.6, 2.3). What can you conclude about the results of a significance test of the null
hypothesis that the population means are equal versus the two-sided alternative?
(b) Will larger samples generally give a larger or smaller margin of error for the
difference between two sample means?
7.99 For each of the following, answer the question and give a short explanation of
your reasoning.
(a) A significance test for comparing two means gave t = −3.11 with 23 degrees
of freedom. Can you reject the null hypothesis that the µ’s are equal versus the
two-sided alternative at the 5% significance level?
(b) Answer part (a) for the one-sided alternative that the difference in means is
positive.
7.100 You want to compare the daily sales for two different designs of Web pages
for your Internet business. You assign the next 60 days to either Design A or Design
B, 30 days to each.
(a) Would you use a one-sided or two-sided significance test for this problem? Ex-
plain your choice.
(b) If you use Table D to find the critical value, what are the degrees of freedom?
(c) The t statistic for comparing the mean sales is 2.06. If you use Table D, what
P -value would you report? What would you conclude?
7.101 If you perform the significance test in the previous exercise using level α =
0.05, how large (positive or negative) must the t statistic be to reject the null
hypothesis that the two designs give the same average sales?
7.102 Do piano lessons improve the spatial-temporal reasoning of preschool chil-

dren? We examined this question in Exercises 7.58 and 7.59 by analyzing the
change in spatial-temporal reasoning of 34 preschool children after six months of
piano lessons. Here we examine the same question by comparing the changes of
those students with the changes of 44 children in a control group. Here are the data
for the children who took piano lessons:
2 5 7 –2 2 7 4 1 0 7 3 4 3 4 9 4 5
2 9 6 0 3 6 –1 3 4 6 7 –2 7 –3 3 4 4
The control group scores are
1 –1 0 1 –4 0 0 1 0 –1 0 1 1 –3 –2
4 –1 2 4 2 2 2 –3 –3 0 2 0 –1 3 –1
5 –1 7 0 4 0 2 1 –6 0 2 –1 0 –2
Section 7.2 197
(a) Display the data and summarize the distributions.

(b) Make a table with the sample size, the mean, the standard deviation, and the
standard error of the mean for each of the two groups.
(c) Translate the question of interest into hypotheses, test them, and summarize
your conclusions.
7.103 Refer to the previous exercise. Give a 95% confidence interval that describes
the comparison between the children who took piano lessons and the controls.
7.104 Refer to Exercises 7.58 and 7.59 and the previous two exercises. We have
used four ways to address the question of interest. Discuss the relative merits of
each approach.
7.105 In what ways are companies that fail different from those that continue to do
business? A study compared various characteristics of 68 healthy and 33 failed firms.
One of the variables was the ratio of current assets to current liabilities. Roughly
speaking, this is the amount that the firm is worth divided by what it owes. The
data are given in the following table.
Ratio of current assets to current liabilities
Healthy firms Failed firms
1.50 0.10 1.76 1.14 1.84 2.21 0.82 0.89 1.31
2.08 1.43 0.68 3.15 1.24 2.03 0.05 0.83 0.90
2.23 2.50 2.02 1.44 1.39 1.64 1.68 0.99 0.62
0.89 0.23 1.20 2.16 1.80 1.87 0.91 0.52 1.45
1.91 1.67 1.87 1.21 2.05 1.06 1.16 1.32 1.17
0.93 2.17 2.61 3.05 1.52 1.93 0.42 0.48 0.93
1.95 2.61 1.11 0.95 0.96 2.25 0.88 1.10 0.23
2.73 1.56 2.73 0.90 2.12 1.42 1.11 0.19 0.13
1.62 1.76 2.22 2.80 1.85 0.96 2.03 0.51 1.12
1.71 1.02 2.50 1.55 1.69 1.64 0.92 0.26 1.15
1.03 1.80 0.67 2.44 2.30 2.21 0.13 0.88 0.09
1.96 1.81
(a) Display the data so that the two distributions can be compared. Describe the
shapes of the distributions and any important characteristics.
(b) We expect that failed firms will have a lower ratio. Describe and test appropriate
hypotheses for these data. What do you conclude?
(c) It is not possible to do a randomized experiment for this kind of question. Explain
why.
7.106 Does cocaine use by pregnant women cause their babies to have low birth
weight? To study this question, birth weights of babies of women who tested positive
for cocaine/crack during a drug-screening test were compared with the birth weights
of babies whose mothers either tested negative or were not tested, a group we call
“other.” Here are the summary statistics. The birth weights are measured in grams.
Group n x s
Positive test 134 2733 599
Other 5974 3118 672
(a) Formulate appropriate hypotheses and carry out the test of significance for these
data.
(b) Give a 95% confidence interval for the mean difference in birth weights.
(c) Discuss the limitations of the study design. What do you believe can be con-
cluded from this study?
7.107 The Survey of Study Habits and Attitudes (SSHA) is a psychological test
designed to measure the motivation, study habits, and attitudes toward learning
of college students. These factors, along with ability, are important in explaining
success in school. Scores on the SSHA range from 0 to 200. A selective private
college gives the SSHA to an SRS of both male and female first-year students. The
data for the women are as follows:
154 109 137 115 152 140 154 178 101
103 126 126 137 165 165 129 200 148
Here are the scores of the men:
108 140 114 91 180 115 126 92 169 146
109 132 75 88 113 151 70 115 187 104
(a) Examine each sample graphically, with special attention to outliers and skewness.
Is use of a t procedure acceptable for these data?
(b) Most studies have found that the mean SSHA score for men is lower than the
mean score in a comparable group of women. Test this supposition here. That is,
state hypotheses, carry out the test and obtain a P -value, and give your conclusions.
(c) Give a 90% confidence interval for the mean difference between the SSHA scores
of male and female first-year students at this college.
7.108 Does bread lose its vitamins when stored? Small loaves of bread were prepared
with flour that was fortified with a fixed amount of vitamins. After baking, the
vitamin C content of two loaves was measured. Another two loaves were baked at
the same time, stored for three days, and then the vitamin C content was measured.
The units are milligrams per hundred grams of flour (mg/100 g) (Helen Park et al.,
“Fortifying bread with each of three antioxidants,” Cereal Chemistry, 74 (1997), pp.
202–206). Here are the data:
Immediately after baking: 47.62, 49.79
Three days after baking: 21.25, 22.34
(a) When bread is stored, does it lose vitamin C? To answer this question, perform a
two-sample t test for these data. Be sure to state your hypotheses, the test statistic
with degrees of freedom, and the P -value.
(b) Give a 90% confidence interval for the amount of vitamin C lost.
7.109 Suppose that the researchers in the previous exercise could have measured
the same two loaves of bread immediately after baking and again after three days.
Assume that the data given had come from this study design. (Assume that the
values given in the previous exercise are for first loaf and second loaf from left to
right.)
(a) Explain carefully why your analysis in the previous exercise is not correct now,
Section 7.3 199
even though the data are the same.

(b) Redo the analysis for the design based on measuring the same loaves twice.
7.110 Refer to the previous two exercises. The amount of vitamin E (in mg/100 g
of flour) in the same loaves was also measured. Here are the data:
Immediately after baking: 94.6, 96.0

Three days after baking: 97.4, 94.3
(a) When bread is stored, does it lose vitamin E? To answer this question, perform a
two-sample t test for these data. Be sure to state your hypotheses, the test statistic
with degrees of freedom, and the P -value.
(b) Give a 90% confidence interval for the amount of vitamin E lost.
7.111 Refer to the previous three exercises. Some people claim that significance
tests with very small samples never lead to rejection of the null hypothesis. Discuss
this claim using the results of these two exercises.
7.112 Refer to the previous four exercises. The analysis of the loss of vitamin C
when bread is stored is a rather unusual case involving very small sample sizes.
There are only two observations per condition (immediately after baking and three
days later). When the samples are so small, we have very little information to
make a judgment about whether the population standard deviations are equal. The
potential gain from pooling is large when the sample sizes are very small. Assume
that we will perform a two-sided test using the 5% significance level.
(a) Find the critical value for the unpooled t test statistic that does not assume
equal variances. Use the minimum of n1 − 1 and n2 − 1 for the degrees of freedom.
(b) Find the critical value for the pooled t test statistic.
(c) How does comparing these critical values show an advantage of the pooled test?
Section 7.3
7.113 The F statistic F = s21 /s22 is calculated from samples of size n1 = 10 and
n2 = 21. (Remember that n1 is the numerator sample size.)
(a) What is the upper 5% critical value for this F ?
(b) In a test of equality of standard deviations against the two-sided alternative,
this statistic has the value F = 2.45. Is this value significant at the 10% level? Is it
significant at the 5% level?
7.114 The F statistic for equality of standard deviations based on samples of sizes
n1 = 21 and n2 = 26 takes the value F = 2.88.
(a) Is this significant evidence of unequal population standard deviations at the 5%
level?
(b) Use Table E to give an upper and a lower bound for the P -value.
7.115 Exercise 7.77 records the results of comparing a measure of rowing style
for skilled and novice female competitive rowers. Is there significant evidence of
inequality between the standard deviations of the two populations?
(a) State H0 and Ha .

(b) Calculate the F statistic. Between which two levels does the P -value lie?
7.116 Answer the same questions for the weights of the two groups, recorded in
Exercise 7.78.
7.117 The observed inequality between the sample standard deviations of male and
female SAT Mathematics scores in Exercise 7.79 is clearly significant. You can say
this without doing any calculations. Find F and look in Table E. Then explain why
the significance of F could be seen without arithmetic.
7.118 An F statistic will be used to compare two variances. The sample sizes are
both 20. How large does the ratio of the largest to the smallest variance need to be
for the significance test to reject the null hypothesis that the population variances
are the same?
7.119 The F statistic F = s21 /s22 is calculated from samples of size n1 = 16 and
n2 = 20. (Remember that n1 is the numerator sample size.)
(a) What is the upper 5% critical value for this F ?
(b) In a test of equality of standard deviations against the two-sided alternative,
this statistic has the value F = 2.71. Is this value significant at the 5% level? Is it
significant at the 1% level?
7.120 The F statistic for equality of standard deviations based on samples of sizes
n1 = 31 and n2 = 28 takes the value F = 1.72.
(a) Is this significant evidence of unequal population standard deviations at the 5%
level?
(b) Use Table E to give an upper and a lower bound for the P -value.
7.121 Exercise 7.82 compares the rents of one-bedroom and two-bedroom apart-
ments. Is there any evidence in the data that would lead us to conclude that the
standard deviations are different? State the appropriate hypotheses, calculate the
test statistic, and write a short summary of the results.
7.122 A USDA survey used to estimate wheat prices in July and September is
described in Exercise 7.85. Using the standard deviations you calculated there,
perform the test for equality of standard deviations and summarize your conclusion.
7.123 The data for VOTs of children and adults in Exercise 7.91 show quite different
sample standard deviations. How statistically significant is the observed inequality?
7.124 Suppose that you wanted to compare intramural basketball players and in-
tramural soccer players on the “ego strength” personality factor described in Ex-
ercise 7.84. With the data from that exercise, you will use σ = 0.7 for planning
purposes. The pooled two-sample t test with α = 0.05 will be used to make the
comparison. Based on Exercise 7.84, you judge a difference of 0.5 points to be of
interest. Pick several values of n and find the power. Plot the power versus n and
use the plot to find a value of n that will give approximately 80% power. Calculate
the power for the value of n that you found.
7.125 An F statistic will be used to compare two variances. How large does the
ratio of the largest to the smallest variance need to be for the significance test to
reject the null hypothesis that the population variances are the same in the following
settings? Use the 5% level of significance.
(a) The two sample sizes are 5.
(b) The two sample sizes are 10.
(c) The two sample sizes are 26.
(d) What do you conclude?
7.126 Return to the SSHA data in Exercise 7.107. SSHA scores are generally less
variable among women than among men. We want to know whether this is true for
this college.
(a) State H0 and Ha . Note that Ha is one-sided in this case.
(b) Because Table E contains only upper critical values for F , a one-sided test
requires that in calculating F the numerator s2 belongs to the group that Ha claims
to have the larger σ. Calculate this F .
(c) Compare F to the entries in Table E (no doubling of p) to obtain the P -value.
Be sure the degrees of freedom are in the proper order. What do you conclude about
the variation in SSHA scores?
7.127 In Exercise 7.106 data on cocaine use and birth weight are summarized. The
study has been criticized because of several design problems. Suppose that you are
designing a new study. Based on the results in Exercise 7.106, you think that the
true difference in mean birth weights may be about 350 grams (g); a difference this
large is clinically important. For planning purposes assume that you will have 75
women in each group and that the common standard deviation is 650 g, a guess
that is between the two standard deviations in Exercise 7.106. If you use a pooled
two-sample t test with a Type I error of 0.05, what is the power of the test for this
design?
7.128 Refer to the previous exercise. Repeat the power calculation for 20, 40, 60,
80, 100, and 120 women in each group. Plot the power versus the sample size and
write a short summary of the results.
7.129 Refer to the previous two exercises. For each of the sample sizes considered,
what is your guess at the margin of error for the 95% confidence interval for the
difference in mean weights? Display these results with a graph or a sketch.
7.130 We studied the loss of vitamin C when bread is stored in Exercise 7.108.
Recall that two loaves were measured immediately after baking and another two
loaves were measured after three days of storage. These are very small sample sizes.
(a) Use Table E to find the value that the ratio of variances would have to exceed
for us to reject the null hypothesis (at the 5% level) that the standard deviations
are equal. What does this suggest about the power of the test?
(b) Perform the test and state your conclusion.
7.131 Data on the numbers of manatees killed by boats each year are given in
Exercise 2.36. After a long period of increasing numbers of deaths, the pattern
flattens somewhat. In fact, the total for 1990 is 47, less than the total of 50 for
1989. Perhaps the trend has now reversed. We would like to do a significance test
to compare these two counts. Theoretical considerations suggest that the standard
√
errors (σ/ n) for these types of counts can be approximated by the square root
of the count. √ So, for example, the 1990 count, 47, has a standard error that is
approximately 47. Use this approximation to perform an approximate two-sample
z test for the difference between the 1989 and 1990 deaths. Find an approximate
95% confidence interval for the difference. What do you conclude?
7.132 In a study of the effectiveness of weight-loss programs, 47 subjects who were

at least 20% overweight took part in a group support program for 10 weeks. Private
weighings determined each subject’s weight at the beginning of the program and 6
months after the program’s end. The matched pairs t test was used to assess the
significance of the average weight loss. The paper reporting the study said, “The
subjects lost a significant amount of weight over time, t(46) = 4.68, p < 0.01.” It
is common to report the results of statistical tests in this abbreviated style. (Based
loosely on D. R. Black et al., “Minimal interventions for weight control: a cost-
effective alternative,” Addictive Behaviors, 9 (1984), pp. 279–285.)
(a) Why was the matched pairs statistic appropriate?
(b) Explain to someone who knows no statistics but is interested in weight-loss
programs what the practical conclusion is.
(c) The paper follows the tradition of reporting significance only at fixed levels such
as α = 0.01. In fact, the results are more significant than “p < 0.01” suggests.
What can you say about the P -value of the t test?
7.133 Nitrites are often added to meat products as preservatives. In a study of the
effect of these chemicals on bacteria, the rate of uptake of a radiolabeled amino acid
was measured for a number of cultures of bacteria, some growing in a medium to
which nitrites had been added. Here are the summary statistics from this study:
Group n x s
Nitrite 30 7880 1115
Control 30 8112 1250
Carry out a test of the research hypothesis that nitrites decrease amino acid uptake,
and report your results.
7.134 The one-hole test is used to test the manipulative skill of job applicants.
This test requires subjects to grasp a pin, move it to a hole, insert it, and return
for another pin. The score on the test is the number of pins inserted in a fixed time
interval. In one study, male college students were compared with experienced female
industrial workers. Here are the data for the first minute of the test: (G. Salvendy,
“Selection of industrial operators: the one-hole test,” International Journal of Pro-
duction Research, 13 (1973), pp. 303–321.)
Group n x s
Students 750 35.12 4.31
Workers 412 37.32 3.83
(a) It was expected that the experienced workers would outperform the students,
at least during the first minute, before learning occurs. State the hypotheses for a
statistical test of this expectation and perform the test. Give a P -value and state
your conclusions.
(b) The distribution of scores is slightly skewed to the left. Explain why the proce-
dure you used in (a) is nonetheless acceptable.
(c) One purpose of the study was to develop performance norms for job applicants.
Based on the data above, what is the range that covers the middle 95% of experi-
enced workers? (Be careful! This is not the same as a 95% confidence interval for
the mean score of experienced workers.)
(d) The five-number summary of the distribution of scores among the workers is
23 33.5 37 40.5 46
for the first minute and

32 39 44 49 59
for the fifteenth minute of the test. Display these facts graphically, and describe
briefly the differences between the distributions of scores in the first and fifteenth
minute.
7.135 The composition of the earth’s atmosphere may have changed over time.
One attempt to discover the nature of the atmosphere long ago studies the gas
trapped in bubbles inside ancient amber. Amber is tree resin that has hardened
and been trapped in rocks. The gas in bubbles within amber should be a sample of
the atmosphere at the time the amber was formed. Measurements on specimens of
amber from the late Cretaceous era (75 to 95 million years ago) give these percents
of nitrogen:
63.4 65.0 64.4 63.3 54.8 64.5 60.8 49.1 51.0
These values are quite different from the present 78.1% of nitrogen in the atmo-
sphere. Assume (this is not yet agreed on by experts) that these observations are
an SRS from the late Cretaceous atmosphere. (Data from R. A. Berner and G. P.
Landis,“Gas bubbles in fossil amber as possible indicators of the major gas compo-
sition of ancient air,” Science, 239 (1988), pp. 1406–1409.)
(a) Graph the data, and comment on skewness and outliers.
(b) The t procedures will be only approximate in this case. Give a 90% t confidence
interval for the mean percent of nitrogen in ancient air.
7.136 The table in Exercise 1.19 gives the number of medical doctors per 100,000
population by state. Is it proper to apply the one-sample t method to these data to
give a 95% confidence interval for the mean number of medical doctors per 100,000
population per state? Explain your answer.
7.137 The amount of lead in a certain type of soil, when released by a standard
extraction method, averages 86 parts per million (ppm). A new extraction method
is tried on 40 specimens of the soil, yielding a mean of 83 ppm lead and a standard
deviation of 10 ppm.
(a) Is there significant evidence at the 5% level that the new method frees less lead
from the soil? What about the 1% level?
(b) A critic argues that because of variations in the soil, the effectiveness of the new
method is confounded with characteristics of the particular soil specimens used.
Briefly describe a better data production design that avoids this criticism.
7.138 High levels of cholesterol in the blood are not healthy in either humans or
dogs. Because a diet rich in saturated fats raises the cholesterol level, it is plau-
sible that dogs owned as pets have higher cholesterol levels than dogs owned by a
veterinary research clinic. “Normal” levels of cholesterol based on the clinic’s dogs
would then be misleading. A clinic compared healthy dogs it owned with healthy
pets brought to the clinic to be neutered. (V. D. Bass, W. E. Hoffmann, and J. L.
Dorner, “Normal canine lipid profiles and effects of experimentally induced pancre-
atitis and hepatic necrosis on lipids,” American Journal of Veterinary Research, 37
(1976), pp. 1355–1357.) The summary statistics for blood cholesterol levels (mil-
ligrams per deciliter of blood) appear below:
Group n x s
Pets 26 193 68
Clinic 23 174 44
(a) Is there strong evidence that pets have a higher mean cholesterol level than clinic
dogs? State the H0 and Ha and carry out an appropriate test. Give the P -value
and state your conclusion.
(b) Give a 95% confidence interval for the difference in mean cholesterol levels be-
tween pets and clinic dogs.
(c) Give a 95% confidence interval for the mean cholesterol level in clinic dogs.
(d) What assumptions must be satisfied to justify the procedures you used in (a),
(b), and (c)? Assuming that the cholesterol measurements have no outliers and are
not strongly skewed, what is the chief threat to the validity of the results of this
study?
7.139 Elite distance runners are thinner than the rest of us. Here are data on
skinfold thickness, which indirectly measures body fat, for 20 elite runners and 95
ordinary men in the same age group. (M. L. Pollock et al., “Body composition of
elite class distance runners,” in P. Milvey (ed.), The Marathon: Physiological, Med-
ical, Epidemiological, and Psychological Studies, New York Academy of Sciences,
1977, p. 366.) The data are in millimeters and are given in the form “mean (stan-
dard deviation).”
Runners Others
Abdomen 7.1 (1.0) 20.6 (9.0)
Thigh 6.1 (1.8) 17.4 (6.6)
Use confidence intervals to describe the difference between runners and typical young
men.
7.140 The following table gives the levels of three pollutants in the exhaust of
46 randomly selected vehicles of the same type. You will investigate emissions of
nitrogen oxides (NOX).
Amounts of three pollutants emitted by light-duty egines (grams per mile)
EN HC CO NOX EN HC CO NOX
1 0.50 5.01 1.28 2 0.65 14.67 0.72
3 0.46 8.60 1.17 4 0.41 4.42 1.31
5 0.41 4.95 1.16 6 0.39 7.24 1.45
7 0.44 7.51 1.08 8 0.55 12.30 1.22
9 0.72 14.59 0.60 10 0.64 7.98 1.32
11 0.83 11.53 1.32 12 0.38 4.10 1.47
13 0.38 5.21 1.24 14 0.50 12.10 1.44
15 0.60 9.62 0.71 16 0.73 14.97 0.51
17 0.83 15.13 0.49 18 0.57 5.04 1.49
19 0.34 3.95 1.38 20 0.41 3.38 1.33
21 0.37 4.12 1.20 22 1.02 23.53 0.86
23 0.87 19.00 0.78 24 1.10 22.92 0.57
25 0.65 11.20 0.95 26 0.43 3.81 1.79
27 0.48 3.45 2.20 28 0.41 1.85 2.27
29 0.51 4.10 1.78 30 0.41 2.26 1.87
31 0.47 4.74 1.83 32 0.52 4.29 2.94
33 0.56 5.36 1.26 34 0.70 14.83 1.16
35 0.51 5.69 1.73 36 0.52 6.35 1.45
37 0.57 6.02 1.31 38 0.51 5.79 1.51
39 0.36 2.03 1.80 40 0.48 4.62 1.47
41 0.52 6.78 1.15 42 0.61 8.43 1.06
43 0.58 6.02 0.97 44 0.46 3.99 2.01
45 0.47 5.22 1.12 46 0.55 7.47 1.39
(a) Make a stemplot and, if your software allows, a Normal quantile plot of the NOX
levels. Do the plots suggest that the distribution of NOX emissions is approximately
Normal? Can you safely employ t procedures to analyze these data?
(b) Give a 95% confidence interval for the mean NOX level in vehicles of this type.
(c) Your supervisor hopes the average NOX level is less than 1 gram per mile. You
will have to tell him that it’s not so. Carry out a significance test to assess the
strength of the evidence that the mean NOX level is greater than 1, and then write
a short report to your supervisor based on your work in (b) and (c). (Your supervisor
has never heard of P -values, so you must use plain language.)
7.141 Refer to the previous exercise. Take a simple random sample of one-half of
the data. Analyze the data for these 112 computer science majors. Compare your
results with those you obtained in the previous exercise and comment on the effect
of the sample size on these procedures.
7.142 In Exercises 7.127 and 7.128 you found the power for a study designed to
compare birth weights of children born to cocaine users with those born to controls.
Fix the sample size at 50 in each group and assume the standard deviation is 650
grams and the significance level is 0.05. Pick a set of alternatives that will give values
of power ranging from fairly low values to fairly high values. Plot your results versus
that alternative and give a short summary of what you have found.
7.143 Healthy bones are continually being renewed by two processes. Through bone
formation, new bone is built; through bone resorption, old bone is removed. If one
or both of these processes are disturbed, by disease, aging, or space travel, for exam-
ple, bone loss can be the result. Osteocalcin (OC) is a biochemical marker for bone
formation: higher levels of bone formation are associated with higher levels of OC.
A blood sample is used to measure OC, and it is much less expensive to obtain than
direct measures of bone formation. The units are milligrams of OC per milliliter of
blood (mg/ml). One study examined various biomarkers of bone turnover (C. M.
Weaver et al., “Quantification of biochemical markers of bone turnover by kinetic
measures of bone formation and resorption in young healthy females,” Journal of
Bone and Mineral Research, 12 (1997), pp. 1714–1720). Here are the OC measure-
ments on 31 healthy females aged 11 to 32 years who participated in this study:
68.9 56.3 54.6 31.2 36.4 31.4 52.8 38.4

35.7 76.5 44.4 40.2 77.9 54.6 9.9 20.6
20.0 17.2 24.2 20.9 17.9 19.7 15.9 20.8
8.1 19.3 16.9 10.1 47.7 30.2 17.2
(a) Display the data with a stemplot or histogram and a boxplot. Describe the
distribution.
(b) Find a 95% confidence interval for the mean OC. Comment on the suitability of
using this procedure for these data.
7.144 Refer to the previous exercise. Tartrate resistant acid phosphatase (TRAP)
is a biochemical marker for bone resorption that is also measured in blood. Here
are the TRAP measurements, in units per liter (U/l), for the same 31 females:
19.4 25.5 19.0 9.0 19.1 14.6 25.2 14.6

28.8 14.9 10.7 5.9 23.7 19.0 6.9 8.1
9.5 6.3 10.1 10.5 9.0 8.8 8.2 10.3
3.3 10.1 9.5 8.1 18.6 14.4 9.6
(a) Display the data with a stemplot or histogram and a boxplot. Describe the
distribution.
(b) Find a 95% confidence interval for the mean TRAP. Comment on the suitability
of using this procedure for these data.
7.145 Refer to Exercise 7.143 and the OC data for 31 females. Variables that
measure concentrations such as this often have distributions that are skewed to the
right. For this reason it is common to work with the logarithms of the measured
values. Here are the OC values transformed with the (natural) log:
4.23 4.03 4.00 3.44 3.59 3.45 3.97 3.65

3.58 4.34 3.79 3.69 4.36 4.00 2.29 3.03
3.00 2.84 3.19 3.04 2.88 2.98 2.77 3.03
2.09 2.96 2.83 2.31 3.86 3.41 2.84
(a) Display the data with a stemplot and a boxplot. Describe the distribution.
(b) Find a 95% confidence interval for the mean OC. Comment on the suitability of
using this procedure for these data.
(c) Transform the mean and the endpoints of the confidence interval back to the
original scale, mg/ml. Compare this interval with the one you computed in Exer-
cise 7.144.
7.146 Refer to Exercise 7.144 and the TRAP data for 31 females. Variables that
measure concentrations such as this often have distributions that are skewed to the
right. For this reason it is common to work with the logarithms of the measured
values. Here are the TRAP values transformed with the (natural) log:
2.97 3.24 2.94 2.20 2.95 2.68 3.23 2.68

3.36 2.70 2.37 1.77 3.17 2.94 1.93 2.09
2.25 1.84 2.31 2.35 2.20 2.17 2.10 2.33
1.19 2.31 2.25 2.09 2.92 2.67 2.26
(a) Display the data with a stemplot and a boxplot. Describe the distribution.
(b) Find a 95% confidence interval for the mean TRAP. Comment on the suitability
of using this procedure for these data.
(c) Transform the mean and the endpoints of the confidence interval back to the orig-
inal scale, U/l. Compare this interval with the one you computed in Exercise 7.145.
7.147 Polychlorinated biphenyls (PCBs) are a collection of compounds that are no

longer produced in the United States but are still found in the environment. Evi-
dence suggests that they can cause harmful health effects when consumed. Because
PCBs can accumulate in fish, efforts have been made to identify areas where fish
contain excessive amounts so that recommendations concerning consumption limits
can be made. There are over 200 types of PCBs. Data from the Environmental
Protection Agency National Study of Residues in Lake Fish are given in the data
set PCB. More details about this data set can be found in the Data Appendix.
Various lakes in the United States were sampled and the amounts of PCBs in fish
were measured. The variable PCB is the sum of the amounts of all PCBs found in
the fish. The units are parts per billion (ppb).
(a) Use graphical and numerical summaries to describe the distribution of this vari-
able. Include a histogram with the location of the mean and the median clearly
marked.
(b) Do you think it is appropriate to use methods based on Normal distributions
for these data? Explain why or why not.
(c) Find a 95% confidence interval for the mean. Will this interval contain approx-
imately 95% of the observations in the data set? Explain your answer.
(d) Transform the PCB variable with a logarithm. Analyze the transformed data
and summarize your results. Do you prefer to work with the raw data or with logs
for this variable? Give reasons for your answer.
(e) Visit the Web site http://epa.gov/waterscience/fishstudy/ to find details
about how the data were collected. Write a summary describing these details and
discuss how the results from this study can be generalized to other settings.
7.148 Refer to the previous exercise. Not all types of PCBs are equally harmful.
A scale has been developed to convert the raw amount of each type of PCB to a
toxic equivalent score (TEQ). The PCB data set contains a variable TEQPCB that
is the total TEQ from all PCBs found in each sample. Using the questions in the
previous exercise, analyze these data and summarize the results.
7.149 In Exercise 7.70 you analyzed the C-reactive protein (CRP) scores for a
random sample of 40 children who participated in a study in Papua New Guinea.
Serum retinol for the same children was analyzed in Exercise 7.72. Data for all 90
children who participated in the study are given in the data set PNG, described
in the Data Appendix. Researchers who analyzed these data along with data from
several other countries were interested in whether or not infections (as indicated by
high CRP values) were associated with lower levels of serum retinol. A child with a
value of CRP greater than 5.0 mg/l is classified as recently infected. Those whose
CRP is less than or equal to 5.0 mg/l are not. Compare the serum retinol levels of
the infected and noninfected children. Include graphical and numerical summaries,
comments on all assumptions, and details of your analyses. Write a short report
summarizing your results.
7.150 Refer to the previous exercise. The researchers in this study also measured
α1-acid glycoprotein (AGP). This protein is similar to CRP in that it is an indicator
of infection. However, it rises more slowly than CRP and reaches a maximum 2 to 3
days after an infection. The units for AGP are grams per liter (g/l), and any value
greater than 1.0 g/l is an indication of infection. Analyze the data on AGP in the
data set PNG and write a report summarizing your results.
CHAPTER 8
Section 8.1
8.1 In each of the following cases state whether or not the Normal approximation
to the binomial should be used for a significance test on the population proportion
p.
(a) n = 10 and H0: p = 0.4.
(b) n = 100 and H0: p = 0.6.
(c) n = 1000 and H0: p = 0.996.
(d) n = 500 and H0: p = 0.3.
8.2 The Gallup Poll asked a sample of 1785 U.S. adults, “Did you, yourself, happen
to attend church or synagogue in the last 7 days?” Of the respondents, 750 said
“Yes.” Suppose (it is not, in fact, true) that Gallup’s sample was an SRS.
(a) Give a 99% confidence interval for the proportion of all U.S. adults who attended
church or synagogue during the week preceding the poll.
(b) Do the results provide good evidence that less than half of the population at-
tended church or synagogue?
(c) How large a sample would be required to obtain a margin of error of ±0.01 in a
99% confidence interval for the proportion who attend church or synagogue? (Use
Gallup’s result as the guessed value of p.)
8.3 Leroy, a starting player for a major college basketball team, made only 38.4%
of his free throws last season. During the summer he worked on developing a softer
shot in the hope of improving his free-throw accuracy. In the first eight games of
this season Leroy made 25 free throws in 40 attempts. Let p be his probability of
making each free throw he shoots this season.
(a) State the null hypothesis H0 that Leroy’s free-throw probability has remained
the same as last year and the alternative Ha that his work in the summer resulted
in a higher probability of success.
(b) Calculate the z statistic for testing H0 versus Ha .
(c) Do you accept or reject H0 for α = 0.05? Find the P -value.
(d) Give a 90% confidence interval for Leroy’s free-throw success probability for the
new season. Are you convinced that he is now a better free-throw shooter than last
season?
(e) What assumptions are needed for the validity of the test and confidence interval
calculations that you performed?
8.4 To profitably produce a planned upgrade of a software product you make, you
must charge customers $100. Are your customers willing to pay this much? You
contact a random sample of 40 customers and find that 11 would pay $100 for the
upgrade. Find a 95% confidence interval for the proportion of all of your customers
(the population) who would be willing to buy the upgrade for $100.
8.5 In the previous exercise we found that 11 customers from a random sample of
40 would be willing to buy a software upgrade that costs $100. If the upgrade is to
209
be profitable, you will need to sell it to more than 20% of your customers. Do the
sample data give good evidence that more than 20% are willing to buy?
(a) Formulate this problem as a hypothesis test. Give the null and alternative
hypotheses. Will you use a one-sided or a two-sided alternative? Why?
(b) Carry out the significance test. Report the test statistic and the P -value.
(c) Should you proceed with plans to produce and market the upgrade?
8.6 A poll of 811 adults aged 18 or older asked about purchases that they intended to
make for the upcoming holiday season. (The poll is part of the “American Express
Retail Index Project” and is reported in Stores, December 2000, pp. 38–40.) One of
the questions asked about what kind of gift they intended to buy for the person on
whom they will spend the most. Clothing was the first choice of 487 people. Give a
99% confidence interval for the proportion of people in this population who intend
to buy clothing as their first choice.
8.7 When trying to hire managers and executives, companies sometimes verify the
academic credentials described by the applicants. One company that performs these
checks summarized its findings for a six-month period. Of the 84 applicants whose
credentials were checked, 15 lied about having a degree. (Data provided by Jude M.
Werra & Associates, Brookfield, Wisconsin.)
(a) Find the proportion of applicants who lied about having a degree and the stan-
dard error.
(b) Consider these data to be a random sample of credentials from a large collec-
tion of similar applicants. Give a 95% confidence interval for the true proportion of
applicants who lie about having a degree.
8.8 Refer to the previous exercise. Suppose that 10 applicants lied about their
major. Can we conclude that a total of 25 = 15 + 10 applicants lied about having
a degree or about their major? Explain your answer.
8.9 A question in a Christmas tree market survey was “Did you have a Christmas
tree last year?” Of the 500 respondents, 421 answered “Yes.”
(a) Find the sample proportion and its standard error.
(b) Give a 90% confidence interval for the proportion of Indiana households who
had a Christmas tree this year.
8.10 Of the 500 respondents in the Christmas tree market survey, 44% had no
children at home and 56% had at least one child at home. The corresponding figures
for the most recent census are 48% with no children and 52% with at least one child.
Test the null hypothesis that the telephone survey technique has a probability of
selecting a household with no children that is equal to the value obtained by the
census. Give the z statistic and the P -value. What do you conclude?
8.11 Refer to the previous exercise. There we arbitrarily chose to state the hypothe-
ses in terms of the proportion of households that have children. We could as easily
have used the proportion of households that do not have children.
(a) Write hypotheses in terms of the proportion of households that do not have
children to examine how well the sample represents the state in regard to having
children in the household or not.
Section 8.1 211
(b) Perform the test of significance and summarize the results.

(c) Compare your results with the results of the previous exercise. Summarize and
generalize your conclusion.
8.12 As part of a quality improvement program, your mail-order company is study-

ing the process of filling customer orders. According to company standards, an order
is shipped on time if it is sent within 3 working days of the time it is received. You
select an SRS of 200 of the 5000 orders received in the past month for an audit. The
audit reveals that 185 of these orders were shipped on time. Find a 95% confidence
interval for the true proportion of the month’s orders that were shipped on time.
8.13 Large trees growing near power lines can cause power failures during storms
when their branches fall on the lines. Power companies spend a great deal of time
and money trimming and removing trees to prevent this problem. Researchers are
developing hormone and chemical treatments that will stunt or slow tree growth. If
the treatment is too severe, however, the tree will die. In one series of laboratory
experiments on 216 sycamore trees, 41 trees died. Give a 95% confidence interval for
the proportion of sycamore trees that would be expected to die from this particular
treatment.
8.14 In recent years over 70% of first-year college students responding to a national
survey have identified “being well-off financially” as an important personal goal. A
state university finds that 103 of an SRS of 150 of its first-year students say that
this goal is important. Give a 95% confidence interval for the proportion of all first-
year students at the university who would identify being well-off as an important
personal goal.
8.15 An entomologist samples a field for egg masses of a harmful insect by placing
a yard-square frame at random locations and carefully examining the ground within
the frame. An SRS of 75 locations selected from a county’s pastureland found egg
masses in 13 locations. Give a 90% confidence interval for the proportion of all
possible locations that are infested.
8.16 Shereka, a starting player for a major college basketball team, made only 36.2%
of her free throws last season. During the summer she worked on developing a softer
shot in the hope of improving her free-throw accuracy. In the first eight games of
this season Shereka made 22 free throws in 42 attempts. Let p be her probability of
making each free throw she shoots this season.
(a) State the null hypothesis H0 that Shereka’s free-throw probability has remained
the same as last year and the alternative Ha that her work in the summer resulted
in a higher probability of success.
(b) Calculate the z statistic for testing H0 versus Ha .
(c) Do you accept or reject H0 for α = 0.05? Find the P -value.
(d) Give a 90% confidence interval for Shereka’s free-throw success probability for
the new season. Are you convinced that she is now a better free-throw shooter than
last season?
(e) What assumptions are needed for the validity of the test and confidence interval
calculations that you performed?
8.17 Land’s Beginning is a company that sells its merchandise through the mail.
It is considering buying a list of addresses from a magazine. The magazine claims
that at least 25% of its subscribers have high incomes (they define this to be house-
hold income in excess of $100,000). Land’s Beginning would like to estimate the
proportion of high-income people on the list. Checking income is very difficult and
expensive but another company offers this service. Land’s Beginning will pay to find
incomes for an SRS of people on the magazine’s list. They would like the margin of
error of the 95% confidence interval for the proportion to be 0.05 or less. Use the
guessed value p∗ = 0.25 to find the required sample size.
8.18 Refer to the previous exercise. For each of the following variations on the
design specifications, state whether the required sample size will be higher, lower,
or the same as that found above.
(a) Use a 90% confidence interval.
(b) Change the allowable margin of error to 0.10.
(c) Use a planning value of p∗ = 0.30.
(d) Use a different company to do the income checks.
8.19 A student organization wants to start a nightclub for students under the age
of 21. To assess support for this proposal, they will select an SRS of students and
ask each respondent if he or she would patronize this type of establishment. They
expect that about 60% of the student body would respond favorably. What sample
size is required to obtain a 95% confidence interval with an approximate margin of
error of 0.08? Suppose that 50% of the sample responds favorably. Calculate the
margin of error of the 95% confidence interval.
8.20 In each of the following circumstances state whether you would use the large-
sample confidence interval, the plus four method, or neither for a 95% confidence
interval.
(a) n = 20, X = 15
(b) n = 100, X = 15
(c) n = 10, X = 2
(d) n = 5, X = 2
(e) n = 50, X = 20
sample confidence interval, the plus-four method, or neither for a 95% confidence
interval.
(a) n = 8, X = 4
(b) n = 1000, X = 12
(c) n = 40, X = 18
(d) n = 15, X = 2
(e) n = 500, X = 225
8.22 Explain what is wrong with each of the following:

(a) An approximate 95% confidence interval for an unknown proportion p is p̂ plus
or minus its standard error.
(b) You can use a significance test to evaluate the hypothesis H0: p̂ = 0.3 versus the
Section 8.1 213
two-sided alternative.
(c) The large-sample significance test for a population proportion is based on a t
statistic.
8.23 Dogs are big and expensive. Rats are small and cheap. Can rats be trained to
replace dogs in sniffing out illegal drugs? One study trained six male albino Sprague-
Dawley rats to rear up on their hind legs in response to the smell of cocaine. After
training, each rat was tested 80 times. In the test a rat was presented with a large
number of cups, one of which smelled like cocaine. A success was recorded if the
rat correctly identified the cup containing cocaine by rearing up in front of it. The
numbers of successes for the six rats were 80, 80, 73, 80, 74, and 80. You want to
estimate the success rate in the future for each of the six rats. Compare the use of
the large-sample estimates with the plus four estimates for this problem and make a
recommendation concerning which is better. Write a short summary giving reasons
for your recommendation.
8.24 The National Congregations Study collected data in a one-hour interview with
a key informant—that is, a minister, priest, rabbi, or other staff person or leader.
One question asked concerned the length of the typical sermon. For this question
390 out of 1191 congregations reported that the typical sermon lasted more than 30
minutes.
(a) Use the large-sample inference procedures to estimate the true proportion for
this question with a 95% confidence interval.
(b) Compute the interval using the plus four method. Compare these results with
those from part (a) and summarize what this example tells you about the two
methods.
(c) There were 1236 congregations surveyed in this study. Calculate the nonresponse
rate for this question. Does this influence how you interpret the results? Write a
short discussion of this issue.
(d) The respondents to this question were not asked to use a stopwatch to record
the lengths of a random sample of sermons at their congregations. They responded
based on their impressions of the sermons. Do you think that ministers, priests,
rabbis, or other staff persons or leaders might perceive sermon lengths differently
from the people listening to the sermons? Discuss how your ideas would influence
your interpretation of the results of this study.
8.25 The study described in the previous exercise also asked each respondent to
classify his or her congregation according to theological orientation. For this ques-
tion, 707 out of 1191 congregations were classified as “more conservative.” Using
the questions in the previous exercise as a guide, analyze and interpret these data.
Compare your answers to parts (c) and (d) and discuss reasons why you think the
answers should be similar or different.
8.26 A survey of 1280 student loan borrowers found that 448 had loans totaling more
than $20,000 for their undergraduate education. Give a 95% confidence interval for
the proportion of all student loan borrowers who have loans of $20,000 or more for
their undergraduate education.
8.27 In the survey described in the previous exercise, there were 1050 borrowers
whose total debt was $10,000 or more. Of these, 192 left school without completing
a degree. Consider the population to be borrowers whose total debt was $10,000
or more. Find a 95% confidence interval for the proportion of borrowers who left
school without completing a degree in this population.
8.28 Refer to Exercise 8.13. Would a 99% confidence interval be wider or narrower
than the one that you found in that exercise? Verify your results by computing the
interval.
8.29 Refer to Exercise 8.14. Would a 90% confidence interval be wider or narrower
than the one that you found in that exercise? Verify your results by computing the
interval.
8.30 Yesterday, your top salesperson called on 10 customers and obtained orders
for your new product from all 10. Suppose that it is reasonable to view these 10
customers as a random sample of all of her customers.
(a) Give the plus four estimate of the proportion of her customers who would buy
the new product. Notice that we don’t estimate that all customers will buy, even
though all 10 in the sample did.
(b) Give the margin of error for 95% confidence. (You may see that the upper
endpoint of the confidence interval is greater than 1. In that case, take the upper
endpoint to be 1.)
(c) Do the results apply to all of your sales force? Explain why or why not.
8.31 In each of the following cases state whether or not the Normal approximation
to the binomial should be used for a significance test on the population proportion
p.
(a) n = 30 and H0: p = 0.3
(b) n = 30 and H0: p = 0.6
(c) n = 1000 and H0: p = 0.5
(d) n = 500 and H0: p = 0.01
8.32 You are planning an evaluation of an alcohol awareness program at your college
that will take place six months after the program. Previous evaluations indicate
that about 30% of the students who participate will respond “Yes” to the question
“Do you think your behavior toward alcohol consumption has changed since the
program?” How large a sample should you take if you want the margin of error for
95% confidence to be about 0.1?
8.33 An automobile manufacturer would like to know what proportion of its cus-
tomers are dissatisfied with the service received from their local dealer. The cus-
tomer relations department will survey a random sample of customers and compute
a 95% confidence interval for the proportion that are dissatisfied. From past studies,
they believe that this proportion will be about 0.25. Find the sample size needed if
the margin of error of the confidence interval is to be about 0.02. Suppose 15% of
the sample say that they are dissatisfied. What is the margin of error of the 95%
confidence interval?
Section 8.1 215
8.34 You have been asked to survey students at a large college to determine the
proportion who favor an increase in student fees to support an expansion of the
student newspaper. Each student will be asked whether he or she is in favor of the
proposed increase. Using records provided by the registrar you can select a random
sample of students from the college. After careful consideration of your resources,
you decide that it is reasonable to conduct a study with a sample of 10 students.
For this sample size, construct a table of the margins of error for 95% confidence
intervals when p̂ takes the values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.
8.35 A former editor of the student newspaper agrees to underwrite the study in
the previous exercise because she believes the results will demonstrate that most
students support an increase in fees. She is willing to provide funds for a sample of
size 400. Write a short summary for your benefactor of why the increased sample
size will provide better results.
8.36 Owning a cell phone. In a 2004 survey of 1200 undergraduate students

throughout the United States, 89% of the respondents said they owned a cell
phone. (2005 press release from The Student Monitor. It is available online at
studentmonitor.com/press/02.pdf.) For 90% confidence, what is the margin of
error?
8.37 Importance of cell phone “features and functions.” Refer to the pre-
vious exercise, In that same survey, one question asked what aspect was most im-
portant when buying a cell phone. “Features and functions” was the choice for 336
students. Give a 95% confidence interval for the proportion of U.S. students who
find “features and functions” the most important aspect when buying a phone.
8.38 Owning a cell phone, continued. Refer to the previous two exercises. It
was reported that cell phone ownership by undergraduate students in 2003 was 83%.
Do the sample data in 2004 give good evidence that this percent has increased?
(a) Give the null and alternative hypotheses.
(b) Carry out the significance test. Report the test statistic and the P -value.
(c) State your conclusion using α = 0.05.
8.39 Working while enrolled in school. A 1993 nationwide survey by the

National Center for Education Statistics reports that 72% of all undergraduates work
while enrolled in school. (From the U.S. Department of Education. National Center
for Education Statistics. Postsecondary Financing Strategies: How Undergraduates
Combine Work, Borrowing, and Attendance, NCES 98088, by Stephanie Cuccaro-
Alamin and Susan P. Choy. Project Officer, C. Dennis Carroll. Washington, DC:
1998.) You decide to test whether this percent is different at your university. In
your random sample of 100 students, 77 said they were currently working.
(a) Give the null and alternative hypotheses for this study.
(b) Carry out the significance test. Report the test statistic and P -value.
(c) Does it appear that the percent of students working at your university is different
at the α = 0.05 level?
8.40 Can we use the large-sample confidence interval? In each of the follow-
ing circumstances state whether you would use the large-sample confidence interval.
(a) n = 50, X = 30
(b) n = 90, X = 15
(c) n = 10, X = 2
(d) n = 60, X = 50
(e) n = 25, X = 15
8.41 More on whether to use the large-sample confidence interval. In

each of the following circumstances state whether you would use the large-sample
confidence interval.
(a) n = 8, X = 4
(b) n = 500, X = 13
(c) n = 40, X = 18
(d) n = 15, X = 15
(e) n = 50, X = 22
8.42 Gambling and college athletics. Gambling is an issue of great concern to

those involved in intercollegiate athletics. Because of this, the National Collegiate
Athletic Association (NCAA) surveyed student-athletes concerning their gambling-
related behaviors. (Based on information in “NCAA 2003 national study of colle-
giate sports wagering and associated health risks,” which can be found at the NCAA
Web site, ncaa.org.) There were 5594 Division I male athletes in the survey. Of
these, 3547 reported participation in some gambling behavior. This included play-
ing cards, betting on games of skill, buying lottery tickets, and betting on sports.
(a) Find the sample proportion and the large-sample margin of error for 95% confi-
dence. Explain in simple terms the meaning of the 95%.
(b) Because of the way that the study was designed to protect the anonymity of
the student-athletes who responded, it was not possible to calculate the number of
students who were asked to respond but did not. Does this fact affect the way that
you interpret the results? Write a short paragraph explaining your answer.
8.43 Gambling and female athletes. In the study described in the previous
exercise, 1447 out of a total of 3469 female student-athletes reported participation
in some gambling activity.
(a) Use the large-sample methods to find an estimate of the true proportion with a
95% confidence interval.
(b) The margin of error for this sample is not the same as the margin of error
calculated for the previous exercise. Explain why.
Section 8.2
8.44 In the 1996 regular baseball season, the World Series Champion New York
Yankees played 80 games at home and 82 games away. They won 49 of their home
games and 43 of the games played away. We can consider these games as samples
from potentially large populations of games played at home and away. How much
advantage does the Yankee home field provide?
Section 8.2 217
(a) Find the proportion of wins for the home games. Do the same for the away
games.
(b) Find the standard error needed to compute a confidence interval for the difference
in the proportions.
(c) Compute a 90% confidence interval for the difference between the probability
that the Yankees win at home and the probability that they win when on the road.
Are you convinced that the 1996 Yankees were more likely to win at home?
8.45 Return to the New York Yankees baseball data in the previous exercise.
(a) Combining all of the games played, what proportion did the Yankees win?
(b) Find the standard error needed for testing that the probability of winning is the
same at home and away.
(c) Most people think that it is easier to win at home than away. Formulate null
and alternative hypotheses to examine this idea.
(d) Compute the z statistic and its P -value. What conclusion do you draw?
8.46 The 1958 Detroit Area Study was an important sociological investigation of the
influence of religion on everyday life. It is described in Gerhard Lenski, The Religious
Factor, Doubleday, New York, 1961. The sample “was basically a simple random
sample of the population of the metropolitan area.” Of the 656 respondents, 267
were white Protestants and 230 were white Catholics. One question asked whether
the government was doing enough in areas such as housing, unemployment, and
education; 161 of the Protestants and 136 of the Catholics said “No.” Is there
evidence that white Protestants and white Catholics differed on this issue?
8.47 The respondents in the Detroit Area Study (see the previous exercise) were
also asked whether they believed that the right of free speech included the right to
make speeches in favor of communism. Of the white Protestants, 104 said “Yes,”
while 75 of the white Catholics said “Yes.” Give a 95% confidence interval for the
amount by which the proportion of Protestants who agreed that communist speeches
are protected exceeds the proportion of Catholics who held this opinion.
8.48 A university financial aid office polled an SRS of undergraduate students to

study their summer employment. Not all students were employed the previous
summer. Here are the results for men and women:
Men Women
Employed 718 593
Not employed 79 139
Total 797 732
(a) Is there evidence that the proportion of male students employed during the
summer differs from the proportion of female students who were employed? State
H0 and Ha , compute the test statistic, and give its P -value.
(b) Give a 99% confidence interval for the difference between the proportions of male
and female students who were employed during the summer. Does the difference
seem practically important to you?
8.49 Refer to the study of undergraduate student summer employment described

in the previous exercise. Similar results from a smaller number of students may
not have the same statistical significance. Specifically, suppose that 72 of 80 men
surveyed were employed and 59 of 73 women surveyed were employed. The sample
proportions are essentially the same as in the earlier exercise.
(a) Compute the z statistic for these data and report the P -value. What do you
conclude?
(b) Compare the results of this significance test with your results in Exercise 8.48.
What do you observe about the effect of the sample size on the results of these
significance tests?
8.50 The power takeoff driveline on farm tractors is a potentially serious hazard to
farmers. A shield covers the driveline on new tractors, but for a variety of reasons,
the shield is often missing on older tractors. Two types of shield are the bolt-on and
the flip-up. A study initiated by the National Safety Council took a sample of older
tractors to examine the proportions of shields removed. The study found that 35
shields had been removed from the 83 tractors having bolt-on shields and that 15
had been removed from the 136 tractors with flip-up shields. (Data from W. E. Sell
and W. E. Field, “Evaluation of PTO master shield usage on John Deere tractors,”
paper presented at the American Society of Agricultural Engineers 1984 Summer
Meeting.)
(a) Test the null hypothesis that there is no difference between the proportions of
the two types of shields removed. Give the z statistic and the P -value. State your
conclusion in words.
(b) Give a 90% confidence interval for the difference in the proportions of removed
shields for the bolt-on and the flip-up types. Based on the data, what recommen-
dation would you make about the type of shield to be used on new tractors?
8.51 Is lying about credentials by job applicants changing? In Exercise 8.7 we looked
at the proportion of applicants who lied about having a degree in a six-month pe-
riod. To see if there is a change over time, we can compare that period with the
following six months. Here are the data:
Period n X(lied)
1 84 15
2 106 21
Use a 95% confidence interval to address the question of interest.
8.52 Data on the proportion of applicants who lied about having a degree in two
consecutive six-month periods are given in the previous exercise. Formulate appro-
priate null and alternative hypotheses that can be addressed with these data, carry
out the significance test, and summarize the results.
8.53 In a Christmas tree market survey, respondents who had a tree during the
holiday season were asked whether the tree was natural or artificial. Respondents
were also asked if they lived in an urban area or in a rural area. Of the 421 households
displaying a Christmas tree, 160 lived in rural areas and 261 were urban residents.
The tree growers want to know if there is a difference in preference for natural trees
versus artificial trees between urban and rural households. Here are the data:
Section 8.2 219
Population n X(natural)
1 (rural) 160 64
2 (urban) 261 89
(a) Give the null and alternative hypotheses that are appropriate for this problem
assuming that we have no prior information suggesting that one population would
have a higher preference than the other.
(b) Test the null hypothesis. Give the test statistic and the P -value, and summarize
the results.
(c) Give a 95% confidence interval for the difference in proportions.
8.54 In the 2000 regular baseball season, the World Series Champion New York
Yankees played 80 games at home and 81 games away. They won 44 of their home
games and 43 of the games played away. We can consider these games as samples
from potentially large populations of games played at home and away. How much
advantage does the Yankee home field provide?
(a) Find the Wilson estimate of proportion of wins for all home games. Do the same
for away games.
in the proportions.
(c) Compute a 90% confidence interval for the difference between the probability
that the Yankees win at home and the probability that they win when on the road.
Are you convinced that the Yankees were more likely to win at home in 2000?
8.55 Refer to the New York Yankees baseball data in the previous exercise.
(a) Combining all of the games played, what proportion did the Yankees win?
(b) Find the standard error needed for testing that the probability of winning is the
same at home and away.
(c) Most people think that it is easier to win at home than away. Formulate null
and alternative hypotheses to examine this idea.
8.56 In the 2000 World Series the New York Yankees played the New York Mets.
The previous two exercises examine the Yankees’ home and away victories. During
the regular season the Mets won 55 of the 84 home games that they played and 39
of the 81 games that they played away. Perform the same analyses for the Mets
and write a short summary comparing these results with those you found for the
Yankees.
8.57 The state agriculture department asked random samples of Indiana farmers
in each county whether they favored a mandatory corn checkoff program to pay for
corn product marketing and research. In Tippecanoe County, 263 farmers were in
favor of the program and 252 were not. In neighboring Benton County, 260 were in
favor and 377 were not.
(a) Find the proportions of farmers in favor of the program in each of the two
counties.
in the proportions.
(c) Compute a 95% confidence interval for the difference between the proportions
of farmers favoring the program in Tippecanoe County and in Benton County. Do

you think opinions differed in the two counties?
8.58 Return to the survey of farmers described in the previous exercise.

(a) Combine the two samples and find the overall proportion of farmers who favor
the corn checkoff program.
(b) Find the standard error needed for testing that the population proportions of
farmers favoring the program are the same in the two counties.
(c) Formulate null and alternative hypotheses for comparing the two counties.
8.59 A study of chromosome abnormalities and criminality examined data on 4124

Danish males born in Copenhagen. (H. A. Witkin et al., “Criminality in XYY
and XXY men,” Science, 193 (1976), pp. 547–555.) The study used the penal
registers maintained in the offices of the local police chiefs and classified each man
as having a criminal record or not. Each was also classified as having the normal
male XY chromosome pair or one of the abnormalities XYY or XXY. Of the 4096
men with normal chromosomes, 381 had criminal records, while 8 of the 28 men
with chromosome abnormalities had criminal records. Some experts believe that
chromosome abnormalities are associated with increased criminality. Do these data
lend support to this belief? Report your analysis and draw a conclusion.
8.60 A university financial aid office polled an SRS of undergraduate students to

study their summer employment. Not all students were employed the previous
summer. Here are the results for men and women:
Men Women
Employed 728 603
Not employed 89 149
Total 817 752
(a) Is there evidence that the proportion of male students employed during the
summer differs from the proportion of female students who were employed? State
H0 and Ha , compute the test statistic, and give its P -value.
(b) Give a 95% confidence interval for the difference between the proportions of male
and female students who were employed during the summer. Does the difference
seem practically important to you?
8.61 Refer to the study of undergraduate student summer employment described

in the previous exercise. Similar results from a smaller number of students may
not have the same statistical significance. Specifically, suppose that 73 of 82 men
surveyed were employed and 60 of 75 women surveyed were employed. The sample
proportions are essentially the same as in the earlier exercise.
(a) Compute the z statistic for these data and report the P -value. What do you
conclude?
(b) Compare the results of this significance test with your results in Exercise 8.60.
significance tests?
Section 8.2 221
8.62 A clinical trial examined the effectiveness of aspirin in the treatment of cere-
bral ischemia (stroke). Patients were randomized into treatment and control groups.
The study was double-blind in the sense that neither the patients nor the physicians
who evaluated the patients knew which patients received aspirin and which received
the placebo tablet. (William S. Fields et al., “Controlled trial of aspirin in cere-
bral ischemia,” Stroke, 8 (1977), pp. 301–315.) After six months of treatment, the
attending physicians evaluated each patient’s progress as either favorable or unfa-
vorable. Of the 78 patients in the aspirin group, 63 had favorable outcomes; 43 of
the 77 control patients had favorable outcomes.
(a) Compute the sample proportions of patients having favorable outcomes in the
two groups.
(b) Give a 90% confidence interval for the difference between the favorable propor-
tions in the treatment and control groups.
(c) The physicians conducting the study had concluded from previous research that
aspirin was likely to increase the chance of a favorable outcome. Carry out a signifi-
cance test to confirm this conclusion. State hypotheses, find the P -value, and write
a summary of your results.
8.63 The pesticide diazinon is in common use to treat infestations of the German
cockroach, Blattella germanica. A study investigated the persistence of this pesti-
cide on various types of surfaces. (Elray M. Roper and Charles G. Wright, “German
cockroach (Orthoptera: Blatellidae) mortality on various surfaces following appli-
cation of diazinon,” Journal of Economic Entomology, 78 (1985), pp. 733–737.)
Researchers applied a 0.5% emulsion of diazinon to glass and plasterboard. After
14 days, they placed 18 cockroaches on each surface and recorded the number that
died within 48 hours. On glass, 9 cockroaches died, while on plasterboard, 13 died.
(a) Calculate the mortality rates (sample proportion that died) for the two surfaces.
(b) Find a 95% confidence interval for the difference in the two population propor-
tions.
(c) Chemical analysis of the residues of diazinon suggests that it may persist longer
on plasterboard than on glass because it binds to the paper covering on the plas-
terboard. The researchers therefore expected the mortality rate to be greater on
plasterboard than on glass. Conduct a significance test to assess the evidence that
this is true.
8.64 Suppose that the experiment in the previous exercise placed more cockroaches
on each surface and observed similar mortality rates. Specifically, suppose that 36
cockroaches were placed on each surface and that 26 died on the plasterboard, while
18 died on the glass.
(a) Compute the z statistic for these data and report its P -value. What do you
conclude?
(b) Compare the results of this significance test with those you gave in Exercise 8.63.
significance tests?
interval.
(a) n1 = 25, n2 = 25, X1 = 10, and X2 = 15

(b) n1 = 5, n2 = 5, X1 = 2, and X2 = 5
(c) n1 = 25, n2 = 25, X1 = 8, and X2 = 20
(d) n1 = 4, n2 = 8, X1 = 2, and X2 = 7
(e) n1 = 100, n2 = 10, X1 = 40, and X2 = 2
interval.
(a) n1 = 4, n2 = 100, X1 = 1, and X2 = 65
(b) n1 = 500, n2 = 300, X1 = 175, and X2 = 208
(c) n1 = 6, n2 = 10, X1 = 4, and X2 = 2
(d) n1 = 60, n2 = 55, X1 = 24, and X2 = 37
(e) n1 = 200, n2 = 100, X1 = 128, and X2 = 94
8.67 Suppose there are two binomial populations. For the first, the true proportion
of successes is 0.4; for the second, it is 0.5. Consider taking independent samples
from these populations, 50 from the first and 60 from the second.
(a) Find the mean and the standard deviation of the distribution of p̂1 − p̂2 .
(b) This distribution is approximately Normal. Sketch this Normal distribution and
mark the location of the mean.
(c) Find a value d for which the probability is 0.95 that the difference in sample
proportions is within ±d. Mark these values on your sketch.
8.68 In a study of pet ownership, the 595 pet owners and 1939 non–pet owners were
classified according to gender. For the pet owners, there were 285 women, while for
the non–pet owners, there were 1024 women. Find the proportion of pet owners
who were women. Do the same for the non–pet owners. Give a 95% confidence
interval for the difference in the two proportions.
8.69 Refer to the previous exercise. Redo it in terms of the proportions of men
in each classification. Explain how you could have obtained these results from the
calculations you did in Exercise 8.68.
8.70 A study was designed to find reasons why patients leave a health maintenance
organization (HMO). Patients were classified as to whether or not they had filed a
complaint with the HMO. We want to compare the proportion of complainers who
leave the HMO with the proportion of those who do not file complaints but who
also leave the HMO. In the year of the study, 639 patients filed complaints, and
54 of these patients left the HMO voluntarily. For comparison, the HMO chose an
SRS of 743 patients who had not filed complaints. Twenty-two of these patients left
voluntarily. Give an estimate of the difference in the two proportions with a 95%
8.71 In the previous exercise you examined data from a study designed to find rea-
sons why patients leave an HMO. There you compared the proportion of complainers
who leave the HMO with the proportion of noncomplainers who leave. In the year
of the study, 639 patients filed complaints and 54 of these patients left the HMO
voluntarily. For comparison, the HMO chose an SRS of 743 patients who had not
Section 8.2 223
filed complaints. Twenty-two of those patients left voluntarily. We expect a higher

proportion of complainers to leave. Do the data support this expectation? State
hypotheses, find the test statistic and its P -value, and state your conclusion.
8.72 Can we use the large-sample confidence interval? In each of the follow-
ing circumstances state whether you would use the large-sample confidence interval.
(a) n1 = 30, n2 = 30, X1 = 10, and X2 = 15
(b) n1 = 15, n2 = 10, X1 = 10, and X2 = 5
(c) n1 = 25, n2 = 20, X1 = 11, and X2 = 8
(d) n1 = 40, n2 = 40, X1 = 20, and X2 = 12
(e) n1 = 50, n2 = 50, X1 = 40, and X2 = 45
8.73 More on whether to use the large-sample confidence interval. In

each of the following circumstances state whether you would use the large-sample
(a) n1 = 25, n2 = 25, X1 = 12, and X2 = 8
(b) n1 = 25, n2 = 25, X1 = 17, and X2 = 12
(c) n1 = 60, n2 = 30, X1 = 30, and X2 = 15
(d) n1 = 60, n2 = 55, X1 = 45, and X2 = 37
(e) n1 = 200, n2 = 100, X1 = 128, and X2 = 94
8.74 Comparing cell phone ownership in 2003 and 2004. In Exercise 8.38,
you were asked to compare the 2004 proportion of cell phone owners (89%) with
the 2003 estimate (83%). It would be more appropriate to compare these two
proportions using the methods of this section. Given that the sample size of each
SRS is 1200 students, compare these two years with a significance test, and give an
estimate of the difference in proportions of undergraduate cell phone owners with a
95% margin of error. Write a short summary of your results.
8.75 Gender and gambling behaviors among student-athletes. Gambling

behaviors of Division I intercollegiate student-athletes were analyzed in Exercises 8.42
and 8.43. Use the methods of this section to compare the males and females with
a significance test, and give an estimate of the difference in proportions of student-
athletes who participate in any gambling activity with a 95% margin of error. In
Exercise 8.42 it is noted that we do not have any information available to assess
nonresponse. Consider the possibility that the response rates differ by gender and
by whether or not the person participates in any gambling activity. Write a short
summary of how these differences might affect inference.
8.76 Pet ownership and marital status. In a study of pet ownership, the 595
pet owners and 1939 non–pet owners were classified according to whether or not
they were married. For the pet owners, 53.3% were married, while for the non–
pet owners, 57.7% were married. Find a 95% confidence interval for the difference.
Write a short summary of your work.
8.77 Pet ownership and marital status: the significance test. In the previous
exercise we compared the proportion of pet owners who were married with the
proportion of non–pet owners who were married in the Health ABC Study. Use a
significance test to make the comparison and summarize the results of your analysis.
8.78 A comparison of the proportion of frequent binge drinkers. In the

published report on binge drinking, survey results from both 1993 and 1999 are
presented. Using the table below, test whether the proportions of frequent binge
drinkers are different at the 5% level. Also construct a 95% confidence interval for
the difference. Write a short summary of your results.
Year n X
1993 14,995 2,973
1999 13,819 3,140
8.79 A comparison of the proportion of frequent binge drinkers, revis-

ited. Refer to the previous exercise. Redo the exercise in terms of the proportion
of nonfrequent binge drinkers in each classification. Explain how you could have
obtained these results from the calculations you did in the previous exercise.
8.80 Many colleges that once enrolled only male or only female students have become
coeducational. Some administrators and alumni were concerned that the academic
standards of the institutions would decrease with the change. One formerly all-male
college undertook a study of the first class to contain women. The class consisted
of 851 students, 214 of whom were women. An examination of first-semester grades
revealed that 15 of the top 30 students were female.
(a) What is the proportion of women in the class? Call this value p0 .
(b) Assume that the number of females in the top 30 is approximately a binomial
random variable with n = 30 and unknown probability p of success. In this case
success corresponds to the student being female. What is the value of p̂?
(c) Are women more likely to be top students than their proportion in the class
would suggest? State hypotheses that ask this question, carry out a significance
test, and report your conclusion.
8.81 In Section 6.1 we studied the effect of the sample size on the margin of error
of the confidence interval for a single proportion. In this exercise we perform some
calculations to observe this effect for the two-sample problem. Suppose that p̂1 =
0.6, p̂2 = 0.4, and n represents the common value of n1 and n2 . Compute the 95%
confidence intervals for the difference in the two proportions for n = 15, 25, 50,
75, 100, and 500. For each interval calculate the margin of error. Summarize and
explain your results.
8.82 For a single proportion the margin of error of a confidence interval is largest
for any given sample size n and confidence level C when p̂ = 0.5. This led us to use
p∗ = 0.5 for planning purposes. The same kind of result is true for the two-sample
problem. The margin of error of the confidence interval for the difference between
two proportions is largest when p̂1 = p̂2 = 0.5. Use these conservative values in
the following calculations, and assume that the sample sizes n1 and n2 have the
common value n. Calculate the margins of error of the 99% confidence intervals for
the difference in two proportions for the following choices of n: 10, 30, 50, 100, 200,
and 500. Present the results in a table or with a graph. Summarize your conclusions.
8.83 You are planning a survey in which a 90% confidence interval for the difference
between two proportions will present the results. You will use the conservative
guessed value 0.5 for p̂1 and p̂2 in your planning. You would like the margin of
error of the confidence interval to be less than or equal to 0.1. It is very difficult
to sample from the first population, so that it will be impossible for you to obtain
more than 20 observations from this population. Taking n1 = 20, can you find a
value of n2 that will guarantee the desired margin of error? If so, report the value;
if not, explain why not.
8.84 “The nature of work is changing at whirlwind speed. Perhaps now more than
ever before, job stress poses a threat to the health of workers and, in turn, to the
health of organizations.” (National Institute for Occupational Safety and Health,
Stress at Work, 2000, www.cdc.gov/niosh/stresswk.html.) So says the National
Institute for Occupational Safety and Health. Employers are concerned about the
effect of stress on their employees. Stress can lower morale and efficiency and increase
medical costs. A large survey of restaurant employees found that 75% reported that
work stress had a negative impact on their personal lives. (Results of this survey
were reported in Restaurant Business, September 15, 1999, pp. 45–49.) The human
resources manager of a chain of restaurants is concerned that work stress may be
affecting the chain’s employees. She asks a random sample of 100 employees to
respond Yes or No to the question “Does work stress have a negative impact on
your personal life?” Of these, 68 say “Yes.” Give a 95% confidence interval for the
proportion of employees who work for this chain of restaurants who believe that
work stress has a negative impact on their personal lives.
8.85 Refer to the previous exercise. Is there evidence to conclude that the proportion
for this chain of restaurants differs from the value given for the national survey?
For this exercise, assume that there is no error associated with the estimate for the
national survey.
8.86 A Gallup Poll used telephone interviews to survey a sample of 1025 U.S.
residents over the age of 18 regarding their use of credit cards. (Based on a Gallup
Poll conducted April 6–8, 2001.) The poll reported that 76% of Americans said that
they had at least one credit card. Give the 95% margin of error for this estimate.
8.87 The Gallup Poll in the previous exercise reported that 41% of those who have
credit cards do not pay the full balance each month. Find the number of people in
the survey who said that they had at least one credit card, using the information in
the previous exercise. Combine this number with the reported 41% to give a margin
of error for the proportion of credit card owners who do not pay their full balance.
8.88 A television news program conducts a call-in poll about a proposed city ban on
handgun ownership. Of the 2372 calls, 1921 oppose the ban. The station, following
recommended practice, makes a confidence statement:“81% of the Channel 13 Pulse
Poll sample opposed the ban. We can be 95% confident that the true proportion
of citizens opposing a handgun ban is within 1.6% of the sample result.” Is this
conclusion justified?
8.89 Eleven percent of the products produced by an industrial process over the
past several months fail to conform to the specifications. The company modifies
the process in an attempt to reduce the rate of nonconformities. In a trial run, the
modified process produces 16 nonconforming items out of a total of 300 produced.
Do these results demonstrate that the modification is effective? Support your con-
clusion with a clear statement of your assumptions and the results of your statistical
calculations.
8.90 In the setting of the previous exercise, give a 95% confidence interval for the
proportion of nonconforming items for the modified process. Then, taking p0 = 0.11
to be the old proportion and p the proportion for the modified process, give a 95%
confidence interval for p − p0 .
8.91 In a study on blood pressure and diet, a random sample of Seventh-Day Ad-
ventists were interviewed at a national meeting. Because many people who belong
to this denomination are vegetarians, they are a very useful group for studying the
effects of a meatless diet. (Data provided by Chris Melby and David Goldflies,
Department of Physical Education, Health, and Recreation Studies, Purdue Uni-
versity.) Blacks in the population as a whole have a higher average blood pressure
than whites. A study of this type should therefore take race into account in the
analysis. The 312 people in the sample were categorized by race and whether or not
they were vegetarians. The data are given in the following table:
Black White
Vegetarian 42 135
Not vegetarian 47 88
Are the proportions of vegetarians the same among all black and white Seventh-
Day Adventists who attended this meeting? Analyze the data, paying particular
attention to this question. Summarize your analysis and conclusions. What can
you infer about the proportions of vegetarians among black and white Seventh-Day
Adventists in general? What about blacks and whites in general?
8.92 A study examined the association between high blood pressure and increased
risk of death from cardiovascular disease. There were 2676 men with low blood
pressure and 3338 men with high blood pressure. In the low-blood-pressure group,
21 men died from cardiovascular disease; in the high-blood-pressure group, 55 died.
(a) Compute the 95% confidence interval for the difference in proportions.
(b) Do the study data confirm that death rates are higher among men with high
blood pressure? State hypotheses, carry out a significance test, and give your con-
clusions.
8.93 An experiment designed to assess the effects of aspirin on cardiovascular disease

studied 5139 male British medical doctors. The doctors were randomly assigned to
two groups. One group of 3429 doctors took one aspirin daily, and the other group
did not take aspirin. After 6 years, there were 148 deaths from heart attack or
stroke in the first group and 79 in the second group. A similar experiment used male
American medical doctors as subjects. These doctors were also randomly assigned
to one of two groups. The 11,037 doctors in the first group took one aspirin every
other day, and the 11,034 doctors in the second group took no aspirin. After nearly
5 years, there were 104 deaths from heart attacks in the first group and 189 in the
second.(The first study is reported in an article in the New York Times of January
30, 1988; the second was described in the New York Times on January 27, 1988.)
Analyze the data from these two studies and summarize the results. How do the
conclusions of the two studies differ, and why?
8.94 Different kinds of companies compensate their key employees in different ways.
Established companies may pay higher salaries, while new companies may offer stock
options that will be valuable if the company succeeds. Do high-tech companies tend
to offer stock options more often than other companies? One study looked at a
random sample of 200 companies. Of these, 91 were listed in the Directory of Public
High Technology Corporations and 109 were not listed. Treat these two groups as
SRSs of high-tech and non-high-tech companies. Seventy-three of the high-tech
companies and 75 of the non-high-tech companies offered incentive stock options to
key employees. Give a 95% confidence interval for the difference in the proportions
of the two types of companies that offer stock options. Then compare the two types
of companies using a significance test. Be sure to state your hypotheses, the test
statistic and the P -value. Write a short summary of your conclusions.
8.95 A Gallup Poll used telephone interviews to survey a sample of 1006 U.S.
residents over the age of 18 regarding their ideal family size.The poll reported that
38% of Americans said that their ideal family would include three or more children.
Assuming that this is an SRS of U.S. residents over the age of 18, give the 95%
margin of error for this estimate.
8.96 Many new products introduced into the market are targeted toward children.
The choice behavior of children with regard to new products is of particular interest
to companies that design marketing strategies for these products. As part of one
study, children in different age groups were compared on their ability to sort new
products into the correct product category (milk or juice). Here are some of the
data:
Age group n Number who sorted correctly
4- to 5-year-olds 50 10
6- to 7-year-olds 53 28
Test the null hypothesis that the two age groups are equally skilled at sorting. Justify
your choice of an alternative hypothesis. Also, give a 90% confidence interval for
the difference. Summarize your results in a short paragraph.
8.97 The Gallup Poll in Exercise 8.95 reported that in a similar poll in 1973, 43%
of Americans said that their ideal family would include three or more children. Give
the margin of error for this poll, assuming that the sample size was the same. Then
compare the proportions with a significance test and give a 95% confidence interval
for the difference. Write a summary of your results.
8.98 In a random sample of 875 students from a large public university, it was found
that 411 of the students changed majors during their college years.
(a) Give a 95% confidence interval for the proportion of students at this university
who change majors.

(b) Express your results from (a) in terms of the percent of students who change
majors.
(c) University officials concerned with counseling students are interested in the num-
ber of students who change majors rather than the proportion. The university has
37,000 undergraduate students. Convert the confidence interval you found in (a)
to a confidence interval for the number of students who change majors during their
college years.
8.99 Gastric freezing was once a recommended treatment for ulcers in the upper
intestine. A randomized comparative experiment found that 28 of the 82 patients
who were subjected to gastric freezing improved, while 30 of the 78 patients in the
control group improved.
(a) State the appropriate null hypothesis and a two-sided alternative. Carry out a
z test. What is the P -value?
(b) What do you conclude about the effectiveness of gastric freezing as a treatment
for ulcers?
8.100 In this exercise we examine the effect of the sample size on the significance test
for comparing two proportions. In each case suppose that p̂1 = 0.6 and p̂2 = 0.5, and
take n to be the common value of n1 and n2 . Use the z statistic to test H0: p1 = p2
versus the alternative Ha: p1 6= p2. Compute the statistic and the associated P -value
for the following values of n: 10, 20, 40, 50, 80, 100, 500, and 1000. Summarize the
results in a table. Explain what you observe about the effect of the sample size on
statistical significance when the sample proportions p̂1 and p̂2 are unchanged.
8.101 In the first section of this chapter, we studied the effect of the sample size on
the margin of error of the confidence interval for a single proportion. In this exercise
we perform some calculations to observe this effect for the two-sample problem.
Suppose that p̂1 = 0.6 and p̂2 = 0.5, and n represents the common value of n1 and
n2 . Compute the 95% margins of error for the difference in the two proportions for
n = 10, 20, 40, 50, 80, 100, 500, and 1000. Be sure to use the plus four method
where appropriate. Present the results in a table and with a graph. Write a short
summary of your findings.
8.102 For a single proportion the margin of error of a confidence interval is largest
for any given sample size n and confidence level C when p̂ = 0.5. This led us to use
p∗ = 0.5 for planning purposes. The same kind of result is true for the two-sample
problem. The margin of error of the confidence interval for the difference between
two proportions is largest when p̂1 = p̂2 = 0.5. Use these conservative values in
the following calculations, and assume that the sample sizes n1 and n2 have the
common value n. Calculate the margins of error of the 95% confidence intervals for
the difference in two proportions for the following choices of n: 10, 20, 40, 50, 80,
100, 500, and 1000. Be sure to use the plus four method where appropriate. Present
the results in a table and with a graph. Summarize your conclusions.
8.103 As the previous problem noted, using the guessed value 0.5 for both p̂1 and p̂2
gives a conservative margin of error in confidence intervals for the difference between
two population proportions. You are planning a survey and will calculate a 95%
confidence interval for the difference in two proportions when the data are collected.
You would like the margin of error of the interval to be less than or equal to 0.10.
You will use the same sample size n for both populations.
(a) How large a value of n is needed?
(b) Give a general formula for n in terms of the desired margin of error m and the
critical value z ∗ .
8.104 What’s wrong? For each of the following, explain what is wrong and why.
(a) A 90% confidence interval for the difference in two proportions includes errors
due to nonresponse.
(b) A z statistic is used to test the null hypothesis that H0: p̂1 = p̂2 .
(c) If two sample proportions are equal, then the sample counts must be equal.
8.105 Proportion of male heavy lottery players. A study of state lotteries

included a random digit dialing (RDD) survey conducted by the National Opinion
Research Center (NORC). The survey asked 2406 adults about their lottery spend-
ing. (From Charles T. Clotfelter et al., “State lotteries at the turn of the century:
report to the National Gambling Impact Study Commission,” 1999.) A total of 248
individuals were classified as “heavy” players. Of these, 152 were male. The study
notes that 48.5% of U.S. adults are male. For this analysis, assume that the 248
heavy lottery players are a random sample of all heavy lottery players and that the
margin of error for the 48.5% estimate of the percent of males in the U.S. adult
population is so small that it can be neglected. Use a significance test to compare
the proportion of males among heavy lottery players with the proportion of males in
the U.S. adult population. Construct a 95% confidence interval for the proportion.
Write a summary of what you have found. Be sure to comment on the possibil-
ity that some people may be reluctant to provide information about their lottery
spending and how this might affect the results.
8.106 Cell phone ownership: 2000 versus 2004. Refer to Exercise 8.74. The
estimated proportion of undergraduates owning a cell phone in 2000 was 43%. We
want to test whether the proportion of undergraduate cell phone owners has more
than doubled in the last 4 years.
(a) Compute the quantity p̂1 − 2p̂2 where p̂1 is the 2004 estimate and p̂2 is the 2000
estimate
(b) Using the rules for variances, compute the standard error of this estimate.
(c) Compute the z statistic and P -value. What is your conclusion at the 5% level?
8.107 “No Sweat” garment labels. Following complaints about the working
conditions in some apparel factories both in the United States and abroad, a joint
government and industry commission recommended in 1998 that companies that
monitor and enforce proper standards be allowed to display a “No Sweat” label on
their products. Does the presence of these labels influence consumer behavior? A
survey of U.S. residents aged 18 or older asked a series of questions about how likely
they would be to purchase a garment under various conditions. For some conditions,
it was stated that the garment had a “No Sweat” label; for others, there was no
mention of such a label. On the basis of the responses, each person was classified as
a “label user” or a “label nonuser.” (Marsha A. Dickson, “Utility of no sweat labels

for apparel customers: profiling label users and predicting their purchases,” Journal
of Consumer Affairs, 35 (2001), pp. 96–119.) There were 296 women surveyed. Of
these, 63 were label users. On the other hand, 27 of 251 men were classified as users.
(a) Give a 95% confidence interval for the difference in the proportions.
(b) You would like to compare the women with the men. Set up appropriate hy-
potheses, and find the test statistic and the P -value. What do you conclude?
8.108 Education of the customers. To devise effective marketing strategies

it is helpful to know the characteristics of your customers. A study compared
demographic characteristics of people who use the Internet for travel arrangements
and of people who do not. (From Karin Weber and Weley S. Roehl, “Profiling people
searching for and purchasing travel products on the World Wide Web,” Journal of
Travel Research, 37 (1999), pp. 291–298.) Of 1132 Internet users, 643 had completed
college. Among the 852 nonusers, 349 had completed college.
(a) Do users and nonusers differ significantly in the proportion of college graduates?
(b) Give a 95% confidence interval for the difference in the proportions.
8.109 Income of the customers. The study mentioned in the previous exercise
also asked about income. Among Internet users, 493 reported income of less than
$50,000 and 378 reported income of $50,000 or more. (Not everyone answered
the income question.) The corresponding numbers for nonusers were 477 and 200.
Perform a significance test to compare the incomes of users with nonusers and also
give an estimate of the difference in proportions with a 95% margin of error.
8.110 Nonresponse for the income question. Refer to the previous two ex-
ercises. Give the total number of users and the total number of nonusers for the
analysis of education. Do the same for the analysis of income. The difference is
due to respondents who chose “Rather not say” for the income question. Give the
proportions of “Rather not say” individuals for users and nonusers. Perform a sig-
nificance test to compare these and give a 95% confidence interval for the difference.
People are often reluctant to provide information about their income. Do you think
that this amount of nonresponse for the income question is a serious limitation for
this study?
CHAPTER 9
Chapter 9 Exercises
9.1 Investors use many “indicators” in their attempts to predict the behavior of the
stock market. One of these is the “January indicator.” Some investors believe that
if the market is up in January, then it will be up for the rest of the year. On the
other hand, if it is down in January, then it will be down for the rest of the year.
The following table gives data for the Standard & Poor’s 500 stock index for the 75
years from 1916 to 1990:
Rest January
of year Up Down
Up 35 13
Down 13 14
A chi-square analysis is valid for this problem if we assume that the yearly data are
independent observations of a process that generates either an “up” or a “down”
both in January and for the rest of the year.
(a) Calculate the column percents for this table. Explain briefly what they express.
(b) Do the same for the row percents.
(c) State appropriate null and alternative hypotheses for this problem. Use words
rather than symbols.
(d) Find the table of expected counts under the null hypothesis. In which cells do
the expected counts exceed the observed counts? In what cells are they less than
the observed counts? Explain why the pattern suggests that the January indicator
is valid.
(e) Give the value of the X 2 statistic, its degrees of freedom, and the P -value. What
do you conclude?
(f) Write a short discussion of the evidence for the January indicator, referring to
your analysis for substantiation.
9.2 In January 1975, the Committee on Drugs of the American Academy of Pedi-
atrics recommended that tetracycline drugs not be given to children under the age
of 8. A two-year study conducted in Tennessee investigated the extent to which
physicians had prescribed these drugs between 1973 and 1975. The study catego-
rized family practice physicians according to whether the county of their practice
was urban, intermediate, or rural. The researchers examined how many doctors in
each of these categories prescribed tetracycline to at least one patient under the age
of 8. Here is the table of observed counts (data from Wayne A. Ray et al., “Pre-
scribing of tetracycline to children less than 8 years old,” Journal of the American
Medical Association, 237 (1977), pp. 2069–2074):
County type
Urban Intermediate Rural
Tetracycline 65 90 172
No tetracycline 149 136 158
231
(a) Find the row and column sums and put them in the margins of the table.
(b) For each type of county find the percent of physicians who prescribed tetracy-
cline and the percent of those who did not. Do the same for the combined sample.
Display the percents in a table and describe briefly what they show.
(c) Write null and alternative hypotheses to assess whether county type and pre-
scription practices are unrelated.
(d) Carry out a significance test, give a full report of the results, and interpret them
in plain language.
9.3 Alcohol and nicotine consumption during pregnancy may harm children. Be-
cause drinking and smoking behaviors may be related, it is important to understand
the nature of this relationship when assessing the possible effects on children. One
study classified 452 mothers according to their alcohol intake prior to pregnancy
recognition and their nicotine intake during pregnancy. The data are summarized
in the following table (from Ann P. Streissguth et al., “Intrauterine alcohol and nico-
tine exposure: attention and reaction time in 4-year-old children,” Developmental
Psychology, 20 (1984), pp. 533–541):
Nicotine (milligrams/day)
Alcohol (ounces/day) None 1–15 16 or more
None 105 7 11
0.01–0.10 58 5 13
0.11–0.99 84 37 42
1.00 or more 57 16 17
Carry out a complete analysis of the association between alcohol and nicotine con-
sumption. That is, describe the nature and strength of this association and assess
its statistical significance. Include charts or figures to display the association.
9.4 Nutrition and illness are related in a complex way. If the diet is inadequate,
the ability to resist infections can be impaired and illness results. On the other
hand, some illnesses cause lack of appetite, so that poor nutrition can be the result
of illness. In a study of morbidity and nutritional status in 1165 preschool children
living in poor conditions in Delhi, India, data were obtained on nutrition and illness.
Nutrition was described by a standard method as normal or as one of four levels
of inadequate: I, II, III, and IV. For the purpose of analysis, the two most severely
undernourished groups, III and IV, were combined. One part of the study examined
four categories of illness during the past year: upper respiratory infection (URI),
diarrhea, URI and diarrhea, and none. The following table gives the data (data from
Vimlesh Seth et al., “Profile of morbidity and nutritional status and their effect on
the growth potentials in preschool children in Delhi, India,” Tropical Pediatrics and
Environmental Health, 25 (1979), pp. 23–29):
Nutritional status
Illness Normal I II III and IV
URI 95 143 144 70
Diarrhea 53 94 101 48
URI and diarrhea 27 60 76 27
None 113 48 44 22
Total 288 345 365 167
Exercises 233
Carry out a complete analysis of the association between nutritional status and type
of illness. That is, describe the association numerically, assess its significance, and
write a brief summary of your findings that refers to your analysis for substantiation.
9.5 Aluminum is suspected as a factor in the development of Alzheimer’s disease.

In one study, researchers compared a group of Alzheimer’s patients with a carefully
selected control group of people who did not have Alzheimer’s but were similar in
other ways. (Selection of a matching control group is a difficult task. In epidemi-
ological studies such as this, however, experiments are not possible.) The focus of
the study was on the use of antacids that contain aluminum. Each subject was clas-
sified according to the use of these antacids. The two-way following table gives the
data (data from Amy Borenstein Graves et al., “The association between aluminum-
containing products and Alzheimer’s disease,” Journal of Clinical Epidemiology, 43
(1990), pp. 35–44):
Aluminum-containing antacid use

None Low Medium High
Alzheimer’s patients 112 3 5 8
Control group 114 9 3 2
Analyze the data and summarize your results. Does the use of aluminum-containing
antacids appear to be associated with Alzheimer’s disease?
9.6 Are there gender differences in the progress of students in doctoral programs?
A major university classified all students entering PhD programs in a given year
by their status 6 years later. The categories used were as follows: completed the
degree, still enrolled, and dropped out. Here are the data:
Status Men Women

Completed 423 98
Still enrolled 134 33
Dropped out 238 98
Assume that these data can be viewed as a random sample giving us information
on student progress. Describe the data using whatever percents are appropriate.
State and test a null hypothesis and alternative that address the question of gender
differences. Summarize your conclusions. What factors not given might be relevant
to this study?
9.7 An article in the New York Times of January 30, 1988, described the results
of an experiment on the effects of aspirin on cardiovascular disease. The subjects
were 5139 male British medical doctors. The doctors were randomly assigned to two
groups. One group of 3429 doctors took one aspirin daily, and the other group did
not take aspirin. After 6 years, there were 148 deaths from heart attack or stroke
in the first group and 79 in the second group. The Physicians’ Health Study was a
similar experiment using male American medical doctors as subjects. These doctors
were also randomly assigned to one of two groups. The 11,037 doctors in the first
group took one aspirin every other day, and the 11,034 doctors in the second group
took no aspirin. After nearly 5 years there were 104 deaths from heart attacks in
the first group and 189 in the second. Analyze the data from these two studies and
summarize the results. How do the conclusions of the two studies differ, and why?
9.8 An article in the New York Times of April 24, 1991, discussed data from the
Centers for Disease Control that showed an increase in cases of measles in the
United States. Of particular concern are complications from measles that can lead
to death. The article noted that young children, who do not have fully developed
immune systems, face an increased risk of death from complications of measles. Here
are data on the 23,067 cases of measles reported in 1990. For each age group, the
probability of death from measles is a parameter of interest. A comparison of the
estimates of these parameters across age groups will provide information about the
relationship between age and survival of an attack of measles.
Survival
Age group Dead Survived
Under 1 year 17 3806
1–4 years 37 7113
5–9 years 3 2208
10–14 years 3 1888
15–19 years 8 2715
20–24 years 6 2209
25–29 years 9 1492
30 years and over 14 1636
Summarize the death rates by age group. Prepare a plot to illustrate the pattern.
Test the hypothesis that survival and age are related, report the results, and sum-
marize your conclusion. From the data given, is it possible to study the association
between catching measles and age? Explain why or why not.
9.9 Refer to Exercise 9.5, where we examined the relationship between use of
aluminum-containing antacid and Alzheimer’s disease. In that exercise the P -value
was 0.068, failing to achieve the traditional standard for statistical significance
(0.05). Suppose that we did a similar study with more data. In particular, let’s
double each of the counts in the original table. Perform the analysis on these counts
and summarize the effect of increasing the sample size.
9.10 The Census Bureau collects data on years of school completed by Americans
of different ages. The following table gives the years of education for three different
age groups. People under the age of 25 are not included because many have not
yet completed their education. Note that the unit of measure for each entry in the
table is thousands of persons.
Years of school completed, by age (thousands of persons)
Age group
Education 25 to 34 35 to 54 55 and over Total
Did not complete high school 5,325 9,152 16,035 30,512
Completed high school 14,061 24,070 18,320 56,451
College, 1 to 3 years 11,659 19,926 9,662 41,247
College, 4 or more years 10,342 19,878 8,005 38,225
Total 41,388 73,028 52,022 166,438
Exercises 235
(a) Give the joint distribution of education and age for this table.
(a) What is the marginal distribution of age?
(c) What is the marginal distribution of education?
9.11 Refer to the previous exercise. Find the conditional distribution of education
for each of the three age categories. Make a bar graph for each distribution and
summarize their differences and similarities.
9.12 Refer to the previous exercise. Compute the conditional distribution of age for
each of the four education categories. Summarize the distributions graphically and
write a short paragraph describing the distributions and how they differ.
9.13 The National Center for Education Statistics collects data on undergraduate
students enrolled in U.S. colleges and universities. The following table gives counts
of undergraduate enrollment in four different classifications:
Undergraduate college enrollment (thousands of students)
2-year 2-year 4-year 4-year
Age full-time part-time full-time part-time
Under 18 36 98 75 37
18 to 21 1126 711 3270 223
22 to 34 634 1575 2267 1380
35 and over 221 1092 390 915
Total 2017 3476 6002 2555
(a) How many undergraduate students were enrolled in colleges and universities?
(b) What percent of all undergraduate students were under 18 years old?
(c) Find the percent of the undergraduates enrolled in each of the four types of
program who were 22 to 34 years old. Make a bar graph to compare these percents.
(d) The 18 to 21 group is the traditional age group for college students. Briefly
summarize what you have learned from the data about the extent to which this
group predominates in different kinds of college programs.
9.14 Refer to the previous exercise. Find the marginal distribution of college type.
Describe the distribution graphically and write a short summary.
9.15 Take the counts for the four college types that you used in the previous exercise
and write these in a 2 × 2 table with year (2-year or 4-year) as the columns and time
(full-time or part-time) as the rows. The marginal distribution that you calculated
for the previous exercise is the joint distribution for this exercise. Compute the
conditional distribution of time for each year category. Describe and contrast these
distributions. Does there appear to be a relationship? If so, describe it.
9.16 For each type of college, find the conditional distributions of age for the 4 × 3
table in Exercise 9.13. Display the distributions with bar graphs and describe how
the age profile of the students varies with the type of college.
9.17 The following two-way table describes the age and marital status of American
women. The table entries are in thousands of women.
Marital status
Never
Age (years) married Married Widowed Divorced Total
18–24 9,289 3,046 19 260 12,613
25–39 6,948 21,437 206 3,408 32,000
40–64 2,307 26,679 2,219 5,508 36,713
≥ 65 768 7,767 8,636 1,091 18,264
Total 19,312 58,931 11,080 10,266 99,588
(a) Find the sum of the entries in the “Married” column. Why does this sum differ
from the “Total” entry for that column?
(b) Give the marginal distribution of marital status for all adult women (use per-
cents). Draw a bar graph to display this distribution.
(c) Compare the conditional distributions of marital status for women aged 18 to 24
and women aged 40 to 64. Briefly describe the most important differences between
the two groups of women, and back up your description with percents.
(d) You are planning a magazine aimed at women who have never been married.
Find the conditional distribution of age among single women and display it in a bar
graph. What age group or groups should your magazine aim to attract?
9.18 Here is a two-way table of suicides committed, categorized by the gender of

the victim and the method used. (“Hanging” also includes suffocation.) Write a
brief account of differences in suicide between males and females. Use calculations
and a graph to justify your statements.
Gender
Method Male Female
Firearms 16,381 2,559
Poison 3,569 2,110
Hanging 3,824 803
Other 1,641 623
9.19 Here are the numbers of flights on time and delayed for two airlines at five
airports. (These data, from reports submitted by airlines to the Department of
Transportation, appear in A. Barnett, “How numbers can trick you,” Technology
Review, October 1994, pp. 38–45.) Overall on-time percents for each airline are
often reported in the news. Lurking variables can make such reports misleading.
Alaska Airlines America West

On time Delayed On time Delayed
Los Angeles 497 62 694 117
Phoenix 221 12 4840 415
San Diego 212 20 383 65
San Francisco 503 102 320 129
Seattle 1841 305 201 61
(a) What percent of all Alaska Airlines flights were delayed? What percent of all
America West flights were delayed? These are the numbers usually reported.
Exercises 237
(b) Now find the percent of delayed flights for Alaska Airlines at each of the five
airports. Do the same for America West.
(c) America West does worse at every one of the five airports, yet does better
overall. That sounds impossible. Explain carefully, referring to the data, how this
can happen. (The weather in Phoenix and Seattle lies behind this example of
Simpson’s paradox.)
9.20 Psychological and social factors can influence the survival of patients with seri-
ous diseases. One study examined the relationship between survival of patients with
coronary heart disease (CHD) and pet ownership. (Erika Friedmann et al., “Ani-
mal companions and one-year survival of patients after discharge from a coronary
care unit,” Public Health Reports, 96 (1980), pp. 307–312.) Each of 92 patients was
classified as having a pet or not and by whether they survived for one year. Here
are the data:
Pet ownership
Patient status No Yes
Alive 28 50
Dead 11 3
(a) Was this study an experiment? Why or why not?

(b) The researchers thought that having a pet might improve survival, so pet own-
ership is the explanatory variable. Compute appropriate percents to describe the
data and state your preliminary findings.
(c) State in words the null hypothesis for this problem. What is the alternative
hypothesis?
(d) Find the X 2 statistic, its degrees of freedom, and the P -value.
(e) What do you conclude? Do the data give convincing evidence that owning a pet
is an effective treatment for increasing the survival of CHD patients?
9.21 The baseball player Reggie Jackson had a reputation for hitting better in the
World Series than during the regular season. In his 21-year career, Jackson was
at bat 9864 times in regular-season play and had 2584 hits. During World Series
games, he was at bat 98 times and had 35 hits. We can view Jackson’s regular-
season at-bats as a random sample from a population of potential at-bats (he might
have batted many more times if the season were longer, for example), and his World
Series at-bats as a sample from a second population.
(a) Display the data in a 2 × 2 table of counts with “regular season” and “World
Series” as the column headings, and fill in the marginal sums.
(b) Calculate appropriate percents to compare Jackson’s regular-season and World
Series performances. Did he hit better in World Series games?
(c) Is there a significant difference between Jackson’s regular-season and World Series
performances? State hypotheses (in words), and then calculate the X 2 statistic, its
degrees of freedom, and its P -value. What is your conclusion?
9.22 If the performance of a stock fund is due to the skill of the manager, then we
would expect a fund that does well this year to perform well next year also. This
is called persistence of fund performance. One study classified funds as losers or
winners depending on whether their rate of return was less than or greater than the
median of all funds. (Burton G. Malkeil, “Returns from investing in equity mutual
funds, 1971 to 1991,” Journal of Finance, 50 (1995), pp. 549–572.) To examine the
question of interest we form a two-way table that classifies each fund as a loser or
winner in each of two successive years. Here are the data for one such table:
Next year
This year Winner Loser
Winner 85 35
Loser 37 83
Is there evidence in favor of persistence of fund performance in this table? Support

your conclusion with a complete analysis of the data.
9.23 Refer to the previous exercise. Rerun the analysis using the method for com-
paring two proportions of Chapter 8. Verify that the X 2 statistic is the square of
the z statistic and that the P -values for both analyses are the same.
9.24 In the previous exercise, it is natural to use “this year” to define the two
samples. If we drew separate random samples of winners and losers this year and we
recorded the outcome next year, we would call this a prospective study (forward
looking). On the other hand, if we drew separate random samples of winners and
losers “next year” and looked back historically to determine if they were winners
or losers in the previous year, we would have a retrospective study (backward
looking). Verify that you get the same value of z (and therefore the same P -value)
using these two different approaches.
9.25 If we find evidence in favor of an effect in one set of circumstances, it is natural

to want to conclude that it holds in many others. Unfortunately, this reasoning can
sometimes lead us to incorrect conclusions. For example, here is another table from
the study described in Exercise 9.22:
Next year
This year Winner Loser
Winner 96 148
Loser 145 99
Analyze these data in the same way. What do you conclude?
9.26 There is much evidence that high blood pressure is associated with increased
risk of death from cardiovascular disease. A major study of this association exam-
ined 2676 men with low blood pressure and 3338 men with high blood pressure. (J.
Stamler, “The mass treatment of hypertensive disease: defining the problem,” Mild
Hypertension: To Treat or Not to Treat, New York Academy of Sciences, 1978, pp.
333–358.) During the period of the study, 21 men in the low-blood-pressure and 55
in the high-blood-pressure group died from cardiovascular disease.
(a) What is the explanatory variable? Describe the association in these data nu-
merically and in words.
(b) Do the study data confirm that death rates are higher among men with high
blood pressure? State hypotheses, carry out a significance test, and give your con-
clusions.
Exercises 239
(c) Present the data in a two-way table. Is the chi-square test appropriate for the
hypotheses you stated in (b)?
(d) Give a 95% confidence interval for the difference between the death rates for the
low- and high-blood-pressure groups.
9.27 It is traditional practice in Egypt to withhold food from children with diar-
rhea. Because it is known that feeding children with this illness reduces mortality,
medical authorities undertook a nationwide program designed to promote feeding
sick children. To evaluate the impact of the program, surveys were taken before
and after the program was implemented. (O. M. Galal et al., “Feeding the child
with diarrhea: a strategy for testing a health education message within the primary
health care system in Egypt,” Socio-economic Planning Sciences, 21 (1987), pp.
139–147.) In the first survey, 457 of 1003 surveyed mothers followed the practice of
feeding children with diarrhea. For the second survey, 437 of 620 surveyed followed
this practice.
(a) Assume that the data come from two independent samples. Test the hypothesis
that the program was effective, that is, that the practice of feeding children with
diarrhea increased between the time of the first study and the time of the second.
State H0 and Ha , give the test statistic and its P -value, and summarize your con-
clusion.
(b) Present the data in a two-way table. Can the X 2 statistic test your hypotheses?
(c) Describe the results using a 95% confidence interval for the difference in propor-
tions.
9.28 PTC is a compound that has a strong bitter taste for some people and is
tasteless for others. The ability to taste this compound is an inherited trait. Many
studies have assessed the proportions of people in different populations who can
taste PTC. The following table gives results for samples from several countries (A. E.
Mourant et al., The Distribution of Human Blood Groups and Other Polymorphisms,
Oxford University Press, 1976):
Country
Taster Ireland Portugal Norway Italy
Yes 558 345 185 402
No 225 109 81 134
Complete the table and describe the data. Do they provide evidence that the pro-
portion of PTC tasters varies among the four countries? Give a complete summary
of your analysis.
9.29 There are four major blood types in humans: O, A, B, and AB. In a study
conducted using blood specimens from the Blood Bank of Hawaii, individuals were
classified according to blood type and ethnic group. The ethnic groups were Hawai-
ian, Hawaiian-white, Hawaiian-Chinese, and white. (A. E. Mourant et al., The
Distribution of Human Blood Groups and Other Polymorphisms, Oxford University
Press, 1976.) Assume that the blood bank specimens are random samples from the
Hawaiian populations of these ethnic groups.
Ethnic group
Hawaiian- Hawaiian-
Blood type Hawaiian white Chinese White
O 1,903 4,469 2,206 53,759
A 2,490 4,671 2,368 50,008
B 178 606 568 16,252
AB 99 236 243 5,001
Summarize the data numerically and with a graph. Is there evidence to conclude
that blood type and ethnic group are related? Explain how you arrived at your
conclusion.
9.30 In healthy individuals the concentration of various substances in the blood

remains within relatively narrow bounds. One such substance is potassium. A
person is said to be hypokalemic if the potassium level is too low (less than 3.5 mil-
liequivalents per liter) and hyperkalemic if the level is too high (above 5.5 meq/l).
Hypokalemia is associated with a variety of symptoms such as excessive tiredness,
while hyperkalemia is generally an indication of a serious problem. Patients be-
ing treated with diuretics (pharmaceuticals that help the body to eliminate water)
sometimes have abnormal potassium concentrations. In a large study of patients on
chronic diuretic therapy, several risk factors were studied to see if they were associ-
ated with abnormal potassium levels. (William M. Tierney, Clement J. McDonald,
and George P. McCabe, “Serum potassium testing in diuretic-treated outpatients,”
Medical Decision Making, 5 (1985), pp. 91–104.) Of the 5810 patients studied, 1094
were hypokalemic, 4689 had normal potassium levels, and 27 were hyperkalemic.
The following table gives the percents of patients having each of four risk factors in
the three potassium groups:
Potassium group
Hypokalemic Normal Hyperkalemic
n 1094 4689 27
Hypertension 88.3% 78.1% 40.7%
Heart failure 16.5% 24.7% 55.6%
Diabetes 20.6% 25.5% 29.6%
Gender (% female) 72.5% 68.0% 48.1%
For example, 88.3% of the 1094 hypokalemic patients had hypertension, and 78.1%
of the 4689 normal patients had hypertension. For each of the four risk factors, use
the percents and n’s given to compute the counts for the 2 × 3 table needed to study
the association between the factor and potassium. Then analyze each table using
the methods presented in this chapter. Note that there are very few patients in the
hyperkalemic group. Therefore, reanalyze the data dropping this category from the
tables. Write a short summary explaining what you have found.
9.31 The proportion of women entering many professions has undergone consider-
able change in recent years. A study of students enrolled in pharmacy programs
describes the changes in this field. A random sample of 700 students in their third
or higher year of study at colleges of pharmacy was taken in each of nine years. (The
data are based on Seventh Report to the President and Congress on the Status of
Exercises 241
Health Personnel in the United States, Public Health Service, 1990.) The following
table gives the numbers of women in each of these samples:
Year 1970 1972 1974 1976 1978 1980 1982 1984 1986
Women 164 195 226 283 302 342 369 385 412
Use the chi-square test to assess the change in the percent of women pharmacy
students over time, and summarize your results. (You will need to calculate the
number of male students for each year using the fact that the sample size each year
is 700.) Plot the percent of women versus year. Describe the plot. Is it roughly
linear? Find the least-squares line that summarizes the relation between time and
the percent of women pharmacy students.
9.32 Refer to the previous exercise. Here are the percents of women pharmacy
students for the years 1987 to 2000 (data provided by Dr. Susan Meyer, Senior Vice
President of the American Association of Colleges of Pharmacy):
Year 1987 1988 1989 1990 1991 1992 1993
Women 60.0% 60.6% 61.6% 62.4% 63.0% 63.4% 63.2%
Year 1994 1995 1996 1997 1998 1999 2000

Women 63.3% 63.4% 63.8% 64.2% 64.4% 64.9% 65.9%
Plot these percents versus year and summarize the pattern. Using your analysis of
the data in this and the previous exercise, write a report summarizing the changes
that have occurred in the percent of women pharmacy students from 1970 to 2000.
Include an estimate of the percent for the year 2010 with an explanation of why you
chose this estimate.
9.33 The Census Bureau provides estimates of numbers of people in the United
States classified in various ways. Let’s look at college students. The following table
gives us data to examine the relation between age and full-time or part-time status.
The numbers in the table are expressed as thousands of U.S. college students.
U.S. college students by age and status

Status
Age Full-time Part-time
15–19 3388 389
20–24 5238 1164
25–34 1703 1699
35 and over 762 2045
(a) What is the U.S. Census Bureau estimate of the number of full-time college
students aged 15 to 19?
(b) Give the joint distribution of age and status for this table.
(c) What is the marginal distribution of age? Display the results graphically.
(d) What is the marginal distribution of status? Display the results graphically.
9.34 Refer to the previous exercise. Find the conditional distribution of status for
each of the four age categories. Display the distributions graphically and summarize
their differences and similarities.
9.35 Refer to the previous two exercises. Compute the conditional distribution of
age for each of the two status categories. Summarize the distributions graphically
and write a short paragraph describing the distributions and how they differ.
9.36 The following table gives some census data concerning the enrollment status of
recent high school graduates aged 16 to 24 years. The table entries are in thousands
of students.
Enrollment and gender

Status Men Women
Two-year college, full-time 890 969
Two-year college, part-time 340 403
Four-year college, full-time 2897 3321
Four-year college, part-time 249 383
Graduate school 306 366
Vocational school 160 137
(a) How many male recent high school graduates aged 16 to 24 years were enrolled
full-time in two-year colleges?
(b) Give the marginal distribution of gender for these students. Display the results
graphically.
(c) What is the marginal distribution of status for these students? Display the
results graphically.
9.37 Refer to the previous exercise. Find the conditional distribution of gender
for each status. Describe the distributions graphically and write a short summary
comparing the major features of these distributions.
9.38 Refer to the previous two exercises. Find the conditional distribution of status
for each gender. Describe the distributions graphically and write a short summary
comparing the major features of these distributions.
9.39 Here are the row and column totals for a two-way table with two rows and two
columns:
a b 200
c d 200
200 200 400
Find two different sets of counts a, b, c, and d for the body of the table that give
these same totals. This shows that the relationship between two variables cannot
be obtained from the two individual distributions of the variables.
9.40 Construct a 3 × 3 table of counts where there is no apparent association

between the row and column variables.
9.41 A marketing research firm conducted a survey of companies in its state. They
mailed a questionnaire to 300 small companies, 300 medium-sized companies, and
300 large companies. The rate of nonresponse is important in deciding how reliable
survey results are. Here are the data on response to this survey:
Exercises 243
Response
Size of company Yes No Total
Small 175 125 300
Medium 145 155 300
Large 120 180 300
(a) What was the overall percent of nonresponse?

(b) Describe how nonresponse is related to the size of the business. (Use percents
to make your statements precise.)
(c) Draw a bar graph to compare the nonresponse percents for the three size cate-
gories.
(d) Using the total number of responses as a base, compute the percent of responses
that come from each of small, medium, and large businesses.
(e) The sampling plan was designed to obtain equal numbers of responses from small,
medium, and large companies. In preparing an analysis of the survey results, do you
think it would be reasonable to proceed as if the responses represented companies
of each size equally?
9.42 A study of the career plans of young women and men sent questionnaires to
all 722 members of the senior class in the College of Business Administration at the
University of Illinois. One question asked which major within the business program
the student had chosen. Here are the data from the students who responded:
Gender
Major Female Male
Accounting 68 56
Administration 91 40
Economics 5 6
Finance 61 59
(a) Describe the differences between the distributions of majors for women and men
with percents, with a graph, and in words.
(b) What percent of the students did not respond to the questionnaire? The non-
response weakens conclusions drawn from these data.
9.43 Asia has become a major competitor of the United States and Western Europe
in education as well as economics. Here are counts of first university degrees in
science and engineering in the three regions:
Region
United Western
Field States Europe Asia
Engineering 61,941 158,931 280,772
Natural science 111,158 140,126 242,879
Social science 182,166 116,353 236,018
Direct comparison of counts of degrees would require us to take into account Asia’s
much larger population. We can, however, compare the distribution of degrees by
field of study in the three regions. Do this using calculations and graphs, and write
a brief summary of your findings.
9.44 Mountain View University has professional schools in business and law. Here is
a three-way table of applicants to these professional schools, categorized by gender,
school, and admission decision.
Business Law
Admit Admit
Gender Yes No Gender Yes No
Male 400 200 Male 90 110
Female 200 100 Female 200 200
(a) Make a two-way table of gender by admission decision for the combined profes-
sional schools by summing entries in the three-way table.
(b) From your two-way table, compute separately the percents of male and female
applicants admitted. Male applicants are admitted to Mountain View’s professional
schools at a higher rate than female applicants.
(c) Now compute separately the percents of male and female applicants admitted
by the business school and by the law school.
(d) Explain carefully, as if speaking to a skeptical reporter, how it can happen that
Mountain View appears to favor males when this is not true within each of the
professional schools.
9.45 Refer to the previous exercise. Make up a similar table for a hypothetical
university having four different schools that illustrates the same point. Carefully
summarize your table with the appropriate percents.
9.46 A university classifies its classes as either “small” (fewer than 40 students) or
“large.” A dean sees that 62% of Department A’s classes are small, while Depart-
ment B has only 40% small classes. She wonders if she should cut Department A’s
budget and insist on larger classes. Department A responds to the dean by pointing
out that classes for third- and fourth-year students tend to be smaller than classes
for first- and second-year students. The three-way following table gives the counts
of classes by department, size, and student audience. Write a short report for the
dean that summarizes these data. Start by computing the percents of small classes
in the two departments, and include other numerical and graphical comparisons as
needed. (Do not perform any statistical significance tests.) Here are the numbers
of classes to be analyzed:
Department A Department B
Year Large Small Total Large Small Total
First 2 0 2 18 2 20
Second 9 1 10 40 10 50
Third 5 15 20 4 16 20
Fourth 4 16 20 2 14 16
9.47 For each of the following situations give the degrees of freedom and an appro-
priate bound on the P -value (give the exact value if you have software available) for
the X 2 statistic for testing the null hypothesis of no association between the row
and column variables.
Exercises 245
(a) a 4 by 5 table with X 2 = 26.23

(b) a 4 by 3 table with X 2 = 26.23
(c) a 5 by 4 table with X 2 = 26.23
(d) a 7 by 6 table with X 2 = 26.23
9.48 For each of the following situations give the degrees of freedom and an appro-
priate bound on the P -value (give the exact value if you have software available) for
the X 2 statistic for testing the null hypothesis of no association between the row
and column variables.
(a) a 2 by 2 table with X 2 = 1.31
(b) a 4 by 4 table with X 2 = 17.54
(c) a 2 by 8 table with X 2 = 22.10
(d) a 5 by 3 table with X 2 = 12.61
9.49 To be competitive in global markets, many U.S. corporations are undertaking

major reorganizations. Often these involve “downsizing,” sometimes called a “re-
duction in force” (RIF), where substantial numbers of employees are terminated.
Federal and various state laws require that employees be treated equally regardless
of their age. In particular, employees over the age of 40 years are in a “protected”
class, and many allegations of discrimination focus on comparing employees over 40
with their younger coworkers. Here are the data for a recent RIF:
Over 40
Terminated No Yes
Yes 16 82
No 585 771
(a) Make a table that includes the following information for each group (over 40 or
not): total number of employees, the proportion of employees who were terminated,
and the standard error for the proportion.
(b) Perform the chi-square test for this two-way table. Give the test statistic, the
degrees of freedom, the P -value, and your conclusion.
9.50 A major issue that arises in the kind of case described in the previous exercise
concerns the extent to which the employees are similar. This often involves exam-
ining other variables. For example, suppose that the employees over 40 did not do
as good a job as the younger workers. Let’s examine the last performance appraisal
to explore this idea. The possible values are as follows: “partially meets expecta-
tions,” “fully meets expectations,” “usually exceeds expectations,” and “continu-
ally exceeds expectations.” Because there were very few employees who partially
exceeded expectations, we combine the first two categories. Here are the data:
Over 40
Performance appraisal No Yes
Partially or fully meets expectations 82 237
Usually exceeds expectations 357 492
Continually exceeds expectations 63 32
Analyze the data. Do the older employees appear to have lower performance evalu-
ations?
9.51 Can you increase the response rate for a mail survey by contacting the respon-
dents before they receive the survey? A study designed to address this question
compared three groups of subjects. The first group received a preliminary letter
about the survey, the second group was phoned, and the third received no prelim-
inary contact. A positive response was defined as returning the survey within two
weeks. Here are the counts:
Intervention
Response Letter Phone call None
Yes 171 146 118
No 220 68 455
Total 391 214 573
Another study also attempted to evaluate the effect of a letter sent before the survey
on the response rate. In this study subjects who received a prenotification letter
were compared with subjects who received no letter. Here are the data:
Response Letter No letter

Yes 2570 2645
No 2448 2384
Total 5018 5029
(a) Summarize the results of each study graphically and numerically.

(b) Perform the appropriate significance tests, and report the results and the con-
clusions.
(c) Based on these two studies, write a report about methods to improve responses
for surveys. The subjects in the first study were students from three Houston uni-
versities. The subjects in the second study were physicians in the United States.
Be sure to include comments on the extent to which you think the results can be
generalized to other populations of people from other places.
9.52 Gastric freezing was once a recommended treatment for ulcers in the upper
intestine. A randomized comparative experiment found that 28 of the 82 patients
who were subjected to gastric freezing improved, while 30 of the 78 patients in the
control group improved. The hypothesis of “no difference” between the two groups
can be tested in two ways: using a z statistic or using the X 2 statistic.
(a) State the appropriate hypothesis and a two-sided alternative, and carry out a z
test. What is the P -value?
(b) Present the data in a 2 × 2 table. State the appropriate hypotheses and carry
out the chi-square test. What is the P -value? Verify that the X 2 statistic is the
square of the z statistic.
(c) What do you conclude about the effectiveness of gastric freezing as a treatment
for ulcers?
9.53 At your college 29% of the students are in their first year, 27% in their second,
25% in their third, and 19% in their fourth year. You have taken a survey of
students, and when you classify them by year of study, you have 54, 66, 56, and 30
students in the first, second, third, and fourth years, respectively. Use a goodness
of fit test to examine how well your sample reflects the population of your college.
Exercises 247
9.54 Computer software generated 500 random numbers that should look like they
are from the standard Normal distribution. They are categorized into five groups:
(1) less than or equal to −0.8, (2) greater than −0.8 and less than or equal to −0.2,
(3) greater than −0.2 and less than or equal to 0.2, (4) greater than 0.2 and less
than or equal to 0.8, and (5) greater than 0.8. The counts in the five groups are
98, 112, 79, 111, and 100, respectively. Find the probabilities for these five intervals
using Table A. Then compute the expected number for each interval for a sample
of 500. Finally, perform the goodness of fit test and summarize your results.
9.55 Computer software generated 500 random numbers that should look like they
are from the uniform distribution on the interval 0 to 1. They are categorized into
five groups: (1) less than or equal to 0.2, (2) greater than 0.2 and less than or equal
to 0.4, (3) greater than 0.4 and less than or equal to 0.6, (4) greater than 0.6 and
less than or equal to 0.8, and (5) greater than 0.8. The counts in the five groups
are 113, 95, 108, 99, and 85, respectively. The probabilities for these five intervals
are all the same. What is this probability? Compute the expected number for each
interval for a sample of 500. Finally, perform the goodness of fit test and summarize
your results.
9.56 Does using Rodham matter? In April 2006, the Opinion Research Cor-
poration conducted a telephone poll for CNN of 1012 adult Americans. (See
i.a.cnn.net/cnn/2006/images/04/27/rel11e.pdf for a summary of the poll.)
Half those polled were asked their opinion of Hillary Rodham Clinton. The other
half were asked their opinion of Hillary Clinton. The table below summarizes the
results. A chi-square test was used to determine if opinions differed based on the
name.
Opinion
Name Favorable Unfavorable Never heard of No opinion
Hillary Rodham Clinton 50% 42% 2% 6%
Hillary Clinton 46% 43% 2% 9%
(a) Computer software gives X 2 = 4.23. Can we comfortably use the chi-square
distribution to compute the P -value? Explain.
(b) What are the degrees of freedom for X 2?
(c) Give an appropriate bound for the P -value using Table F and state your con-
clusions.
9.57 New treatment for cocaine addiction. Cocaine addiction is difficult to

overcome. Addicts have been reported to have a significant depletion of stimulating
neurotransmitters and thus continue to take cocaine to avoid feelings of depression
and anxiety. A 3-year study with 72 chronic cocaine users compared an antidepres-
sant drug called desipramine with lithium and a placebo. (Lithium is a standard
drug to treat cocaine addiction. A placebo is a substance containing no medica-
tion, used so that the effect of being in the study but not taking any drug can
be seen.) One-third of the subjects, chosen at random, received each treatment.
(Data from D. M. Barnes, “Breaking the cycle of addiction,” Science, 241 (1988),
pp. 1029–1030.) Here are the results:
Cocaine relapse?
Treatment Yes No
Desipramine 10 14
Lithium 18 6
Placebo 20 4
(a) Compare the effectiveness of the three treatments in preventing relapse using
percents and a bar graph. Write a brief summary.
(b) Can we comfortably use the chi-square test to test the null hypothesis that there
is no difference between treatments? Explain.
(c) Perform the significance test and summarize the results.
9.58 Drinking status and class attendance. As part of the 1999 College Alco-
hol Study, students who drank alcohol in the last year were asked if drinking ever
resulted in missing a class. (Results of this survey are reported in Henry Wech-
sler et al., “College Binge Drinking in the 1990s: A Continuing Problem,” Journal
of American College Health, 48 (2000), pp. 199–210.) The data are given in the
following table:
Drinking status
Missed a class Nonbinger Occasional binger Frequent binger
No 4617 2047 1176
Yes 446 915 1959
(a) Summarize the results of this table graphically and numerically.

(b) What is the marginal distribution of drinking status? Display the results graph-
ically.
(c) Compute the relative risk of missing a class for occasional bingers versus non-
bingers and for frequent bingers versus nonbingers. Summarize these results.
(d) Perform the chi-square test for this two-way table. Give the test statistic, degrees
of freedom, the P -value, and your conclusion.
9.59 Air pollution from a steel mill. One possible effect of air pollution is ge-
netic damage. A study designed to examine this problem exposed one group of mice
to air near a steel mill and another group to air in a rural area and compared the
numbers of mutations in each group. (Christopher Somers et al., “Air pollution in-
duces heritable DNA mutations,” Proceedings of the National Academy of Sciences,
99 (2002), pp. 15,904–15,907.) Here are the data for a mutation at the Hm-2 gene
locus:
Location
Mutation Steel mill air Rural air
Yes 30 23
No
Total 96 150
(a) Fill in the missing entries in the table.

(b) Summarize the data numerically and graphically.
Exercises 249
(c) Is there evidence to conclude that the location is related to the occurrence of
mutations? Perform the significance test and summarize the results.
9.60 Secondhand stores. Shopping at secondhand stores is becoming more pop-

ular and has even attracted the attention of business schools. A study of customers’
attitudes toward secondhand stores interviewed samples of shoppers at two second-
hand stores of the same chain in two cities. The breakdown of the respondents
by gender is as follows (William D. Darley, “Store-choice behavior for pre-owned
merchandise,” Journal of Business Research, 27 (1993), pp. 17–31):
Gender City 1 City 2

Men 38 68
Women 203 150
Total 241 218
Is there a significant difference between the proportions of women customers in the

two cities?
(a) State the null hypothesis, find the sample proportions of women in both cities,
do a two-sided z test, and give a P -value using Table A.
(b) Calculate the X 2 statistic and show that it is the square of the z statistic. Show
that the P -value from Table F agrees (up to the accuracy of the table) with your
result from (a).
(c) Give a 95% confidence interval for the difference between the proportions of
women customers in the two cities.
9.61 More on secondhand stores. The study of shoppers in secondhand stores

cited in the previous exercise also compared the income distributions of shoppers in
the two stores. Here is the two-way table of counts:
Income City 1 City 2

Under $10,000 70 62
$10,000 to $19,999 52 63
$20,000 to $24,999 69 50
$25,000 to $34,999 22 19
$35,000 or more 28 24
Verify that the X 2 statistic for this table is X 2 = 3.955. Give the degrees of
freedom and the P -value. Is there good evidence that customers at the two stores
have different income distributions?
9.62 Why are animals brought to animal shelters? Euthanasia of healthy

but unwanted pets by animal shelters is believed to be the leading cause of death for
cats and dogs. A study designed to find factors associated with bringing a cat to an
animal shelter compared data on cats that were brought to an animal shelter with
data on cats from the same county that were not brought in. (Gary J. Patronek
et al., “Risk factors for relinquishment of cats to an animal shelter,” Journal of the
American Veterinary Medical Association, 209 (1996), pp. 582–588.) One of the
factors examined was the source of the cat: the categories were private owner or
breeder, pet store, and other (includes born in home, stray, and obtained from a
shelter). This kind of study is called a case-control study by epidemiologists. Here

are the data:
Source
Group Private Pet store Other
Cases 124 16 76
Controls 219 24 203
The same researchers did a similar study for dogs. (Gary J. Patronek et al., “Risk
factors for relinquishment of dogs to an animal shelter,” Journal of the American
Veterinary Medical Association, 209 (1996), pp. 572–581.) The data are given in
the following table:
Source
Group Private Pet store Other
Cases 188 7 90
Controls 518 68 142
(a) Analyze the data for the dogs and the cats separately. Be sure to include graph-
ical and numerical summaries. Is there evidence to conclude that the source of the
animal is related to whether or not the pet is brought to an animal shelter?
(b) Write a discussion comparing the results for the cats with those for the dogs.
(c) These data were collected using a telephone interview with pet owners in Mishawaka,
Indiana. The animal shelter was run by the Humane Society of Saint Joseph County.
The control group data were obtained by a random digit dialing telephone survey.
Discuss how these facts relate to your interpretation of the results.
9.63 More on why animals are brought to animal shelters. Refer to the pre-
vious exercise concerning the case-control study of factors associated with bringing
a cat to an animal shelter and the similar study for dogs. The last category for the
source of the pet was given as “Other” and includes born in home, stray, and ob-
tained from a shelter. The following two-way table lists these categories separately
for cats:
Source
Group Private Pet store Home Stray Shelter
Cases 124 16 20 38 18
Controls 219 24 38 116 49
Here is the same breakdown for dogs:
Source
Group Private Pet store Home Stray Shelter
Cases 188 7 11 23 56
Controls 518 68 20 55 67
Analyze these 2 × 5 tables and compare the results with those that you obtained for
the 2 × 3 tables in Exercise 9.62. With a large number of cells, the chi-square test
sometimes does not have very much power.
Exercises 251
9.64 Student loans. A study of 865 college students found that 42.5% had student
loans. (Data provided by Susan Prohofsky, from her PhD dissertation, “Selection of
undergraduate major: the influence of expected costs and expected benefits,” Pur-
due University, 1991.) The students were randomly selected from the approximately
30,000 undergraduates enrolled in a large public university. The overall purpose of
the study was to examine the effects of student loan burdens on the choice of a
career. A student with a large debt may be more likely to choose a field where
starting salaries are high so that the loan can more easily be repaid. The following
table classifies the students by field of study and whether or not they have a loan:
Student loan
Field of study Yes No
Agriculture 32 35
Child development and family studies 37 50
Engineering 98 137
Liberal arts and education 89 124
Management 24 51
Science 31 29
Technology 57 71
Carry out a complete analysis of the association between having a loan and field of
study, including a description of the association and an assessment of its statistical
significance.
9.65 Altruism and field of study. In one part of the study described in the
previous exercise, students were asked to respond to some questions regarding their
interests and attitudes. Some of these questions form a scale called PEOPLE that
measures altruism, or an interest in the welfare of others. Each student was classified
as low, medium, or high on this scale. Is there an association between PEOPLE
score and field of study? Here are the data:
PEOPLE score
Field of study Low Medium High
Agriculture 5 27 35
Child development and family studies 1 32 54
Engineering 12 129 94
Liberal arts and education 7 77 129
Management 3 44 28
Science 7 29 24
Technology 2 62 64
Analyze the data and summarize your results. Are there some fields of study that
have very large or very small proportions of students in the high-PEOPLE category?
9.66 “No Sweat” label. Following complaints about the working conditions in
some apparel factories both in the United States and abroad, a joint government
and industry commission recommended in 1998 that companies that monitor and
enforce proper standards be allowed to display a “No Sweat” label on their products.
Does the presence of these labels influence consumer behavior? A survey of U.S.
residents aged 18 or older asked a series of questions about how likely they would be
to purchase a garment under various conditions. For some conditions, it was stated
that the garment had a “No Sweat” label; for others, there was no mention of such
a label. On the basis of the responses, each person was classified as a “label user” or
a “label nonuser.” (Marsha A. Dickson, “Utility of no sweat labels for apparel cus-
tomers: profiling label users and predicting their purchases,” Journal of Consumer
Affairs, 35 (2001), pp. 96–119.) There were 296 women surveyed. Of these, 63 were
label users. On the other hand, 27 of 251 men were classified as users.
(a) Construct the 2 × 2 table of counts for this problem. Include the marginal totals
for your table.
(b) Use a X 2 statistic to examine the question of whether or not there is a relation-
ship between gender and use of No Sweat labels. Give the test statistic, degrees of
freedom, the P -value, and your conclusion.
(c) You examined this question using the methods of the previous chapter in Exer-
cise 8.107. Verify that if you square the z statistic you calculated for that exercise,
you obtain the X 2 statistic that you calculated for this exercise.
CHAPTER 10
Chapter 10 Exercises
10.1 Manatees are large sea creatures that live in the shallow water along the coast
of Florida. Many manatees are injured or killed each year by powerboats. Exer-
cise 2.36 gives data on manatees killed and powerboat registrations (in thousands
of boats) in Florida for the period 1977 to 1990.
(a) Make a scatterplot of boats registered and manatees killed. Is there a strong
straight line pattern?
(b) Find the equation of the least-squares regression line. Draw this line on your
scatterplot.
(c) Is there strong evidence that the mean number of manatees killed increases as
the number of powerboats increases? State this question as null and alternative
hypotheses about the slope of the population regression line, obtain the t statistic,
and give your conclusion.
(d) Predict the number of manatees that will be killed if there are 716,000 power-
boats registered. In 1991, 1992, and 1993, the number of powerboats remained at
716,000. The numbers of manatees killed were 53, 38, and 35. Compare your predic-
tion with these data. Does the comparison suggest that measures taken to protect
the manatees in these years were effective?
10.2 Can a pretest on mathematics skills predict success in a statistics course? The
55 students in an introductory statistics class took a pretest at the beginning of the
semester. The least-squares regression line for predicting the score y on the final
exam from the pretest score x was ŷ = 10.5 + 0.82x. The standard error of b1 was
0.38. Test the null hypothesis that there is no linear relationship between the pretest
score and the score on the final exam against the two-sided alternative.
10.3 Exercise 2.53 gives the following data from a study of two methods for mea-
suring the blood flow in the stomachs of dogs:
Spheres 4.0 4.7 6.3 8.2 12.0 15.9 17.4 18.1 20.2 23.9
Vein 3.3 8.3 4.5 9.3 10.7 16.4 15.4 17.6 21.0 21.7
“Spheres” is an experimental method that the researchers hope will predict “Vein,”
the standard but difficult method. Examination of the data gives no reason to doubt
the validity of the simple linear regression model. The estimated regression line is
ŷ = 1.031 + 0.902x, where y is the response variable Vein and x is the explanatory
variable Spheres. The estimate of σ is s = 1.757.
P
(a) Find x and (xi − x)2 from the data.
(b) We expect x and y to be positively associated. State hypotheses in terms of the
slope of the population regression line that express this expectation, and carry out
a significance test. What conclusion do you draw?
(c) Find a 99% confidence interval for the slope.
(d) Suppose that we observe a value of Spheres equal to 15.0 for one dog. Give a
90% interval for predicting the variable Vein for that dog.
253
10.4 Ohm’s law I = V /R states that the current I in a metal wire is proportional
to the voltage V applied to its ends and is inversely proportional to the resistance R
in the wire. Students in a physics lab performed experiments to study Ohm’s law.
They varied the voltage and measured the current at each voltage with an ammeter.
The goal was to determine the resistance R of the wire. We can rewrite Ohm’s law
in the form of a linear regression as I = β0 + β1V , where β0 = 0 and β1 = 1/R. Be-
cause voltage is set by the experimenter, we think of V as the explanatory variable.
The current I is the response. Here are the data for one experiment (data provided
by Sara McCabe):
V 0.50 1.00 1.50 1.80 2.00

I 0.52 1.19 1.62 2.00 2.40
(a) Plot the data. Are there any outliers or unusual points?
(b) Find the least-squares fit to the data, and estimate 1/R for this wire. Then give
a 95% confidence interval for 1/R.
(c) If b1 estimates 1/R, then 1/b1 estimates R. Estimate the resistance R. Similarly,
if L and U represent the lower and upper confidence limits for 1/R, then the corre-
sponding limits for R are given by 1/U and 1/L, as long as L and U are positive.
Use this fact and your answer to (b) to find a 95% confidence interval for R.
(d) Ohm’s law states that β0 in the model is 0. Calculate the test statistic for this
hypothesis and give an approximate P -value.
10.5 Most statistical software systems have an option for doing regressions in which
the intercept is set in advance at 0. If you have access to such software, reanalyze
the Ohm’s law data given in the previous exercise with this option and report the
estimate of R. The output should also include an estimated standard error for 1/R.
Use this to calculate the 95% confidence interval for R. (Note: With this option the
degrees of freedom for t∗ will be 1 greater than for the model with the intercept.)
10.6 Return to the data on current versus voltage given in the Ohm’s law experiment
of Exercise 10.4.
(a) Compute all values for the ANOVA table.
(b) State the null hypothesis tested by the ANOVA F statistic, and explain in plain
language what this hypothesis says.
(c) What is the distribution of this F statistic when H0 is true? Find an approximate
P -value for the test of H0 .
10.7 Here are the golf scores of 12 members of a college women’s golf team in
two rounds of tournament play. (A golf score is the number of strokes required to
complete the course, so that low scores are better.)
Player 1 2 3 4 5 6 7 8 9 10 11 12
Round 1 89 90 87 95 86 81 102 105 83 88 91 79
Round 2 94 85 89 89 81 76 107 89 87 91 88 80
(a) Plot the data and describe the relationship between the two scores.
(b) Find the correlation between the two scores and test the null hypothesis that
the population correlation is 0. Summarize your results.
Exercises 255
(c) The plot shows one outlier. Recompute the correlation and redo the significance
test without this observation. Write a short summary explaining the effect of the
outlier on the correlation and significance test in (b).
10.8 A study reported a correlation r = 0.5 based on a sample size of n = 20;

another reported the same correlation based on a sample size of n = 10. For each,
perform the test of the null hypothesis that ρ = 0. Describe the results and explain
why the conclusions are different.
10.9 Returns on common stocks in the United States and overseas appear to be
growing more closely correlated as economies become more interdependent. Suppose
that this population regression line connects the total annual returns (in percent)
of two indexes of stock prices:
MEAN OVERSEAS RETURN = 4.7 + 0.66 × U.S. RETURN
(a) What is β0 in this line? What does this number say about overseas returns when
the U.S. market is flat (0% return)?
(b) What is β1 in this line? What does this number say about the relationship
between U.S. and overseas returns?
(c) We know that overseas returns will vary during years that have the same return
on U.S. common stocks. Write the regression model based on the population re-
gression line given above. What part of this model allows overseas returns to vary
when U.S. returns remain the same?
10.10 How well does the number of beers a student drinks predict his or her blood
alcohol content? Sixteen student volunteers at Ohio State University drank a ran-
domly assigned number of cans of beer. Thirty minutes later, a police officer mea-
sured their blood alcohol content (BAC). Here are the data (these are part of the
data from the EESEE story “Blood Alcohol Content,” found on the IPS Web site
www.whfreeman.com/ips):
Student 1 2 3 4 5 6 7 8
Beers 5 2 9 8 3 7 3 5
BAC 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06
Student 9 10 11 12 13 14 15 16
Beers 3 5 4 6 5 7 1 4
BAC 0.02 0.05 0.07 0.10 0.085 0.09 0.01 0.05
The students were equally divided between men and women and differed in weight
and usual drinking habits. Because of this variation, many students don’t believe
that number of drinks predicts blood alcohol well.
(a) Make a scatterplot of the data. Find the equation of the least-squares regression
line for predicting blood alcohol from number of beers and add this line to your
plot. What is r2 for these data? Briefly summarize what your data analysis shows.
(b) Is there significant evidence that drinking more beers increases blood alcohol on
the average in the population of all students? State hypotheses, give a test statistic
and P -value, and state your conclusion.
10.11 Your scatterplot in the previous exercise shows one unusual point: student
number 3, who drank 9 beers.
(a) Does student 3 have the largest residual from the fitted line? (You can use the
scatterplot to see this.) Is this observation extreme in the x direction, so that it
may be influential?
(b) Do the regression again, omitting student 3. Add the new regression line to
your scatterplot. Does removing this observation greatly change predicted BAC?
Does r2 change greatly? Does the P -value of your test change greatly? What do
you conclude: did your work in the previous problem depend heavily on this one
student?
10.12 Utility companies need to estimate the amount of energy that will be used
by their customers. The consumption of natural gas required for heating homes
depends on the outdoor temperature. When the weather is cold, more gas will be
consumed. A study of one home recorded the average daily gas consumption y (in
hundreds of cubic feet) for each month during one heating season. The explanatory
variable x is the average number of heating degree-days per day during the month.
One heating degree-day is accumulated for each degree a day’s average temperature
falls below 65◦F. An average temperature of 50◦, for example, corresponds to 15
degree-days. The data for October through June are given in the following table
(data were provided by Professor Robert Dale of the Purdue University Agronomy
Department):
Month Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June
Degree-days 15.6 26.8 37.8 36.4 35.5 18.6 15.3 7.9 0.0
Gas consumption 5.2 6.1 8.7 8.5 8.8 4.9 4.5 2.5 1.1
(a) Find the equation of the least-squares line.

(b) Test the null hypothesis that the slope is zero and describe your conclusion.
(c) Give a 90% confidence interval for the slope.
(d) The parameter β0 corresponds to natural gas consumption for cooking, hot
water, and other uses when there is no demand for heating. Give a 90% confidence
interval for this parameter.
10.13 The previous exercise demonstrates that there is a strong linear relationship
between household consumption of natural gas and outdoor temperature, measured
by heating degree-days. The slope and intercept depend on the particular house
and on the habits of the household living there. Data for two heating seasons (18
months) for another household produce the least-squares line ŷ = 2.405 + 0.26896x
for predicting average daily gas consumption y from average degree-days per day x.
The standard error of the slope is SEb1 = 0.00815.
(a) Explain briefly what the slope β1 of the population regression line represents.
Then give a 90% confidence interval for β1.
(b) This interval is based on twice as many observations as the one calculated in
the previous exercise for a different household, and the two standard errors are of
similar size. How would you expect the margins of error of the two intervals to be
related? Check your answer by comparing the two margins of error.
Exercises 257
10.14 The standard error of the intercept in the regression of gas consumption on
degree-days for the household in the previous exercise is SEb0 = 0.20351.
(a) Explain briefly what the intercept represents in this setting. Find a 90% confi-
dence interval for the intercept.
(b) Compare the width of your interval with the one calculated for a different house-
hold in Exercise 10.12. Explain why it is narrower.
10.15 Exercise 10.12 gives information about the regression of natural gas consump-
tion on degree-days for a particular household.
(a) What is the t statistic for testing H0: β1 = 0?
(b) For the alternative Ha : β1 > 0, what critical value would you use for a test at
the α = 0.05 significance level? Do you reject H0 at this level?
10.16 Can a pretest on mathematics skills predict success in a statistics course?

The 102 students in an introductory statistics class took a pretest at the beginning
of the semester. The least-squares regression line for predicting the score y on the
final exam from the pretest score x was ŷ = 10.5 + 0.73x. The standard error of b1
was 0.42.
(a) Test the null hypothesis that there is no linear relationship between the pretest
score and the score on the final exam against the two-sided alternative.
(b) Would you reject this null hypothesis versus a two-sided alternative? Explain
your answer.
10.17 The human body takes in more oxygen when exercising than when it is at
rest. To deliver the oxygen to the muscles, the heart must beat faster. Heart rate
is easy to measure, but measuring oxygen uptake requires elaborate equipment. If
oxygen uptake (VO2) can be accurately predicted from heart rate (HR), the pre-
dicted values can replace actually measured values for various research purposes.
Unfortunately, not all human bodies are the same, so no single prediction equa-
tion works for all people. Researchers can, however, measure both HR and VO2
for one person under varying sets of exercise conditions and calculate a regression
equation for predicting that person’s oxygen uptake from heart rate. They can then
use predicted oxygen uptakes in place of measured uptakes for this individual in
later experiments. (These data are from experiments conducted in Don Corrigan’s
laboratory at Purdue University and were provided by Paul Waldsmith.) Here are
data for one individual:
HR 94 96 95 95 94 95 94 104 104 106

VO2 0.473 0.753 0.929 0.939 0.832 0.983 1.049 1.178 1.176 1.292
HR 108 110 113 113 118 115 121 127 131

VO2 1.403 1.499 1.529 1.599 1.749 1.746 1.897 2.040 2.231
(a) Plot the data. Are there any outliers or unusual points?
(b) Compute the least-squares regression line for predicting oxygen uptake from
heart rate for this individual.
(c) Test the null hypothesis that the slope of the regression line is 0. Explain in
words the meaning of your conclusion from this test.

(d) Calculate a 95% interval for the oxygen uptake of this individual on a future
occasion when his heart rate is 96. Repeat the calculation for a heart rate of 115.
(e) From what you have learned in (a), (b), (c), and (d) of this exercise, do you
think that the researchers should use predicted VO2 in place of measured VO2 for
this individual under similar experimental conditions? Explain your answer.
10.18 Premature infants are often kept in intensive-care nurseries after they are
born. It is common practice to measure their blood pressure frequently. The oscil-
lometric method of measuring blood pressure is noninvasive and easy to use. The
traditional procedure, called the direct intra-arterial method, is believed to be more
accurate but is invasive and more difficult to perform. Several studies have reported
high correlations between measurements made by the two methods, ranging from
r = 0.49 to r = 0.98. These correlations are statistically significant. One study that
investigated the relation between the two methods reported the regression equation
ŷ = 15+0.83x. Here x represents the easy method and y represents the difficult one.
(John A. Wareham et al., “Prediction of arterial blood pressure on the premature
neonate using the oscillometric method,” American Journal of Diseases of Children,
141 (1987), pp. 1108–1110.) The standard error of the slope is 0.065 and the sample
size is 81. Calculate the t statistic for testing H0: β1 = 0. Specify an appropriate al-
ternative hypothesis for this problem, and give an approximate P -value for the test.
Then explain your conclusion in words a physician can understand. (The authors of
the study calculated and plotted prediction intervals. They found the widths to be
unacceptably large and concluded that statistical significance does not imply that
results are clinically useful.)
10.19 Soil aeration and soil water evaporation involve the exchange of gases between
the soil and the atmosphere. Experimenters have investigated the effect of the airflow
above the soil on this process. One such experiment varied the speed of the air x
and measured the rate of evaporation y. The fitted regression equation based on 18
observations was ŷ = 5.0 + 0.00665x. The standard error of the slope was reported
to be 0.00182.
(a) It is reasonable to suppose that greater airflow will cause more evaporation. State
hypotheses to test this belief and calculate the test statistic. Find an approximate
P -value for the significance test and report your conclusion.
(b) Construct a 90% confidence interval for the additional evaporation experienced
when airflow increases by 1 unit.
10.20 Here are data on the average yield in bushels per acre for corn in the United
States.
Exercises 259
Year Yield Year Yield Year Yield Year Yield

1957 48.3 1968 79.5 1979 109.5 1990 118.5
1958 52.8 1969 85.9 1980 91.0 1991 108.6
1959 53.1 1970 72.4 1981 108.9 1992 131.5
1960 54.7 1971 88.1 1982 113.2 1993 100.7
1961 62.4 1972 97.0 1983 81.1 1994 138.6
1962 64.7 1973 91.3 1984 106.7 1995 113.5
1963 67.9 1974 71.9 1985 118.0 1996 127.1
1964 62.9 1975 86.4 1986 119.4 1997 126.7
1965 74.1 1976 88.0 1987 119.8 1998 134.4
1966 73.1 1977 90.8 1988 84.6 1999 133.8
1967 80.1 1978 101.0 1989 116.3 2000 136.9
(a) Plot the yield versus year. Describe the relationship. Are there any outliers or
unusual years?
(b) Perform the regression analysis and summarize the results. How rapidly has
yield increased over time?
10.21 Refer to the previous exercise. Give a 95% prediction interval for the yield
in the year 2006.
10.22 In the previous exercise you examined the relationship between time and the
yield of corn in the United States. Here are similar data for the yield of soybeans
(in bushels per acre).
Year Yield Year Yield Year Yield Year Yield

1957 23.2 1968 26.7 1979 32.1 1990 34.1
1958 24.2 1969 27.4 1980 26.5 1991 34.2
1959 23.5 1970 26.7 1981 30.1 1992 37.6
1960 23.5 1971 27.5 1982 31.5 1993 32.6
1961 25.1 1972 27.8 1983 26.2 1994 41.4
1962 24.2 1973 27.8 1984 28.1 1995 35.3
1963 24.4 1974 23.7 1985 34.1 1996 37.6
1964 22.8 1975 28.9 1986 33.3 1997 38.9
1965 24.5 1976 26.1 1987 33.9 1998 38.9
1966 25.4 1977 30.6 1988 27.0 1999 36.6
1967 24.5 1978 29.4 1989 32.3 2000 38.1
Give a complete analysis of these data. Include a plot of the data, significance test
results, examination of the residuals, and your conclusions.
10.23 The corn yields of Exercise 10.20 and the soybean yields of Exercise 10.22
both vary over time for similar reasons, including improved technology and weather
conditions. Let’s examine the relationship between the two yields.
(a) Plot the two yields with corn on the x axis and soybeans on the y axis. Describe
the relationship.
(b) Find the correlation. How well does it summarize the relation?
(c) Use corn yield to predict soybean yield. Give the equation and the results of the
significance test for the slope. This test also tests the null hypothesis that the two
yields are uncorrelated.

(d) Obtain the residuals from the model in part (c) and plot them versus time.
Describe the pattern.
10.24 The corn yield data of Exercise 10.20 show a larger amount of scatter about
the least-squares line for the later years, when the yields are higher. This may be
an indication that the standard deviation σ of our model is not a constant but is
increasing with time. Take logs of the yields and rerun the analyses. Prepare a short
report comparing the two analyses. Include plots, a comparison of the significance
test results, and the percent of variation explained by each model.
10.25 Exercise 10.10 gives data from measuring the blood alcohol content (BAC)
of students 30 minutes after they drank an assigned number of cans of beer. Steve
thinks he can drive legally 30 minutes after he drinks 5 beers. The legal limit is
BAC = 0.08. Give a 95% confidence interval for Steve’s BAC. Can he be confident
he won’t be arrested if he drives and is stopped?
10.26 Return to the oxygen uptake and heart rate data given in Exercise 10.17.
(a) Construct the ANOVA table.
(b) What null hypothesis is tested by the ANOVA F statistic? What does this
hypothesis say in practical terms?
(c) Give the degrees of freedom for the F statistic and an approximate P -value for
the test of H0.
(d) Verify that the square of the t statistic that you calculated in Exercise 10.16
is equal to the F statistic in your ANOVA table. (Any difference found is due to
roundoff error.)
(e) What proportion of the variation in oxygen uptake is explained by heart rate for
this set of data?
10.27 A study conducted in the Egyptian village of Kalama examined the relation-
ship between the birth weights of 40 infants and various socioeconomic variables.
(M. El-Kholy, F. Shaheen, and W. Mahmoud, “Relationship between socioeconomic
status and birth weight, a field study in a rural community in Egypt,” Journal of
the Egyptian Public Health Association, 61 (1986), pp. 349–358.)
(a) The correlation between monthly income and birth weight was r = 0.39. Cal-
culate the t statistic for testing the null hypothesis that the correlation is 0 in the
entire population of infants.
(b) The researchers expected that higher birth weights would be associated with
higher incomes. Express this expectation as an alternative hypothesis for the pop-
ulation correlation.
(c) Determine a P -value for H0 versus the alternative that you specified in (b).
What conclusion does your test suggest?
10.28 Chinese students from public schools in Hong Kong were the subjects of a
study designed to investigate the relationship between various measures of parental
behavior and other variables. The sample size was 713. The data were obtained
from questionnaires filled in by the students. One of the variables examined was
parental control, an indication of the amount of control that the parents exercised
Exercises 261
over the behavior of the students. Another was the self-esteem of the students.
(S. Lau and P. C. Cheung, “Relations between Chinese adolescents’ perception of
parental control and organization and their perception of parental warmth,” Devel-
opmental Psychology, 23 (1987), pp. 726–729.)
(a) The correlation between parental control and self-esteem was r = −0.19. Cal-
culate the t statistic for testing the null hypothesis that the population correlation
is 0.
(b) Find an approximate P -value for testing H0 versus the two-sided alternative and
report your conclusion.
10.29 The data on gas consumption and degree-days from Exercise 10.12 are as
follows:
Month Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June
Degree-days 15.6 26.8 37.8 36.4 35.5 18.6 15.3 7.9 0.0
Gas consumption 5.2 6.1 8.7 8.5 8.8 4.9 4.5 2.5 1.1
Suppose that the gas consumption for January was incorrectly recorded as 85 instead
of 8.5.
(a) Calculate the least-squares regression line for the incorrect set of data.
(b) Find the standard error of b1.
(c) Compute the test statistic for H0 : β1 = 0 and find the P -value. How do the
results compare with those for the correct set of data?
10.30 Archaeopteryx is an extinct beast having feathers like a bird but teeth and
a long bony tail like a reptile. Only six fossil specimens are known. Because these
specimens differ greatly in size, some scientists think they are different species rather
than individuals from the same species. Here are data on the lengths in centimeters
of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five
specimens that preserve both bones (Marilyn A. Houck et al., “Allometric scaling in
the earliest fossil bird, Archaeopteryx lithographica,” Science, 247 (1990), pp. 195–
198. The authors conclude from a variety of evidence that all specimens represent
the same species):
Femur 38 56 59 64 74
Humerus 41 63 70 72 84
(a) Plot the data and describe the pattern. Is it reasonable to summarize this kind
of relationship with a correlation?
(b) Find the correlation and perform the significance test. Summarize the results
and report your conclusion.
10.31 How does the fuel consumption of a car change as its speed increases? Here
are data for a British Ford Escort. Speed is measured in kilometers per hour, and
fuel consumption is measured in liters of gasoline used per 100 kilometers traveled.
(Based on T. N. Lam, “Estimating fuel consumption from engine size,” Journal of
Transportation Engineering, 111 (1985), pp. 339–357. The data for 10 to 50 km/h
are measured; those for 60 and higher are calculated from a model given in the paper
and are therefore smoothed.)

(km/h) (liters/100 km) (km/h) (liters/100 km)
10 21.00 90 7.57
20 13.00 100 8.27
30 10.00 110 9.03
40 8.00 120 9.87
50 7.00 130 10.79
60 5.90 140 11.77
70 6.30 150 12.83
80 6.95
You were assigned to analyze these data for a team project. (The other two members
of your team did not take a statistics course based on this text.) The first team
member prepared a draft with the following summary:
Fuel consumption does not depend on speed for this vehicle (t = −0.63,
df = 13, P = 0.54).
The second team member had some trouble with the statistical software. When
the data were read, only the first two columns of the data table were used by the
software. These are the data for speeds between 10 and 80 km/h. This team member
prepared the following draft:
There is a strong relationship between fuel consumption and speed for
this vehicle (t = −3.63, df = 6, P = 0.0109). Speed explains 68.8% of
the variation in fuel consumption.
First, verify that your two teammates have computed the quantities that they re-
ported correctly. Then analyze the data and write a summary of your analysis of
the relationship between fuel consumption and speed.
10.32 Find a 95% confidence interval for the slope in each of the following settings:
(a) n = 25, ŷ = 12.3 + 16.10x, and SEb1 = 8.05
(b) n = 25, ŷ = 1.3 + 6.10x, and SEb1 = 8.05
(c) n = 125, ŷ = 12.3 + 16.10x, and SEb1 = 8.05
10.33 In Exercise 7.70 we examined the distribution of C-reactive protein (CRP)

in a sample of 40 children from Papua New Guinea. Serum retinol values for the
same children were studied in Exercise 7.72. One important question that can be
addressed with these data is whether or not infections, as indicated by CRP, cause
a decrease in the measured values of retinol, low values of which indicate a vitamin
A deficiency. The data are given below.
(a) Examine the distributions of CRP and serum retinol. Use graphical and numer-
ical methods.
(b) Forty percent of the CRP values are zero. Does this violate any assumption that
we need to do a regression analysis using CRP to predict serum retinol? Explain
your answer.
(c) Run the regression, summarize the results, and write a short paragraph explain-
ing you conclusions.
(d) Explain the assumptions needed for your results to be valid. Examine the data
with respect to these assumptions and report your results.
Exercises 263
C-reactive protein and serum retinol
CRP Retinol CRP Retinol CRP Retinol CRP Retinol CRP Retinol
0.00 1.15 30.61 0.97 22.82 0.24 5.36 1.19 0.00 0.83
3.90 1.36 0.00 0.67 0.00 1.00 0.00 0.94 0.00 1.11
5.64 0.38 73.20 0.31 0.00 1.13 5.66 0.34 0.00 1.02
8.22 0.34 0.00 0.99 3.49 0.31 0.00 0.35 9.37 0.56
0.00 0.35 46.70 0.52 0.00 1.44 59.76 0.33 20.78 0.82
5.62 0.37 0.00 0.70 0.00 0.35 12.38 0.69 7.10 1.20
3.92 1.17 0.00 0.88 4.81 0.34 15.74 0.69 7.89 0.87
6.81 0.97 26.41 0.36 9.57 1.90 0.00 1.04 5.53 0.41
10.34 In Exercise 7.143 we looked at the distribution of osteocalcin (OC), a biomarker

for bone formation, in a sample of 31 healthy females aged 11 to 32 years. This
biomarker is relatively inexpensive to measure, requiring only a single sample of
blood. Measuring bone formation (VO+), on the other hand, is very expensive.
Oral and intravenous administration of stable isotopes of calcium are needed, 25
blood samples over a period of two weeks are drawn, and the collection of all urine
and fecal samples for two weeks is required. If a biomarker can reliably predict bone
formation, then we could avoid the cost of the expensive VO+ measures. Studies
designed to assess the effects of interventions intended to increase bone formation
could include many more subjects if only the biomarker measurement is needed.
The measured values of VO+ and OC for the 31 females in this study are given
below.
(a) Use numerical and graphical summaries to describe the distributions of VO+
and OC.
(b) Plot the data. Give a reason for your choice of variables for the x and y axes.
Describe the pattern and note any unusual observations. Do the assumptions needed
for regression analysis appear to be approximately satisfied?
(c) Run the regression using OC to predict VO+. Summarize the results.
VO+ and osteocalcin

VO+ OC VO+ OC VO+ OC VO+ OC
476 8.1 1032 40.2 624 17.2 285 9.9
694 10.1 445 20.6 479 15.9 403 19.7
753 17.9 896 31.2 572 16.9 391 20.0
687 17.2 968 19.3 512 24.2 513 20.8
628 20.9 985 44.4 838 30.2 878 31.4
1100 38.4 1251 76.5 870 47.7 2221 54.6
1303 54.6 2545 36.4 1606 68.9 1126 77.9
1682 52.8 2240 56.3 1557 35.7
10.35 In Exercise 7.144 we looked at the distribution of tartrate resistant acid

phosphatase (TRAP), a biomarker for bone resorption. The following table gives
values for this biomarker and a measure of bone resorption VO–. Analyze these
data using the questions in the previous exercise as a guide.
VO– and TRAP
VO– TRAP VO– TRAP VO– TRAP VO– TRAP

407 3.3 874 5.9 445 6.3 351 6.9
980 8.1 493 8.1 572 8.2 634 8.8
1028 9.0 1116 9.0 857 9.5 536 9.5
701 9.6 934 10.1 477 10.1 254 10.3
766 10.5 496 10.7 924 14.4 954 14.6
918 14.6 1065 14.9 722 18.6 1486 19.0
1018 19.0 2236 19.1 903 19.4 960 23.7
1251 25.2 1761 25.5 1446 28.8
10.36 Refer to the OC and VO+ data in Exercise 10.34. For variables such as these,
it is common to work with the logarithms of the measured values. Reanalyze these
data using the logs of both OC and VO+. Summarize your results and compare
them with those you obtained in Exercise 10.34.
10.37 Refer to the TRAP and VO– data in Exercise 10.35. Reanalyze these data
using the logs of both TRAP and VO–. Summarize your results and compare them
with those you obtained in Exercise 10.35.
10.38 Refer to the data in Exercise 2.11. Metatarsus adductus (call it MA) is a
turning in of the front part of the foot that is common in adolescents and usually
corrects itself. Hallux abducto valgus (call it HAV) is a deformation of the big
toe that is not common in youth and often requires surgery. Perhaps the severity
of MA can help predict the severity of HAV. Using X-rays, doctors measured the
angle of deformity for both MA and HAV. They speculated that there is a positive
association—more serious MA is associated with more serious HAV.
(a) Make a scatterplot of the data. (Which is the explanatory variable?)
(b) Describe the form, direction, and strength of the relationship between MA angle
and HAV angle. Are there any clear outliers in your graph?
(c) Give a statistical model that provides a framework for asking the question of
interest for this problem.
(d) Translate the question of interest into null and alternative hypotheses.
(e) Test these hypotheses and write a short description of the results. Be sure to
include the value of the test statistic, the degrees of freedom, the P -value, and a
clear statement of what you conclude.
10.39 Refer to the previous exercise. Give a 95% confidence interval for the slope.
Explain how this interval can tell you what to conclude from a significance test for
this parameter.
10.40 A study reported correlations between several personality traits and scores
on the Graduate Record Examination (GRE) for a sample of 342 test takers (D.
E. Powers and J. K. Kaufman, “Do standardized multiple choice tests penalize
deep-thinking or creative students?” Educational Testing Service Research Report
RR–02–15 (2002)). Here is a table of the correlations:
Exercises 265
GRE score
Personality trait Analytical Quantitative Verbal
Conscientiousness −0.17 −0.14 −0.12
Rationality −0.06 −0.03 −0.08
Ingenuity −0.06 −0.08 −0.02
Quickness 0.21 0.15 0.26
Creativity 0.24 0.26 0.29
Depth 0.06 0.08 0.15
For each correlation, test the null hypothesis that the corresponding true correlation
is zero. Reproduce the table and mark the correlations that have P < 0.001 with
***, those that have P < 0.01 with **, and those that have P < 0.05 with *. Some
critics of standardized tests have suggested that the tests penalize students who
are “deep thinkers” and those who are very creative. Others have suggested that
students who work quickly do better on these tests. Write a summary of the results
of your significance tests, taking into account these comments.
CHAPTER 11
11.1 One model for subpopulation means for a computer science study is described
as
µGPA = β0 + β1 HSM + β2 HSS + β3HSE
Give the model for the subpopulation mean GPA for students having high school
grade scores HSM = 9 (A−), HSS = 8 (B+), and HSE = 7 (B).
11.2 Use the model given in the previous exercise to express the subpopulation
mean in terms of the parameters βj for students having high school grade scores
HSM = 6 (B−), HSS = 7 (B), and HSE = 8 (B+),
11.3 A multiple regression is used to relate a response variable to a set of 7 explana-

tory variables. There are 120 observations. Outline the analysis of variance table
for this analysis giving the sources of variation and the degrees of freedom.
11.4 A multiple regression analysis of 105 cases was performed with 4 explanatory
variables. Suppose that SSM = 20 and SSE = 200.
(a) Find the value of the F statistic for testing the null hypothesis that the coeffi-
cients of all of the explanatory variables are zero.
(b) Use Table E to determine if the result is significant at the 5% level. Is it signif-
icant at the 1% level?
11.5 Refer to the previous exercise. What proportion of the variation in the response
variable is explained by the explanatory variables?
11.6 An instructor in an introductory statistics class used multiple regression to

predict the score on the final exam using the results of 10 quizzes that were given
during the semester. The F statistic for this analysis was highly significant (P <
0.0001) but none of the t tests for the individual coefficients was significant. Explain
this apparent inconsistency.
The following eight exercises are related to the case study described in this chapter.
They require use of the CSDATA data set described in the Data Appendix.
11.7 Use software to make a plot of GPA versus SATM. Do the same for GPA versus
SATV. Describe the general patterns. Are there any unusual values?
11.8 Make a plot of GPA versus HSM. Do the same for the other two high school
grade variables. Describe the three plots. Are there any outliers or influential
points?
11.9 Regress GPA on the three high school grade variables. Calculate and store the
residuals from this regression. Plot the residuals versus each of the three predictors
and versus the predicted value of GPA. Are there any unusual points or patterns in
these four plots?
Exercises 267
11.10 Use the two SAT scores in a multiple regression to predict GPA. Calculate
and store the residuals. Plot the residuals versus each of the explanatory variables
and versus the predicted GPA. Describe the plots.
11.11 It appears that the mathematics explanatory variables are strong predictors
of GPA in the computer science study. Run a multiple regression using HSM and
SATM to predict GPA.
(a) Give the fitted regression equation.
(b) State the H0 and Ha tested by the ANOVA F statistic, and explain their mean-
ing in plain language. Report the value of the F statistic, its P -value, and your
conclusion.
(c) Give 95% confidence intervals for the regression coefficients of HSM and SATM.
Do either of these include the point 0?
(d) Report the t statistics and P -values for the tests of the regression coefficients of
HSM and SATM. What conclusions do you draw from these tests?
(e) What is the value of s, the estimate of σ?
(f) What percent of the variation in GPA is explained by HSM and SATM in your
model?
11.12 How well do verbal variables predict the performance of computer science
students? Perform a multiple regression analysis to predict GPA from HSE and
SATV. Summarize the results and compare them with those obtained in the previous
exercise. In what ways do the regression results indicate that the mathematics
variables are better predictors?
11.13 The variable SEX has the value 1 for males and 2 for females. Create a
data set containing the values for males only. Run a multiple regression analysis for
predicting GPA from the three high school grade variables for this group. Using the
case study in the text as a guide, interpret the results and state what conclusions
can be drawn from this analysis. In what way (if any) do the results for males alone
differ from those for all students?
11.14 Refer to the previous exercise. Perform the analysis using the data for females
only. Are there any important differences between female and male students in
predicting GPA?
The following three exercises use the CONCEPT data set described in the Data
Appendix.
11.15 Find the correlations between the response variable GPA and each of the
explanatory variables IQ, AGE, SEX, SC, and C1 to C6. Of all the explanatory
variables, IQ does the best job of explaining GPA in a simple linear regression with
only one variable. How do you know this without doing all of the regressions? What
percent of the variation in GPA can be explained by the straight-line relationship
between GPA and IQ?
11.16 Let us look at the role of positive self-concept about one’s physical appearance
(variable C3). We include IQ in the model because it is a known important predictor
of GPA.
(a) Report the fitted regression model
dA = b0 + b1IQ + b2C3
GP
with its R2 , and report the t statistic for significance of self-concept about one’s
physical appearance with its P -value. Does C3 contribute significantly to explaining
GPA when added to IQ? How much does adding C3 to the model raise R2?
(b) Now start with the model that includes overall self-concept SC along with IQ.
Does C3 help explain GPA significantly better? Does it raise R2 enough to be of
practical value? In answering these questions, report the fitted regression model
dA = b0 + b1IQ + b2C3 + b3SC

GP
with its R2 and the t statistic for significance of C3 with its P -value.
(c) Explain carefully in words why the coefficient b2 for the variable C3 takes quite
different values in the regressions of parts (a) and (b). Then explain simply how
it can happen that C3 is a useful explanatory variable in part (a) but worthless in
part (b).
11.17 A reasonable model explains GPA using IQ, C1 (behavior self-concept), and
C5 (popularity self-concept). These three explanatory variables are all significant in
the presence of the other two, and no other explanatory variable is significant when
added to these three. (You do not have to verify these statements.) Let’s use this
model.
(a) What is the fitted regression model, its R2 , and the standard deviation s about
the fitted model? What GPA does this model predict for a student with IQ 109, C1
score 13, and C5 score 8?
(b) If both C1 and C5 could be held constant, how much would GPA increase
for each additional point of IQ score, according to the fitted model? Give a 95%
confidence interval for the mean increase in GPA in the entire population if IQ could
be increased by 1 point while holding C1 and C5 constant.
(c) Compute the residuals for this model. Plot the residuals against the predicted
values and also make a Normal quantile plot. Which observation (value of OBS)
produces the most extreme residual? Circle this observation in both of your plots
of the residuals. For which individual variables does this student have very unusual
values? What are these values? (Look at all the variables, not just those in the
model.)
(d) Repeat part (a) with this one observation removed. How much did this one
student affect the fitted model and the prediction?
The following 7 exercises use the corn and soybean data given in Exercises 10.20
and 10.22.
11.18 Run the simple linear regression using year to predict corn yield.
(a) Summarize the results of your analysis, including the significance test results for
the slope and R2 for this model.
(b) Analyze the residuals with a Normal quantile plot. Is there any indication in
the plot that the residuals are not Normal?
Exercises 269
(c) Plot the residuals versus soybean yield. Does the plot indicate that soybean
yield might be useful in a multiple linear regression with year to predict corn yield?
11.19 Run the simple linear regression using soybean yield to predict corn yield.
(a) Summarize the results of your analysis, including the significance test results for
the slope and R2 for this model.
(b) Analyze the residuals with a Normal quantile plot. Is there any indication in
the plot that the residuals are not Normal?
(c) Plot the residuals versus year. Does the plot indicate that year might be useful
in a multiple linear regression with soybean yield to predict corn yield?
11.20 From the previous two exercises, we conclude that year and soybean yield
may be useful together in a model for predicting corn yield. Run this multiple re-
gression.
(a) Explain the results of the ANOVA F test. Give the null and alternative hy-
potheses, the test statistic with degrees of freedom, and the P -value. What do you
conclude?
(b) What percent of the variation in corn yield is explained by these two variables?
Compare it with the percent explained in the simple linear regression models of the
previous two exercises.
(c) Give the fitted model. Why do the coefficients of year and soybean yield differ
from those in the previous two exercises?
(d) Summarize the significance test results for the regression coefficients for year
and soybean yield.
(e) Give a 95% confidence interval for each of these coefficients.
(f) Plot the residuals versus year and versus soybean yield. What do you conclude?
11.21 We need a new variable to model the curved relation that we see between corn
yield and year in the residual plot of the last exercise. Let year2 = (year − 1978.5)2.
(When adding a squared term to a multiple regression model, it is a good idea to
subtract the mean of the variable being squared before squaring. This avoids all
sorts of messy problems that we cannot discuss here.)
(a) Run the multiple linear regression using year, year2, and soybean yield to predict
corn yield. Give the fitted regression equation.
(b) Give the null and alternative hypotheses for the ANOVA F test. Report the
results of this test, giving the test statistic, degrees of freedom, P -value, and con-
clusion.
(c) What percent of the variation in corn yield is explained by this multiple regres-
sion? Compare this with the model in the previous exercise.
(d) Summarize the results of the significance tests for the individual regression co-
efficients.
(e) Analyze the residuals and summarize your conclusions.
11.22 Run the model to predict corn yield using year and the squared term year2
defined in the previous exercise.
(a) Summarize the significance test results.
(b) The coefficient of year2 is not statistically significant in this run, but it was
highly significant in the model analyzed in the previous exercise. Explain how this
can happen.
(c) Obtain the fitted values for each year in the data set and use these to sketch the
curve on a plot of the data. Plot the least-squares line on this graph for comparison.
Describe the differences between the two regression functions. For what years do
they give very similar fitted values? For what years are the differences between the
two relatively large?
11.23 Use the simple linear regression model with corn yield as the response variable
and year as the explanatory variable to predict the corn yield for the year 2001, and
give the 95% prediction interval. Also, use the multiple regression model where year
and year2 are both explanatory variables to find another predicted value with the
95% interval. Explain why these two predicted values are so different. The actual
yield for 2001 is 138.2 bushels per acre. How well did your models predict this value?
11.24 Repeat the previous exercise doing the prediction for 2002. Compare the
results of this exercise with the previous one. Why are they different?
We assume that our wages will increase as we gain experience and become more
valuable to our employers. Wages also increase because of inflation. By examining
a sample of employees at a given point in time, we can look at part of the picture.
How does length of service (LOS) relate to wages? The following table gives data on
the LOS in months and wages for 60 women who work in Indiana banks. Wages are
yearly total income divided by the number of weeks worked. We have multiplied wages
by a constant for reasons of confidentiality. (The data were provided by Professor
Shelly MacDermid, Department of Child Development and Family Studies, Purdue
University, from a study reported in S. M. MacDermid et al., “Is small beautiful?
Work-family tension, work conditions, and organizational size,” Family Relations,
44 (1994), pp. 159–167.)
Exercises 271
Wages LOS Size Wages LOS Size Wages LOS Size

48.3355 94 Large 64.1026 24 Large 41.2088 97 Small
49.0279 48 Small 54.9451 222 Small 67.9096 228 Small
40.8817 102 Small 43.8095 58 Large 43.0942 27 Large
46.7596 60 Large 61.9893 153 Large 40.5748 7 Large
39.1304 45 Large 50.7143 43 Small 50.1742 204 Large
60.5637 30 Large 38.5579 22 Small 57.2344 184 Small
38.7847 10 Small 45.7875 83 Small 67.9584 36 Large
For these exercises we code the size of the bank as 1 if it is large and 0 if it is
small. There is one outlier in this data set. Delete it and use the remaining 59
observations in these exercises.
11.25 Use length of service (LOS) to predict wages with a simple linear regression.
Write a short summary of your results and conclusions.
11.26 Predict wages using the size of the bank as the explanatory variable. Use the
coded values 0 and 1 for this model.
(a) Summarize the results of your analysis. Include a statement of all hypotheses,
test statistics with degrees of freedom, P -values, and conclusions.
(b) Calculate the t statistic for comparing the mean wages for the large and the small
banks assuming equal standard deviations. Give the degrees of freedom. Verify that
this t is the same as the t statistic for the coefficient of size in the regression. Explain
why this makes sense.
(c) Plot the residuals versus LOS. What do you conclude?
11.27 Use a multiple linear regression to predict wages from LOS and the size of
the bank. Write a report summarizing your work. Include graphs and the results of
significance tests.
11.28 In each of the following settings, give a 95% confidence interval for the coef-
ficients of x1 and x2 .
(a) n = 25, ŷ = 10.6 + 12.1x1 + 17.3x2, SEb1 = 7.2, and SEb2 = 4.1
(b) n = 103, ŷ = 15.6 + 12.1x1 + 7.3x2, SEb1 = 7.2, and SEb2 = 4.1
11.29 In each of the following situations, explain what is wrong and why.
(a) The multiple correlation gives the proportion of the variation in the response
variable that is explained by the explanatory variables.
(b) In a multiple regression with a sample size of 50 and four explanatory variables,
the test statistic for the null hypothesis H0 : b2 = 0 is a t statistic that follows the
t(45) distribution when the null hypothesis is true.
(c) One of the assumptions for multiple regression is that the distribution of each
explanatory variable should be Normal.
11.30 The fitted regression equation for a multiple regression is
ŷ = −2.6 + 4.1x1 − 3.2x2
(a) If x1 = 5 and x2 = 3, what is the predicted value of y?

(b) For the answer to part (a) to be valid, is it necessary that the values x1 = 5 and
x2 = 3 correspond to a case in the data set? Explain why or why not.
(c) If you hold x1 at a fixed value, what is the effect of an increase of one unit in x2
on the predicted value of y?
11.31 Six explanatory variables are used to predict a response variable using a mul-
tiple regression. There are 200 observations.
(a) Write the statistical model that is the foundation for this analysis. Be sure to
include a description of all assumptions.
(b) Outline the analysis of variance table giving the sources of variation and numer-
ical values for the degrees of freedom.
11.32 A multiple regression analysis of 54 cases was performed with 3 explanatory

variables. Suppose that SSM = 18 and SSE = 100.
(a) Find the value of the F statistic for testing the null hypothesis that the coeffi-
cients of all of the explanatory variables are zero.
(b) What are the degrees of freedom for this statistic?
(c) Find bounds on the P -value using Table E. Give the two entries from the table
that you use to determine the bounds.
11.33 Refer to the previous exercise. What proportion of the variation in the
response variable is explained by the explanatory variables?
11.34 Using a new software package, you ran a multiple regression. The output
reported an F statistic with P < 0.001, but none of the t tests for the individual
coefficients were significant. Does this mean that there is something wrong with the
software? Explain your answer.
11.35 Banks charge different interest rates for different loans. A random sample
of 2229 loans made for the purchase of new automobiles was studied to identify
variables that explain the interest rate charged. All of these loans were made directly
by the bank. A multiple regression was run with interest rate as the response variable
and 13 explanatory variables.
(a) The F statistic reported is 71.34. State the null and alternative hypotheses for
this statistic. Give the degrees of freedom and the P -value for this test. What do
Exercises 273
you conclude?
(b) The value of R2 is 0.297. What percent of the variation in interest rates is
explained by the 13 explanatory variables?
(c) The researchers report a t statistic for each of the regression coefficients. State
the null and alternative hypotheses tested by each of these statistics. What are the
degrees of freedom for these t statistics? What values of t will lead to rejection of
the null hypothesis at the 5% level?
(d) The following table gives the explanatory variables and the t statistics (these
are given without the sign, assuming that all tests are two-sided) for the regression
coefficients. Which of the explanatory variables are significantly different from zero
in this model?
Variable b t
Intercept 15.47
Loan size (in dollars) −0.0015 10.30
Length of loan (in months) −0.906 4.20
Percent down payment −0.522 8.35
Cosigner (0 = no, 1 = yes) −0.009 3.02
Unsecured loan (0 = no, 1 = yes) 0.034 2.19
Total payments (borrower’s monthly installment debt) 0.100 1.37
Total income (borrower’s total monthly income) −0.170 2.37
Bad credit report (0 = no, 1 = yes) 0.012 1.99
Young borrower (0 = older than 25, 1 = 25 or younger) 0.027 2.85
Male borrower (0 = female, 1 = male) −0.001 0.89
Married (0 = no, 1 = yes) −0.023 1.91
Own home (0 = no, 1 = yes) −0.011 2.73
Years at current address −0.124 4.21
(e) The signs of many of these coefficients are what we might expect before looking
at the data. For example, the negative coefficient of loan size means that larger
loans get a smaller interest rate. This is very reasonable. Examine the signs of each
of the statistically significant coefficients and give a short explanation of what they
mean.
11.36 Refer to the previous exercise. The researchers also looked at loans made in-
directly, that is, through an auto dealer. They studied 5664 indirect loans. Multiple
regression was used to predict the interest rate using the same set of explanatory
variables.
(a) The F statistic reported is 27.97. State the null and alternative hypotheses for
this statistic. Give the degrees of freedom and the P -value for this test. What do
you conclude?
(b) The value of R2 is 0.141. What percent of the variation in interest rates is
explained by the 13 explanatory variables? Compare this value with the percent
explained for direct loans in the previous exercise.
(c) The researchers report a t statistic for each of the regression coefficients. State
the null and alternative hypotheses tested by each of these statistics. What are the
degrees of freedom for these t statistics? What values of t will lead to rejection of
the null hypothesis at the 5% level?
(d) The following table gives the explanatory variables and the t statistics (these
are given without the sign, assuming that all tests are two-sided) for the regression
coefficients. Which of the explanatory variables are significantly different from zero
in this model?
Variable b t
Intercept 15.89
Loan size (in dollars) −0.0029 17.40
Length of loan (in months) −1.098 5.63
Percent down payment −0.308 4.92
Cosigner (0 = no, 1 = yes) −0.001 1.41
Unsecured loan (0 = no, 1 = yes) 0.028 2.83
Total payments (borrower’s monthly installment debt) −0.513 1.37
Total income (borrower’s total monthly income) 0.078 0.75
Bad credit report (0 = no, 1 = yes) 0.039 1.76
Young borrower (0 = older than 25, 1 = 25 or younger) −0.036 1.33
Male borrower (0 = female, 1 = male) −0.179 1.03
Married (0 = no, 1 = yes) −0.043 1.61
Own home (0 = no, 1 = yes) −0.047 1.59
Years at current address −0.086 1.73
(e) The signs of many of these coefficients are what we might expect before looking
at the data. For example, the negative coefficient of loan size means that larger
loans get a smaller interest rate. This is very reasonable. Examine the signs of each
of the statistically significant coefficients and give a short explanation of what they
mean.
11.37 Refer to the previous two exercises. The authors conclude that banks take
higher risks with indirect loans because they do not take into account borrower
characteristics when setting the loan rate. Explain how the results of the multiple
regressions lead to this conclusion.
11.38 Multiple regressions are sometimes used in litigation. In the case of Cargill,
Inc. v. Hardin, the prosecution charged that the cash price of wheat was manipulated
in violation of the Commodity Exchange Act. In a statistical study conducted for
this case, a multiple regression model was constructed to predict the price of wheat
using three supply and demand explanatory variables. Data for 14 years were used
to construct the regression equation, and a prediction for the suspect period was
computed from this equation. The value of R2 was 0.989. The predicted value was
reported as $2.136 with a standard error of $0.013. Express the prediction with a
95% interval. (The degrees of freedom were large for this analysis, so use 100 as
the df to determine t∗ .) The actual price for the period in question was $2.13. The
judge in this case decided that the analysis provided evidence that the price was
not artificially depressed, and the opinion was sustained by the court of appeals.
Write a short summary of the results of the analysis that relate to the decision and
explain why you agree or disagree with it.
11.39 The prevalence of childhood obesity in industrialized nations is constantly

rising. Since between 30% and 60% of obese children maintain their obesity into
adulthood, there is great interest in better understanding the reasons for this rising
Exercises 275
trend. In one study, researchers looked at the relationship between a child’s percent
fat mass and several explanatory variables (C. Maffeis et al., “Distribution of food
intake as a risk factor for childhood obesity,” International Journal of Obesity, 24
(2000), pp. 75–80). These were the percent of energy intake at dinner, each parent’s
body mass index (BMI), an index of energy intake validity (EI/BMR), and gender.
The following table summarizes the results of the multiple regression analysis:
b s(b)
Intercept 5.13 3.03
Sex (M=0, F=1) 4.69 0.51
Dinner (%) 0.08 0.02
EI/predicted BMR -1.90 0.65
Mother’s BMI (kg/m2) 0.23 0.07
Father’s BMI (kg/m2) 0.27 0.09
In addition, it is reported that R = 0.44 and F (5, 524) = 25.16.

(a) How many children were used in this study?
(b) What percent of the variation in percent fat mass is explained by these explana-
tory variables?
(c) Interpret the sign of each of the regression coefficients given in the table. For
EI/predicted BMR, data values ranged between 1.4 and 2.8 with a low value asso-
ciated with underreporting of energy intake.
(d) Construct a 95% confidence interval for the difference in predicted percent fat
mass when energy intake at dinner differs by 5% (assume all other variables are the
same).
11.40 What factors predict substance abuse among high school students? One
study designed to answer this question collected data from 89 high school seniors
in a suburban Florida high school (Miguel A. Diego et al., “Academic performance,
popularity, and depression predict adolescent substance abuse,” Adolescence, 38
(2003), pp. 35–42). One of the response variables was marijuana use, which was
rated on a four-point scale. A multiple regression analysis used grade point average
(GPA), popularity, and a depression score to predict marijuana use. The results
were reported in a table similar to this:
b t P
GPA −0.597 4.55 < 0.001
Popularity 0.340 2.69 < 0.01
Depression 0.030 2.69 < 0.01
A footnote to the table gives R2 = 0.34, F (3, 85) = 14.83, and P < 0.001.
(a) State the null and alternative hypotheses that are tested by each of the t statis-
tics. Give the results of these significance tests.
(b) Interpret the sign of each of the regression coefficients given in the table.
(c) In the expression F (3, 85), what do the numbers 3 and 85 represent?
(d) State the null and alternative hypotheses that are tested by the F statistic.
What is the conclusion?
(e) Each of the variables in this analysis was measured by having the students com-
plete a questionnaire. Discuss how this might affect the results.
(f) How well do you think that these results can be applied to other populations of
high school students?
11.41 Refer to the previous exercise. The researchers also studied cigarette use,
alcohol use, and cocaine use. Here is a summary of the results for the individual
regression coefficients:
b t P
GPA −0.340 2.16 < 0.05
Cigarette Popularity 0.338 2.24 < 0.05
Depression 0.034 2.60 < 0.05
GPA −0.321 3.83 < 0.001
Alcohol Popularity 0.185 2.29 < 0.05
Depression 0.015 2.19 < 0.05
GPA −0.583 5.99 < 0.001
Cocaine Popularity −1.90 2.25 < 0.05
Depression 0.002 0.27 Not sig.
And here are other relevant results:
Response variable R2 F P
Cigarette 0.18 6.38 < 0.001
Alcohol 0.27 10.37 < 0.001
Cocaine 0.38 12.21 < 0.001
Using the questions given in the previous exercise, summarize the results for each
of these response variables. Then write a short essay comparing the results for the
four different response variables.
11.42 A study designed to determine how willing consumers are to pay a premium
for non-biotech breakfast cereals (cereals that do not include gene-altered ingredi-
ents) included both U.S. and U.K. subjects (Wanki Moon and Siva K. Balasubrama-
nian, “Willingness to pay for non-biotech foods in the U.S. and the U.K.,” Journal
of Consumer Affairs, 37 (2003), pp. 317–339). The response variable was a mea-
sure of how much extra they would be willing to pay, and the explanatory variables
included items related to perceived risks and benefits, demographic variables, and
country. Country was coded as 0 for the U.K. subjects and 1 for the U.S. subjects.
The parameter estimate for country was reported as −0.2304 with t = −4.196. The
total number of subjects was 1810.
(a) Interpret the regression coefficient. Are subjects in the U.S. more willing to pay
extra for non-biotech breakfast cereals than U.K. subjects, or are they less willing?
(b) Use the t statistic to find a bound on the P -value. Explain the hypothesis tested
by this statistic and summarize the result of the significance test.
(c) The U.S. data were collected using questionnaires that were sent to a nation-
ally representative sample of 5200 households enrolled in the National Dairy Panel
(NDP) Group. The response rate was 58%. The same questionnaire was used for
an online survey of the 9000 U.K. customers enrolled in another NDP Group. The
response rate was 28.5%. Several of the items used in the analysis included “Don’t
know” as a possible response. Respondents choosing this option were excluded from
the analysis. Discuss the implications of these considerations on the results.
Exercises 277
11.43 Although the benefits of physical exercise are well known, most people do not
exercise and many who start exercise programs drop out after a short time. A study
designed to determine factors associated with exercise enjoyment collected data from
282 female volunteers who were participants in not-for-credit aerobic dance classes
at two university centers (Steven R. Wininger and David Pargman, “Assessment of
factors associated with exercise enjoyment,” Journal of Music Therapy, 40 (2003),
pp. 57–73). Exercise enjoyment was the response variable, with a possible range of
18 to 136. Three explanatory variables were analyzed: satisfaction with the music
used (range 4 to 28), satisfaction with the instructor (range 6 to 42), and identity,
a variable that measured the extent to which the subject viewed herself as an exer-
ciser. A table of correlations among the four variables was given, and the text noted
that all were significant with P < 0.01. The coefficients for music (1.02), instructor
(0.96), and identity (0.30) were given in another table, where it was noted that
R2 = 0.33.
(a) Can you give the fitted regression equation? If your answer is Yes, write the
equation; if No, explain what additional information you would need.
(b) Does the fact that all of the correlations between the four variables are significant
at P < 0.01 tell us that the regression coefficients for each of the three explanatory
variables will be statistically significant? Explain your answer.
(c) The statistic for testing the null hypothesis that the population regression co-
efficients for the three explanatory variables are all zero is F = 45.64. Give the
degrees of freedom for this statistic, and carry out the significance test. What do
you conclude?
(d) What proportion of the variation in exercise enjoyment is explained by music,
instructor, and identity?
(e) The authors of the study note that males were not included because there were
too few of them in these classes. Do you think that these results would apply to
males? Explain why or why not.
11.44 Labels providing nutrition facts give consumers information about the nutri-
tional value of food products that they buy. A study of these labels collected data
from 152 consumers who were sent information about a frozen chicken dinner. Each
subject was asked to give an overall product nutrition score and also evaluated each
of 10 nutrients on a 9-point scale, with higher values indicating that the product
has a healthy value for the given nutrient. Composite scores for favorable nutrients
(such as protein and fiber) and unfavorable nutrients (such as fat and sodium) were
used in a multiple regression to predict the overall product nutrition score (Scot
Burton et al., “Implications of accurate usage of nutrition facts panel information
for food product evaluations and purchase intentions,” Journal of the Academy of
Marketing Science, 27 (1999), pp. 470–480). The following was reported in a table:
Explanatory variables b se t Model F R2

33.7** 0.31
Unfavorable nutrients 0.82 0.12 6.8**
Favorable nutrients 0.57 0.10 5.5**
Constant 3.33 0.13 26.1**
**p < 0.01
(a) What is the equation of the least-squares line?

(b) Give the null and alternative hypotheses associated with the entry labeled
“Model F ” and interpret this result.
(c) The column labeled “t” contains three entries. Explain what each of these means.
(d) What are the degrees of freedom associated with the t statistics that you ex-
plained in part (c)?
11.45 The product used in the previous exercise was described by the researchers
as a poor-nutrition product. The label information for this product had high values
for unfavorable nutrients such as fat and low values for favorable nutrients such as
fiber. The researchers who conducted this study collected a parallel set of data from
subjects who were provided label information for a good-nutrition product. This
label had low values for the unfavorable nutrients and high values for the favorable
ones. The same type of multiple regression model was run for the 162 consumers
who participated in this part of the study. Here are the regression results:
Explanatory variables b se t Model F R2

44.0** 0.36
Unfavorable nutrients 0.86 0.12 6.9**
Favorable nutrients 0.66 0.10 6.9**
Constant 3.96 0.12 32.8**
**p < 0.01
For this analysis, answer the questions in parts (a) to (d) of the previous exercise.
11.46 Refer to the previous two exercises. When the researchers planned these stud-
ies, they expected both unfavorable nutrients and favorable nutrients to be positively
associated with the overall product nutrition score. They also expected the unfa-
vorable nutrients to have a stronger effect. Examine the regression coefficients and
the associated t statistics for the two regression models. Then, use this information
to discuss how well the researchers’ expectations were fulfilled.
The following five exercises use the data given in the next exercise.
11.47 Online stock trading has increased dramatically during the past several years.
An article discussing this new method of investing provided data on the major Inter-
net stock brokerages who provide this service (Alan Levinsohn, “Online brokerage,
the new core account?” ABA Banking Journal, September 1999, pp. 34–42). Fol-
lowing are some data for the top 10 Internet brokerages. The variables are Mshare,
the market share of the firm; Accts, the number of Internet accounts in thousands;
and Assets, the total assets in billions of dollars. These firms are not a random
sample from any population but we will use multiple regression methods to develop
statistical models that relate assets to the other two variables.
Exercises 279
ID Broker Mshare Accts Assets

1 Charles Schwab 27.5 2500 219.0
2 E*Trade 12.9 909 21.1
3 TD Waterhouse 11.6 615 38.8
4 Datek 10.0 205 5.5
5 Fidelity 9.3 2300 160.0
6 Ameritrade 8.4 428 19.5
7 DLJ Direct 3.6 590 11.2
8 Discover 2.8 134 5.9
9 Suretrade 2.2 130 1.3
10 National Discount Brokers 1.3 125 6.8
(a) Plot assets versus accounts and describe the relationship.
(b) Perform a simple linear regression to predict assets from the number of accounts.
Give the least-squares line and the results of the hypothesis test for the slope.
(c) Obtain the residuals from part (b) and plot them versus accounts. Describe the
plot. What do you conclude?
(d) Construct a new variable that is the square of the number of accounts. Re-
run the regression analysis with accounts and the square as explanatory variables.
Summarize the results.
11.48 In the multiple regression you performed in the previous exercise, the P -value
for the number of accounts was 0.8531, while the P -value for the square was 0.0070.
Unless we have a strong theoretical reason for considering a model with a quadratic
term and no linear term, we prefer not to do this. One problem with these two
explanatory variables is that they are highly correlated. Here is a way to construct
a version of the quadratic term that is less correlated with the linear term. We first
find the mean for accounts, and then we subtract this value from accounts before
squaring. The mean is 793.6, so the new quadratic explanatory variable will be
(Accts − 793.6)2. Run the multiple regression to predict assets using accounts and
the new quadratic term. Compare these results with what you found in the previous
exercise.
11.49 To one person, the plot of assets versus the number of accounts indicates that
the relationship is curved. Another person might see this as a linear relationship
with two outliers. Identify the two outliers and rerun the linear regression and the
multiple regression with the linear and quadratic terms. Summarize your results.
11.50 Sometimes we attempt to model curved relationships by transforming vari-

ables. Take the logarithm of assets and the logarithm of the number of accounts.
Does the relationship between the logs appear to be approximately linear? Analyze
the data and provide a summary of your results. Be sure to include plots along with
the results of your statistical inference.
11.51 Recall that the relationship between an explanatory variable and a response
variable can depend on what other explanatory variables are included in the model.
(a) Use a simple linear regression to predict assets using the number of accounts.
Give the regression equation and the results of the significance test for the regression
coefficient.
(b) Do the same using market share to predict assets.

(c) Run a multiple regression using both the number of accounts and market share
to predict assets. Give the multiple regression equation and the results of the sig-
nificance tests for the two regression coefficients.
(d) Compare the results in parts (a), (b), and (c). If you had to choose one of these
three models, which one do you prefer? Give an explanation for your answer.
CHAPTER 12
12.1 For each of the following situations, identify the response variable and the
populations to be compared, and give I, the ni , and N .
(a) To compare four varieties of tomato plants, 12 plants of each variety are grown
and the yield in pounds of tomatoes is recorded.
(b) A marketing experiment compares five different types of packaging for a laundry
detergent. Each package is shown to 40 different potential consumers, who rate the
attractiveness of the product on a 1 to 10 scale.
(c) To compare the effectiveness of three different weight-loss programs, 20 people
are randomly assigned to each. At the end of the program, the weight loss for each
of the participants is recorded.
(a) In a study on smoking, subjects are classified as nonsmokers, moderate smokers,
or heavy smokers. A sample of size 100 is drawn from each group. Each person is
asked to report the number of hours of sleep he or she gets on a typical night.
(b) The strength of concrete depends upon the formula used to prepare it. One study
compared four different mixtures. Five batches of each mixture were prepared, and
the strength of the concrete made from each batch was measured.
(c) Which of three methods of teaching sign language is most effective? Twenty
students are randomly assigned to each of the methods, and their scores on a final
exam are recorded.
12.3 How do nematodes (microscopic worms) affect plant growth? A botanist pre-
pares 16 identical planting pots and then introduces different numbers of nematodes
into the pots. A tomato seedling is transplanted into each plot. Here are data on
the increase in height of the seedlings (in centimeters) 16 days after planting (data
provided by Matthew Moore):
Nematodes Seedling growth

0 10.8 9.1 13.5 9.2
1,000 11.1 11.1 8.2 11.3
5,000 5.4 4.6 7.4 5.0
10,000 5.8 5.3 3.2 7.5
(a) Make a table of means and standard deviations for the four treatments, and plot
the means. What does the plot of the means show?
(b) State H0 and Ha for an ANOVA on these data, and explain in words what
ANOVA tests in this setting.
(c) Using computer software, run the ANOVA. What are the F statistic and its
P -value? Give the values of sp and R2 . Report your conclusion.

(a) Define the contrast that compares the 0 treatment (the control group) with the
281
average of the other three.

(b) State H0 and Ha for using this contrast to test whether or not the presence of
nematodes causes decreased growth in tomato seedlings.
(c) Perform the significance test and give the P -value. Do you reject H0 ?
(d) Define the contrast that compares the 0 treatment with the treatment with
10,000 nematodes. This contrast is a measure of the decrease in growth due to
having a very large nematode infestation. Give a 95% confidence interval for this
decrease in growth.
12.5 In large classes instructors sometimes use different forms of an examination.

When average scores for the different forms are calculated, students who received
the form with the lowest average score may complain that their examination was
more difficult than the others. Analysis of variance can help determine whether
the variation in mean scores is larger than would be expected by chance. One such
class used three forms. Summary statistics were as follows (data provided by Peter
Georgeoff of the Purdue University Department of Educational Studies):
Form n x s Min. Q1 Median Q3 Max.
1 79 31.78 4.45 18 29 32 35 42
2 81 32.88 4.40 20 30 33 36 42
3 81 34.47 4.29 24 32 35 38 46
Here is the SAS output for a one-way ANOVA run on the exam scores:
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 292.01871 146.00936 7.61 0.0006
Error 238 4566.28004 19.18605
Corrected Total 240 4858.29876
R-Square C.V. Root MSE SCORE Mean

0.060107 13.25164 4.3802 33.054
Bonferroni (Dunn) T tests for variable: SCORE
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey’s for
all pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 238 MSE= 19.18605

Critical value of T= 2.41102
Comparisons significant at the 0.05 level are indicated by ’***’.
Simultaneous Simultaneous
Lower Difference Upper
FORM Confidence Between Confidence
Comparison Limit Means Limit
Exercises 283
3 - 2 -0.0669 1.5926 3.2521

3 - 1 1.0144 2.6843 4.3543 ***
2 - 3 -3.2521 -1.5926 0.0669

2 - 1 -0.5782 1.0917 2.7617
1 - 3 -4.3543 -2.6843 -1.0144 ***

1 - 2 -2.7617 -1.0917 0.5782
(a) Compare the distributions of exam scores for the three forms with side-by-side
boxplots. Give a short summary of the information contained in these plots.
(b) Summarize and interpret the results of the ANOVA, including the multiple
comparisons procedure.
12.6 The presence of lead in the soil of forests is an important ecological concern.
One source of lead contamination is the exhaust from automobiles. In recent years
this source has been greatly reduced by the elimination of lead from gasoline. Can
the effects be seen in our forests? The Hubbard Brook Experimental Forest in West
Thornton, New Hampshire, is the site of an ongoing study of the forest floor. Lead
measurements of samples taken from this forest are available for several years. The
variable of interest is lead concentration recorded as milligrams per square meter.
Because the data are strongly skewed to the right, logarithms of the concentrations
were analyzed. Here are some summary statistics for 5 years (data provided by Tom
Siccama of the Yale University School of Forestry and Environmental Studies):
Year n x s Min. Q1 Median Q3 Max.

76 59 6.80 .58 5.74 6.33 6.73 7.32 8.05
77 58 6.75 .68 3.95 6.39 6.80 7.23 8.10
78 58 6.76 .50 5.01 6.50 6.78 7.10 7.66
82 68 6.50 .55 5.15 6.11 6.53 6.83 7.82
87 70 6.40 .68 4.38 6.09 6.46 6.85 8.15
Here is the SAS output for a one-way ANOVA run on the logs of the lead
concentrations:
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 8.4437799 2.1109450 5.75 0.0002
Error 308 113.1440666 0.3673509
Corrected Total 312 121.5878465
R-Square C.V. Root MSE LLEAD Mean

0.069446 9.143762 0.6061 6.6285
Bonferroni (Dunn) T tests for variable: LLEAD
NOTE: This test controls the type I experimentwise error rate but
generally has a higher type II error rate than Tukey’s for
all pairwise comparisons.
Alpha= 0.05 Confidence= 0.95 df= 308 MSE= 0.367351

Critical value of T= 2.82740
Comparisons significant at the 0.05 level are indicated by ’***’.
Simultaneous Simultaneous
Lower Difference Upper
YEAR Confidence Between Confidence
Comparison Limit Means Limit
76 - 78 -0.2745 0.0424 0.3592

76 - 77 -0.2687 0.0482 0.3650
76 - 82 -0.0046 0.3003 0.6052
76 - 87 0.0995 0.4024 0.7052 ***
78 - 76 -0.3592 -0.0424 0.2745

78 - 77 -0.3124 0.0058 0.3240
78 - 82 -0.0484 0.2579 0.5642
78 - 87 0.0557 0.3600 0.6643 ***
77 - 76 -0.3650 -0.0482 0.2687

77 - 78 -0.3240 -0.0058 0.3124
77 - 82 -0.0542 0.2521 0.5584
77 - 87 0.0499 0.3542 0.6585 ***
82 - 76 -0.6052 -0.3003 0.0046

82 - 78 -0.5642 -0.2579 0.0484
82 - 77 -0.5584 -0.2521 0.0542
82 - 87 -0.1897 0.1021 0.3938
87 - 76 -0.7052 -0.4024 -0.0995 ***

87 - 78 -0.6643 -0.3600 -0.0557 ***
87 - 77 -0.6585 -0.3542 -0.0499 ***
87 - 82 -0.3938 -0.1021 0.1897
(a) Display the data with side-by-side boxplots. Describe the major features of the
data.
(b) Summarize the ANOVA results. Do the data suggest that the (log) concentration
of lead in the Hubbard Forest floor is decreasing?
12.7 A randomized comparative experiment compares three programs designed to

help people lose weight. There are 20 subjects in each program. The sample stan-
dard deviations for the amount of weight lost (in pounds) are 5.2, 8.9, and 10.1.
Can you use the assumption of equal standard deviations to analyze these data?
Compute the pooled variance and find sp .
Exercises 285
12.8 A study of physical fitness collected data on the weight (in kilograms) of men
in four different age groups. The sample sizes for the groups were 92, 34, 35, and
24. The sample standard deviations for the groups were 12.2, 10.4, 9.2, and 11.7.
Can you use the assumption of equal standard deviations to analyze these data?
Compute the pooled variance and find sp .
12.9 For each part of Exercise 12.1, outline the ANOVA table, giving the sources
of variation and the degrees of freedom. (Do not compute the numerical values for
the sums of squares and mean squares.)
12.10 For each part of Exercise 12.2, outline the ANOVA table, giving the sources
of variation and the degrees of freedom. (Do not compute the numerical values for
the sums of squares and mean squares.)
12.11 Return to the nematode experiment described in Exercise 12.3. Suppose

that when entering the data into the computer, you accidentally entered the first
observation as 108 rather than 10.8.
(a) Run the ANOVA with the incorrect observation. Summarize the results.
(b) Compare this run with the results obtained with the correct data set. What
does this illustrate about the effect of outliers in an ANOVA?
(c) Compute a table of means and standard deviations for each of the four treatments
using the incorrect data. How would this table have helped you to detect the
incorrect observation?
12.12 With small numbers of observations in each group, it can be difficult to detect
deviations from Normality and violations of the equal standard deviations assump-
tion for ANOVA. Return to the nematode experiment described in Exercise 12.3.
The log transformation is often used for variables such as the growth of plants.
In many cases this will tend to make the standard deviations more similar across
groups and to make the data within each group look more Normal. Rerun the
ANOVA using the logarithms of the recorded values. Answer the questions given
in Exercise 12.3. Compare these results with those obtained by analyzing the raw
data.
12.13 You are planning a study of the SAT mathematics scores of four groups of
students. From a previous study, we found the pooled standard deviation to be
82.5. Since the power of the F test decreases as the standard deviation increases,
use σ = 90 for the calculations in this exercise. This choice will lead to sample
sizes that are perhaps a little larger than we need but will prevent us from choosing
sample sizes that are too small to detect the effects of interest. You would like
to conclude that the population means are different when µ1 = 620, µ2 = 600,
µ3 = 580, and µ4 = 560.
(a) Pick several values for n (the number of students that you will select from each
group) and calculate the power of the ANOVA F test for each of your choices.
(b) Plot the power versus the sample size. Describe the general shape of the plot.
(c) What choice of n would you choose for your study? Give reasons for your answer.
12.14 Refer to the previous exercise. Repeat all parts for the alternative µ1 = 610,
µ2 = 600, µ3 = 590, and µ4 = 580.
(a) A company wants to compare three different training programs for its new
employees. In a one-month period there are 90 new hires. One-third of these are
randomly assigned to each of the three training programs. At the end of the program
the employees are asked to rate the effectiveness of the program on a 7-point scale.
(b) A marketing experiment compares six different types of packaging for computer
disks. Each package is shown to 50 different potential consumers, who rate the
attractiveness of the product on a 1 to 10 scale.
(c) Four different new formulations for a hand lotion have been produced by your
research and development group, and you want to decide which of these, if any, to
market. Samples of the first new lotion are sent to 100 randomly selected customers
who use your regular product. The same procedure is followed for each of the other
three new lotions. You ask each customer to compare the new lotion sent to them
with the regular product by rating it on a 7-point scale. The middle point of the
scale corresponds to no preference, while higher values indicate that the new product
is preferred and lower values indicate that the regular product is better.
12.16 Refer to the previous exercise. For each situation, give the following:
(a) Degrees of freedom for the model, for error, and for the total.
(b) Null and alternative hypotheses.
(c) Numerator and denominator degrees of freedom for the F statistic.
(a) A company wants to compare three different water treatment devices that can
be attached to a kitchen faucet. From a list of potential customers, they select 225
households who will receive free samples. One-third of the households will receive
each of the devices. The household is asked to rate the likelihood that they would
buy this kind of device on a 5-point scale.
(b) The strength of concrete depends upon the formula used to prepare it. One
study compared five different mixtures. Six batches of each mixture were prepared,
and the strength of the concrete made from each batch was measured.
(c) Which of three methods of teaching statistics is most effective? Twenty students
are randomly assigned to each of the methods, and their scores on a final exam are
recorded.
12.18 Refer to the previous exercise. For each situation, give the following:
12.19 An experiment was run to compare three groups. The sample sizes were 10,
12, and 14, and the corresponding estimated standard deviations were 18, 24, and
20.
(a) Is it reasonable to use the assumption of equal standard deviations when we
analyze these data?
(b) Give the values of the variances for the three groups.
Exercises 287
(c) Find the pooled variance.

(d) What is the value of the pooled standard deviation?
12.20 An experiment was run to compare four groups. The sample sizes were 20,
220, 18, and 15, and the corresponding estimated standard deviations were 62, 40,
52, and 48.
analyze these data?
(b) Give the values of the variances for the four groups.
(e) Explain why your answer in part (c) is much closer to the standard deviation
for the second group than to any of the other standard deviations.
12.21 For each of the following situations find the degrees of freedom for the F
statistic and then use Table E to approximate the P -value or use computer software
to obtain an exact value.
(a) Three groups are being compared, with 8 observations per group. The value of
the F statistic is 5.82.
(b) Six groups are being compared, with 11 observations per group. The value of
12.22 For each of the following situations find the degrees of freedom for the F
statistic and then use Table E to approximate the P -value or use computer software
to obtain an exact value.
(a) Five groups are being compared, with 13 observations per group. The value of
(b) Ten groups are being compared, with 4 observations per group. The value of
12.23 The presence of harmful insects in farm fields is detected by erecting boards
covered with a sticky material and then examining the insects trapped on the board.
To investigate which colors are most attractive to cereal leaf beetles, researchers
placed six boards of each of four colors in a field of oats in July. (M. C. Wilson and
R. E. Shade, “Relative attractiveness of various luminescent colors to the cereal leaf
beetle and the meadow spittlebug,” Journal of Economic Entomology, 60 (1967),
pp. 578–580.) The following table gives data on the number of cereal leaf beetles
trapped:
Color Insects trapped

Lemon yellow 45 59 48 46 38 47
White 21 12 14 17 13 17
Green 37 32 15 25 39 41
Blue 16 11 20 21 14 7
(a) Make a table of means and standard deviations for the four colors, and plot the
means.
(b) State H0 and Ha for an ANOVA on these data, and explain in words what
ANOVA tests in this setting.
(c) Using computer software, run the ANOVA. What are the F statistic and its
P -value? Give the values of sp and R2 . What do you conclude?
12.24 Return to the previous exercise. For the Bonferroni procedure with α = 0.05,
the value of t∗∗ is 2.61. Use this multiple comparisons procedure to decide which
pairs of colors are significantly different. Summarize your results. Which color would
you recommend for attracting cereal leaf beetles?
12.25 A study of the effects of exercise on physiological and psychological variables

compared four groups of male subjects. The treatment group (T) consisted of 10
participants in an exercise program. A control group (C) of 5 subjects volunteered
for the program but were unable to attend for various reasons. Subjects in the
other two groups were selected to be similar to those in the first two groups in age
and other characteristics. These were 11 joggers (J) and 10 sedentary people (S)
who did not regularly exercise. (Data provided by Dennis Lobstein, from his PhD
dissertation, “A multivariate study of exercise training effects on beta-endorphin and
emotionality in psychologically normal, medically healthy men,” Purdue University,
1983.) One of the variables measured at the end of the program was a physical
fitness score. Part of the ANOVA table used to analyze these data is given below:
Degrees
Source of freedom Sum of squares Mean square F
Groups 3 104,855.87
Error 32 70,500.59
Total
(a) Fill in the missing entries in the ANOVA table.

(b) State H0 and Ha for this experiment.
(c) What is the distribution of the F statistic under the assumption that H0 is
true? Using Table E, give an approximate P -value for the ANOVA test. Write a
brief conclusion.
(d) What is s2p , the estimate of the within-group variance? What is sp ?
12.26 Another variable measured in the experiment described in the previous ex-
ercise was a depression score. Higher values of this score indicate more depression.
Part of the ANOVA table for these data appears below:
Degrees
Source of freedom Sum of squares Mean square F
Groups 3 158.96
Error 32 62.81
Total
(a) Fill in the missing entries in the ANOVA table.

(b) State H0 and Ha for this experiment.
(c) What is the distribution of the F statistic under the assumption that H0 is true?
Using Table E, give an approximate P -value for the ANOVA test. What do you
conclude?
(d) What is s2p , the estimate of the within-group variance? What is sp ?
Exercises 289
12.27 The weight gain of women during pregnancy has an important effect on the
birth weight of their children. If the weight gain is not adequate, the infant is
more likely to be small and will tend to be less healthy. In a study conducted in
three countries, weight gains (in kilograms) of women during the third trimester
of pregnancy were measured. (These data were taken from Collaborative Research
Support Program in Food Intake and Human Function, Management Entity Final
Report, University of California, Berkeley, 1988.) The results are summarized in the
following table:
Country n x s
Egypt 46 3.7 2.5
Kenya 111 3.1 1.8
Mexico 52 2.9 1.8
(a) Find the pooled estimate of the within-country variance s2p . What entry in the
ANOVA table gives this quantity?
(b) The sum of squares for countries (groups) is 17.22. Use this information and
that given above to complete all the entries in an ANOVA table.
(c) State H0 and Ha for this study.
(d) What is the distribution of the F statistic under the assumption that H0 is true?
Use Table E to find an approximate P -value for the significance test. Report your
conclusion.
(e) Calculate R2, the coefficient of determination.
12.28 The previous exercise gives data on the weight gains of pregnant women
in Egypt, Kenya, and Mexico. Computer software gives the critical value for the
Bonferroni multiple comparisons procedure with α = 0.05 as t∗∗ = 2.41. Explain
in plain language what α = 0.05 means in the Bonferroni procedure. Use this
procedure to compare the mean weight gains for the three countries. Summarize
your conclusions.
12.29 In another part of the study described in the previous exercise, measurements
of food intake in kilocalories were taken on many individuals several times during the
period of a year. From these data, average daily food intake values were computed
for each individual. The results for toddlers aged 18 to 30 months are summarized
in the following table:
Country n x s
Egypt 88 1217 327
Kenya 91 844 184
Mexico 54 1119 285
(a) Find the pooled estimate of the within-country variance s2p . What entry in the
ANOVA table gives this quantity?
(b) The sum of squares for countries (groups) is 6,572,551. Use this information and
that given above to complete all the entries in an ANOVA table.
(c) State H0 and Ha for this study.
(d) What is the distribution of the F statistic under the assumption that H0 is true?
Use Table E to find an approximate P -value for the significance test. Report your
conclusion.
(e) Calculate R2, the coefficient of determination.
12.30 The previous exercise gives summary statistics for the food intake values for
toddlers in Egypt, Kenya, and Mexico. Computer software gives the critical value
for the Bonferroni multiple comparisons procedure with α = 0.05 as t∗∗ = 2.41.
Explain in plain language what α = 0.05 means in the Bonferroni procedure. Use
this procedure to compare the toddler food intake means for the three countries.
What do you conclude?
12.31 In the exercise program study described in Exercise 12.25, the summary
statistics for physical fitness scores are as follows:
Group n x s
Treatment (T) 10 291.91 38.17
Control (C) 5 308.97 32.07
Joggers (J) 11 366.87 41.19
Sedentary (S) 10 226.07 63.53
The researchers wanted to address the following questions for the physical fitness
scores. In these questions “better” means a higher fitness score. (1) Is T better than
C? (2) Is T better than the average of C and S? (3) Is J better than the average of
the other three groups?
(a) For each of the three questions, define an appropriate contrast. Translate the
questions into null and alternative hypotheses about these contrasts.
(b) Test your hypotheses and give approximate P -values. Summarize your conclu-
sions. Do you think that the use of contrasts in this way gives an adequate summary
of the results?
(c) You found that the groups differ significantly in the physical fitness scores. Does
this study allow conclusions about causation—for example, that a sedentary lifestyle
causes people to be less physically fit? Explain your answer.
12.32 Refer to the physical fitness scores for the four groups in the exercise program
study discussed in the previous exercise. Computer software gives the critical value
for the Bonferroni multiple comparisons procedure with α = 0.05 as t∗∗ = 2.81. Use
this procedure to compare the mean fitness scores for the four groups. Summarize
your conclusions.
12.33 Exercise 12.26 gives the ANOVA table for depression scores from the exercise
program study described in Exercise 12.25. Here are the summary statistics for the
depression scores:
Group n x s
Treatment (T) 10 51.90 6.42
Control (C) 5 57.40 10.46
Joggers (J) 11 49.73 6.27
Sedentary (S) 10 58.20 9.49
In planning the experiment, the researchers wanted to address the following ques-
tions for the depression scores. In these questions “better” means a lower depression
Exercises 291
score. (1) Is T better than C? (2) Is T better than the average of C and S? (3) Is J
better than the average of the other three groups?
(a) For each of the three questions, define an appropriate contrast. Translate the
questions into null and alternative hypotheses about these contrasts.
(b) Test your hypotheses and give approximate P -values. Summarize your conclu-
sions. Do you think that the use of contrasts in this way gives an adequate summary
of the results?
(c) You found that the groups differ significantly in the depression scores. Does this
study allow conclusions about causation—for example, that a sedentary lifestyle
causes people to be more depressed? Explain your answer.
12.34 Refer to the depression scores for the four groups in the exercise program
study discussed in the previous exercises. Computer software gives the critical value
for the Bonferroni multiple comparisons procedure with α = 0.05 as t∗∗ = 2.81.
Use this procedure to compare the mean depression scores for the four groups.
Summarize your conclusions.
12.35 You are planning a study of the weight gains of pregnant women during
the third trimester of pregnancy similar to that described in Exercise 12.27. The
standard deviations given in that exercise range from 1.8 to 2.5. To perform power
calculations, assume that the standard deviation is σ = 2.4. You have three groups,
each with n subjects, and you would like to reject the ANOVA H0 when the alter-
native µ1 = 2.6, µ2 = 3.0, and µ3 = 3.4 is true. Use software to make a table of
powers against this alternative for the following numbers of women in each group:
n = 50, 100, 150, 175, and 200. What sample size would you choose for your study?
12.36 Repeat the previous exercise for the alternative µ1 = 2.7, µ2 = 3.1, and
µ3 = 3.5. Why are the results the same?
12.37 Refer to the color attractiveness experiment described in Exercise 12.23.

Suppose that when entering the data into the computer, you accidentally entered
the first observation as 450 rather than 45.
(a) Run the ANOVA with the incorrect observation. Summarize the results.
(b) Compare this run with the results obtained with the correct data set. What
does this illustrate about the effect of outliers in an ANOVA?
(c) Compute a table of means and standard deviations for each of the four treatments
using the incorrect data. How would this table have helped you to detect the
incorrect observation?
12.38 Refer to the color attractiveness experiment described in Exercise 12.25. The
square root transformation is often used for variables that are counts, such as the
number of insects trapped in this example. In many cases data transformed in this
way will conform more closely to the assumptions of Normality and equal standard
deviations. Rerun the ANOVA using the square roots of the original counts of
insects. Answer the questions given in Exercise 12.23. Compare these results with
those obtained by analyzing the raw data.
12.39 For each of the following, explain what is wrong and why.
(a) Use one-way ANOVA when the response variable has only two possible values.
(b) You cannot use one-way ANOVA when there are more than three means to be
compared.
(c) The pooled estimate sp is a parameter of the ANOVA model.
(a) The ANOVA F statistic tests the null hypothesis that the three sample means
are equal.
(b) The mean squares in an ANOVA table will add, that is, MST = MSG + MSE.
(c) Within-group variation is the variation in the data due to the differences in the
sample means.
12.41 A study compared 4 groups with 11 observations per group. An F statistic

of 3.52 was reported.
(a) Give the degrees of freedom for this statistic and the entries from Table E that
correspond to this distribution.
(b) Sketch a picture of this F distribution with the information from the table
included.
(c) Based on the table information, how would you report the P -value?
(d) Can you conclude that all pairs of means are different? Explain your answer.
12.42 For each of the following situations, state how large the F statistic needs to
be for rejection of the null hypothesis at the 0.05 level.
(a) Compare 6 groups with 2 observations per group.
(b) Compare 6 groups with 4 observations per group.
(c) Compare 6 groups with 11 observations per group.
(d) Summarize what you have learned about F distributions from this exercise.
12.43 For each of the following situations, find the F statistic and the degrees of
freedom. Then draw a sketch of the distribution under the null hypothesis and
shade in the portion corresponding to the P -value. State how you would report the
P -value.
(a) Compare 5 groups with 8 observations per group, MSE = 50, and MSG = 57.
(b) Compare 3 groups with 7 observations per group, SSG = 40, and SSE = 90.
12.44 For each of the following situations, draw a picture of the ANOVA model.
Use numerical values for the µi . To sketch the Normal curves, you may want to
review the 68–95–99.7 rule.
(a) µ1 = 10, µ2 = 25, µ3 = 24, and σ = 5.
(b) µ1 = 10, µ2 = 20, µ3 = 30, µ4 = 30.1, and σ = 5.
(c) µ1 = 10, µ2 = 25, µ3 = 24, and σ = 2.
(a) Last semester, an alcohol awareness program was conducted for three groups of
students at an eastern university. Follow-up questionnaires were sent to the partici-
pants two months after each presentation. There were 220 responses from students
in an elementary statistics course, 145 from a health and safety course, and 76 from
a cooperative housing unit. One of the questions was “Did you discuss the presen-
tation with any of your friends?” The answers were rated on a five-point scale with
Exercises 293
1 corresponding to “not at all” and 5 corresponding to “a great deal.”

(b) A study examined the effects of consuming three different varieties of onions
on blood cholesterol levels. Five cross-bred (Large White by Landrace) pigs were
randomly assigned to each treatment.
(c) A researcher for a video game developer researcher wanted to evaluate a proto-
type of a new game. Free copies were given to 25 eighth-graders, 25 third-year high
school students, and 25 second-year college students who were regular video game
players. Two weeks later they were asked to tell how much they liked the game on
a 1 to 10 scale.
(a) The effects of six different treatments designed to make fabrics stronger were
compared. A batch of 120 samples of cloth was available for the experiment, and
an equal number were randomly assigned to each of the treatments. The breaking
strength of each cloth sample was measured.
(b) A waiter designed a study to see the effects of his behaviors on the amount of
tips that he received. For some customers, he would tell a joke; for others, he would
describe two of the food items as being particularly good that night; and for others
he would behave normally. Using a table of random numbers, he assigned equal
numbers of his next 30 customers to his different behaviors.
(c) A supermarket wants to compare the effects of providing free samples of cheddar
cheese on sales. An experiment will be conducted from 5:00 p.m. to 6:00 p.m. for
the next 20 weekdays. On each day, customers will be offered one of the following:
a small cube of cheese pierced by a toothpick, a small slice of cheese on a cracker, a
cracker with no cheese, or nothing.
12.47 Refer to Exercise 12.45. For each situation, give the following:
12.48 Refer to Exercise 12.46. For each situation, give the following:
12.49 Refer to Exercise 12.45. For each situation, discuss the method of obtain-
ing the data and how this would affect the extent to which the results could be
generalized.
12.50 Refer to Exercise 12.46. For each situation, discuss the method of obtain-
ing the data and how this would affect the extent to which the results could be
generalized.
12.51 An experiment was run to compare three groups. The sample sizes were 20,
18, and 15, and the corresponding estimated standard deviations were 220, 190, and
200.
analyze these data? Give a reason for your answer.

(b) Give the values of the variances for the three groups.
12.52 Does bread lose its vitamins when stored? Small loaves of bread were prepared
with flour that was fortified with a fixed amount of vitamins. After baking, the
vitamin C content of two loaves was measured. Another two loaves were baked at
the same time, stored for one day, and then the vitamin C content was measured.
In a similar manner, two loaves were stored for three, five, and seven days before
measurements were taken. The units are milligrams of vitamin C per hundred
grams of flour (mg/100 g) (H. Park et al., “Fortifying bread with each of three
antioxidants,” Cereal Chemistry, 74 (1997), pp. 202–206). Here are the data:
Condition Vitamin C (mg/100 g)

Immediately after baking 47.62 49.79
One day after baking 40.45 43.46
Three days after baking 21.25 22.34
Five days after baking 13.18 11.65
Seven days after baking 8.51 8.13
(a) Give a table with sample size, mean, standard deviation, and standard error for
each condition.
(b) Perform a one-way ANOVA for these data. Be sure to state your hypotheses,
the test statistic with degrees of freedom, and the P -value.
(c) Summarize the data and the means with a plot. Use the plot and the ANOVA
results to write a short summary of your conclusions.
12.53 Refer to the previous exercise. Use the Bonferroni or another multiple-
comparisons procedure to compare the group means. Summarize the results.
12.54 Refer to Exercise 12.52. Measurements of the amounts of vitamin A (beta-

carotene) and vitamin E in each loaf are given below. Use the analysis of variance
method to study the data for each of these vitamins.
Condition Vitamin A (mg/100 g) Vitamin E (mg/100 g)

Immediately after baking 3.36 3.34 94.6 96.0
One day after baking 3.28 3.20 95.7 93.2
Three days after baking 3.26 3.16 97.4 94.3
Five days after baking 3.25 3.36 95.0 97.7
Seven days after baking 3.01 2.92 92.3 95.1

(a) Explain why it is inappropriate to perform a multiple-comparisons analysis for
the vitamin E data.
(b) Perform the Bonferroni or another multiple-comparisons procedure for the vita-
min A data and summarize the results.
Exercises 295
12.56 Refer to Exercises 12.52 to 12.55. Write a report summarizing what hap-
pens to vitamins A, C, and E after bread is baked. Include appropriate statistical
inference results and graphs.
12.57 The air in poultry-processing plants often contains fungus spores. If the
ventilation is inadequate, this can affect the health of the workers. To measure the
presence of spores, air samples are pumped to an agar plate, and “colony-forming
units (CFUs)” are counted after an incubation period. Here are data from the “kill
room” of a plant that slaughters 37,000 turkeys per day, taken at four seasons of
the year. The units are CFUs per cubic meter of air (Michael Wayne Peugh, “Field
investigation of ventilation and air quality in duck and turkey slaughter plants,” MS
thesis, Purdue University, 1996).
Fall Winter Spring Summer

1231 384 2105 3175
1254 104 701 2526
1088 97 842 1090
(a) Examine the data using graphs and descriptive measures. How do airborne
fungus spores vary with the seasons?
(b) Is the effect of season statistically significant?
12.58 Refer to the previous exercise. There is not sufficient information to examine
the distributions in detail, but it is not unreasonable to expect count data such
as these to be skewed. Reanalyze the data after taking logs of the CFU counts.
Summarize your work and compare the results you have found here with what you
obtained in the previous exercise.
12.59 If a supermarket product is offered at a reduced price frequently, do customers

expect the price of the product to be lower in the future? This question was ex-
amined by researchers in a study conducted on students enrolled in an introductory
management course at a large midwestern university. For 10 weeks 160 subjects
received information about the products. The treatment conditions corresponded
to the number of promotions (1, 3, 5, or 7) that were described during this 10-week
period. Students were randomly assigned to four groups (based on M. U. Kalwani
and C. K. Yim, “Consumer price and promotion expectations: an experimental
study,” Journal of Marketing Research, 29 (1992), pp. 90–100). The table below
gives the data.
Price promotion data

Number of
promotions Expected price (dollars)
1 3.78 3.82 4.18 4.46 4.31 4.56 4.36 4.54 3.89 4.13
3.97 4.38 3.98 3.91 4.34 4.24 4.22 4.32 3.96 4.73
3.62 4.27 4.79 4.58 4.46 4.18 4.40 4.36 4.37 4.23
4.06 3.86 4.26 4.33 4.10 3.94 3.97 4.60 4.50 4.00
3 4.12 3.91 3.96 4.22 3.88 4.14 4.17 4.07 4.16 4.12
3.84 4.01 4.42 4.01 3.84 3.95 4.26 3.95 4.30 4.33
4.17 3.97 4.32 3.87 3.91 4.21 3.86 4.14 3.93 4.08
4.07 4.08 3.95 3.92 4.36 4.05 3.96 4.29 3.60 4.11
5 3.32 3.86 4.15 3.65 3.71 3.78 3.93 3.73 3.71 4.10
3.69 3.83 3.58 4.08 3.99 3.72 4.41 4.12 3.73 3.56
3.25 3.76 3.56 3.48 3.47 3.58 3.76 3.57 3.87 3.92
3.39 3.54 3.86 3.77 4.37 3.77 3.81 3.71 3.58 3.69
7 3.45 3.64 3.37 3.27 3.58 4.01 3.67 3.74 3.50 3.60
3.97 3.57 3.50 3.81 3.55 3.08 3.78 3.86 3.29 3.77
3.25 3.07 3.21 3.55 3.23 2.97 3.86 3.14 3.43 3.84
3.65 3.45 3.73 3.12 3.82 3.70 3.46 3.73 3.79 3.94
(a) Make a Normal quantile plot for the data in each of the four treatment groups.
Summarize the information in the plots and draw a conclusion regarding the Nor-
mality of these data.
(b) Summarize the data with a table containing the sample size, mean, standard
deviation, and standard error for each group.
(c) Is the assumption of equal standard deviations reasonable here? Explain why or
why not.
(d) Run the one-way ANOVA. Give the hypotheses tested, the test statistic with
degrees of freedom, and the P -value. Summarize your conclusion.
12.60 Refer to the previous exercise. Use the Bonferroni or another multiple-
comparisons procedure to compare the group means. Summarize the results and
support your conclusions with a graph of the means.
12.61 Recommendations regarding how long infants in developing countries should

be breast-fed are controversial. If the nutritional quality of the breast milk is in-
adequate because the mothers are malnourished, then there is risk of inadequate
nutrition for the infant. On the other hand, the introduction of other foods carries
the risk of infection from contamination. Further complicating the situation is the
fact that companies that produce infant formulas and other foods benefit when these
foods are consumed by large numbers of customers. One question related to this
controversy concerns the amount of energy intake for infants who have other foods
introduced into the diet at different ages. Part of one study compared the energy
intakes, measured in kilocalories per day (kcal/d), for infants who were breast-fed
exclusively for 4, 5, or 6 months (based on a study by J. E. Stuff and B. L. Nichols
reported in Chelsea Lutter, “Recommended length of exclusive breast-feeding, age
of introduction of complementary foods and the weaning dilemma,” World Health
Organization, 1992). Here are the data:
Exercises 297
Breast-fed for: Energy intake (kcal/d)

4 months 499 620 469 485 660 588 675 517 649 209
404 738 628 609 617 704 558 653 548
5 months 490 395 402 177 475 617 616 587 528 518
370 431 518 639 368 538 519 506
6 months 585 647 477 445 485 703 528 465
(a) Make a table giving the sample size, mean, and standard deviation for each
group of infants. Is it reasonable to pool the variances?
(b) Run the analysis of variance. Report the F statistic with its degrees of freedom
and P -value. What do you conclude?

(a) Examine the residuals. Is the Normality assumption reasonable for these data?
(b) Explain why you do not need to use a multiple-comparisons procedure for these
data.
12.63 Refer to Exercise 12.52, where we studied the effects of storage on the vita-
min C content of bread. In this experiment 64 mg of vitamin C per 100 g of flour
was added to the flour that was used to make each loaf.
(a) Convert the vitamin C amounts (mg/100 g) to percents of the amounts origi-
nally in the loaves by dividing the amounts in Exercise 12.52 by 64 and multiplying
by 100. Calculate the transformed means, standard deviations, and standard errors
and summarize them with the sample sizes in a table.
(b) Explain how you could have calculated the table entries directly from the table
you gave in part (a) of Exercise 12.52.
(c) Analyze the percents using analysis of variance. Compare the test statistic, de-
grees of freedom, P -value, and conclusion you obtain here with the corresponding
values that you found in Exercise 12.52.
12.64 Refer to the previous exercise and Exercise 12.54. The flour used to make
the loaves contained 5 mg of vitamin A per 100 g of flour and 100 mg of vitamin E
per 100 g of flour. Summarize the effects of transforming the data to percents for
all three vitamins.
CHAPTER 13
13.1 Each of the following situations is a two-way study design. For each case,
identify the response variable and both factors, and state the number of levels for
each factor (I and J) and the total number of observations (N ).
(a) A study of the productivity of tomato plants compares five varieties of tomatoes
and two types of fertilizer. Four plants of each variety are grown with each type of
fertilizer. The yield in pounds of tomatoes is recorded for each plant.
(b) A marketing experiment compares six different types of packaging for a laundry
detergent. A survey is conducted to determine the attractiveness of the packaging
in six U.S. cities. Each type of packaging is shown to 50 different consumers in each
city, who rate the attractiveness of the product on a 1 to 10 scale.
(c) To compare the effectiveness of four different weight-loss programs, 10 men and
10 women are randomly assigned to each. At the end of the program, the weight
loss for each of the participants is recorded.
13.2 For each part of the previous exercise, outline the ANOVA table, giving the
sources of variation and the degrees of freedom. (Do not compute the numerical
values for the sums of squares and mean squares.)
(a) A study of smoking classifies subjects as nonsmokers, moderate smokers, or
heavy smokers. Samples of 120 men and 120 women are drawn from each group.
Each person reports the number of hours of sleep he or she gets on a typical night.
(b) The strength of concrete depends upon the formula used to prepare it. An
experiment compares four different mixtures. Six specimens of concrete are poured
from each mixture. Two of these specimens are subjected to 0 cycles of freezing and
thawing, two are subjected to 100 cycles, and two specimens are subjected to 500
cycles. The strength of each specimen is then measured.
(c) Three methods for teaching sign language are to be compared. Seven students
in special education and seven students in linguistics are randomly assigned to each
of the methods and the scores on a final exam are recorded.
13.5 A large research project studied the physical properties of wood materials
constructed by bonding together small flakes of wood. Different species of trees
were used, and the flakes were made in different sizes. One of the physical properties
measured was the tension modulus of elasticity in the direction perpendicular to the
alignment of the flakes, in pounds per square inch (psi). Some of the data are given
in the following table. The sizes of the flakes are S1 = 0.015 inches by 2 inches and
298
Exercises 299
S2 = 0.025 inches by 2 inches. (Data provided by Michael Hunt and Bob Lattanzi
of the Purdue University Forestry Department.)
Size of flakes
Species S1 S2
Aspen 308 278
428 398
426 331
Birch 214 534
433 512
231 320
Maple 272 158
376 503
322 220
(a) Compute means and standard deviations for the three observations in each
species-size group. Find the marginal mean for each species and for each size of
flakes. Display the means and marginal means in a table.
(b) Plot the means of the six groups. Put species on the x axis and modulus of
elasticity on the y axis. For each size connect the three points corresponding to
the different species. Describe the patterns you see. Do the species appear to be
different? What about the sizes? Does there appear to be an interaction?
(c) Run a two-way ANOVA on these data. Summarize the results of the significance
tests. What do these results say about the impressions that you described in part
(b) of this exercise?
13.6 Refer to the previous exercise. Another of the physical properties measured
was the strength, in kilopounds per square inch (ksi), in the direction perpendicular
to the alignment of the flakes. Some of the data are given in the following table. The
sizes of the flakes are S1 = 0.015 inches by 2 inches and S2 = 0.025 inches by 2 inches.
Size of flakes
Species S1 S2
Aspen 1296 1472
1997 1441
1686 1051
Birch 903 1422
1246 1376
1355 1238
Maple 1211 1440
1827 1238
1541 748
(a) Compute means and standard deviations for the three observations in each
species-size group. Find the marginal means for the species and for the flake sizes.
Display the means and marginal means in a table.
(b) Plot the means of the six groups. Put species on the x axis and strength on the
y axis. For each size connect the three points corresponding to the different species.
Describe the patterns you see. Do the species appear to be different? What about
the sizes? Does there appear to be an interaction?
(c) Run a two-way ANOVA on these data. Summarize the results of the significance
tests. What do these results say about the impressions that you described in part
(b) of this exercise?
(a) A company wants to compare three different training programs for its new
employees. Each of these programs takes 8 hours to complete. The training can
be given for 8 hours on one day or for 4 hours on two consecutive days. The next
120 employees that the company hires will be the subjects for this study. After the
training is completed, the employees are asked to evaluate the effectiveness of the
program on a 7-point scale.
(b) A marketing experiment compares four different types of packaging for computer
disks. Each type of packaging can be presented in three different colors. Each
combination of package type with a particular color is shown to 40 different potential
customers, who rate the attractiveness of the product on a 1 to 10 scale.
(c) Five different formulations for your hand lotion product have been produced by
your research and marketing group, and you want to decide which of these, if any, to
market. The lotions can be made with three different fragrances. Samples of each
formulation-by-fragrance lotion are sent to 120 randomly selected customers who
use your regular product. You ask each customer to compare the new lotion with
the regular product by rating it on a 7-point scale. The middle point of the scale
corresponds to no preference, while higher values indicate that the new product is
preferred and lower values indicate that the regular product is better.
(a) A study of smoking classifies subjects as nonsmokers, moderate smokers, or
heavy smokers. Samples of 80 men and 80 women are drawn from each group. Each
person reports the number of hours of sleep he or she gets on a typical night.
(b) The strength of concrete depends upon the formula used to prepare it. An
experiment compares six different mixtures. Nine specimens of concrete are poured
from each mixture. Three of these specimens are subjected to 0 cycles of freezing
and thawing, three are subjected to 100 cycles, and three specimens are subjected
to 500 cycles. The strength of each specimen is then measured.
(c) Four methods for teaching sign language are to be compared. Sixteen students in
special education and sixteen students majoring in other areas are the subjects for
the study. Within each group they are randomly assigned to the methods. Scores
on a final exam are compared.
Exercises 301
13.11 A two-way ANOVA model was used to analyze an experiment with three
levels of one factor, four levels of a second factor, and 6 observations per treatment
combination.
(a) For each of the main effects and the interaction, give the degrees of freedom for
the corresponding F statistic.
(b) Using Table E or statistical software, find the value that each of these F statistics
must exceed for the result to be significant at the 5% level.
(c) Answer part (b) for the 1% level.
13.12 A two-way ANOVA model was used to analyze an experiment with two levels
of one factor, three levels of a second factor, and 6 observations per treatment
combination.
(a) For each of the main effects and the interaction, give the degrees of freedom for
the corresponding F statistic.
(b) Using Table E or statistical software, find the value that each of these F statistics
must exceed for the result to be significant at the 5% level.
(c) Answer part (b) for the 1% level.
13.13 In the course of a clinical trial of measures to prevent coronary heart disease,
blood pressure measurements were taken on 12,866 men. Individuals were classified
by age group and race. (W. M. Smith et al., “The multiple risk factor intervention
trial,” in H. M. Perry, Jr., and W. M. Smith (eds.), Mild Hypertension: To Treat or
Not to Treat, New York Academy of Sciences, 1978, pp. 293–308.) The means for
systolic blood pressure are given in the following table:
35–39 40–44 45–49 50–54 55–59

White 131.0 132.3 135.2 139.4 142.0
Nonwhite 132.3 134.2 137.2 141.3 144.1
(a) Plot the group means, with age on the x axis and blood pressure on the y axis.
For each racial group connect the points for the different ages.
(b) Describe the patterns you see. Does there appear to be a difference between the
two racial groups? Does systolic blood pressure appear to vary with age? If so, how
does it vary? Is there an interaction?
(c) Compute the marginal means. Then find the differences between the white
and nonwhite mean blood pressures for each age group. Use this information to
summarize numerically the patterns in the plot.
13.14 The means for diastolic blood pressure recorded in the clinical trial described
in the previous exercise are:
35–39 40–44 45–49 50–54 55–59

White 89.4 90.2 90.9 91.6 91.4
Nonwhite 91.2 93.1 93.3 94.5 93.5
(a) Plot the group means with age on the x axis and blood pressure on the y axis.
For each racial group connect the points for the different ages.
(b) Describe the patterns you see. Does there appear to be a difference between the
two racial groups? Does diastolic blood pressure appear to vary with age? If so,
how does it vary? Is there an interaction between race and age?
(c) Compute the marginal means. Find the differences between the white and non-
white mean blood pressures for each age group. Use this information to summarize
numerically the patterns in the plot.
13.15 The Chapin Social Insight Test measures how well people can appraise others
and predict what they may say or do. A study administered this test to different
groups of people and compared the mean scores. (This exercise is based on results
reported in H. G. Gough, The Chapin Social Insight Test, Consulting Psychologists
Press, 1968.) Some of the results are given in the following table. Means for males
and females who were psychology graduate students (PG) and liberal arts under-
graduates (LU) are presented. The two factors are labeled Gender and Group.
Group
Gender PG LU
Males 27.56 25.34
Females 29.25 24.94
Plot the means and describe the essential features of the data in terms of main
effects and interactions.
13.16 Refer to the previous exercise. Part of the ANOVA table for these data is
given below:
Degrees Sum of Mean

Source of freedom squares square F
A (Gender) 62.40
B (Group) 1,599.03
AB
Error 13,633.29
Total 15,458.52
(a) There were 150 individuals tested in each of the groups. Fill in the missing
values in the ANOVA table.
(b) What is the value of the F statistic to test the null hypothesis that there is
no interaction? What is its distribution when the null hypothesis is true? Using
Table E, find an approximate P -value for this test.
(c) Answer the questions in part (b) for the main effect of Gender and the main
effect of Group.
(d) What is s2p , the within-group variance? What is sp ?
(e) Using what you have learned in this exercise and your answer to Exercise 13.15,
summarize the results of this study.
(a) The FIT part of the model in a two-way ANOVA represents the variation that
Exercises 303
is sometimes called error or residual.

(b) You should reject the null hypothesis that there is no interaction in a two-way
ANOVA when the test statistic is small.
(c) Sums of squares are equal to mean squares divided by degrees of freedom.
(a) The significance tests for the main effects in a two-way ANOVA have a chi-square
distribution when the null hypothesis is true.
(b) You can perform a two-way ANOVA only when the sample sizes are the same
in all cells.
(c) The cell means x̄ij are parameters of the two-way ANOVA model.
13.19 A 2 × 4 ANOVA was run with 5 observations per cell.

(a) Give the degrees of freedom for the F statistic that is used to test for interaction
in this analysis and the entries from Table E that correspond to this distribution.
(b) Sketch a picture of this distribution with the information from the table included.
(c) The calculated value of the F statistic is 2.59. How would you report the P -
value?
(d) Would you expect a plot of the means to look parallel? Explain your answer.
13.20 For each of the following situations, state how large the F statistic needs to
be for rejection of the null hypothesis at the 5% level. Sketch each distribution and
indicate the region where you would reject.
(a) the main effect for the first factor in a 3 × 3 ANOVA with 4 observations per cell
(b) the interaction in a 3 × 3 ANOVA with 4 observations per cell
(c) the interaction in a 2 × 2 ANOVA with 251 observations per cell
13.21 Analysis of data for a 3 × 2 ANOVA with 5 observations per cell gave the F
statistics in the following table:
Effect F
A 1.21
B 3.63
AB 2.04
What can you conclude from the information given?
13.22 Does repetition of an advertising message increase its effectiveness? One

theory suggests that there are two phases in the process. In the first phase, called
“wearin,” negative or unfamiliar views are transformed into positive views. In the
second phase, called “wearout,” the effectiveness of the ad is decreased because
of boredom or other factors. One study designed to investigate this theory exam-
ined two factors. The first was familiarity of the ad, with two levels, familiar and
unfamiliar; the second was repetition, with three levels, 1, 2, and 3 (Margaret C.
Campbell and Kevin Lane Keller, “Brand familiarity and advertising repetition ef-
fects,” Journal of Consumer Research, 30 (2003), pp. 292–304). One of the response
variables collected was attitude toward the ad. This variable was the average of four
items, each measured on a seven-point scale, anchored by bad–good, low quality–
high quality, unappealing–appealing, and unpleasant–pleasant. Here are the means
for attitude:
Repetition
Familiarity 1 2 3
Familiar 4.56 4.73 5.24
Unfamiliar 4.14 5.26 4.41
(a) Make a plot of the means and describe the patterns that you see.
(b) Does the plot suggest that there is an interaction between familiarity and repe-
tition? If your answer is Yes, describe the interaction.
13.23 Refer to the previous exercise. In settings such as this, researchers collect
data for several response variables. For this study, they also constructed variables
that were called attitude toward the brand, total thoughts, support arguments, and
counterarguments. Here are the means:
Attitude to brand Total Support Counter

Repetition Repetition Repetition Repetition
Familiarity 1 2 3 1 2 3 1 2 3 1 2 3
Familiar 4.67 4.65 5.06 1.33 1.93 2.55 0.63 0.67 0.98 0.54 0.70 0.49
Unfamiliar 3.94 4.79 4.26 1.52 3.06 3.17 0.76 1.40 0.64 0.52 0.75 1.14
For each of the four response variables, give a graphical summary of the means.
Use this summary to discuss any interactions that are evident. Write a short report
summarizing the effect of repetition on the response variables measured, using the
data in this exercise and the previous one.
13.24 Refer to the previous exercise. Here are the standard deviations for attitude
toward brand:
Repetition
Familiarity 1 2 3
Familiar 1.16 1.46 1.16
Find the pooled estimate of the standard deviation for these data. Use the rule
for examining standard deviations in ANOVA from Chapter 12 to determine if it is
reasonable to use a pooled standard deviation for the analysis of these data.
13.25 Refer to Exercise 13.23. Here are the standard deviations for total thoughts:
Repetition
Familiarity 1 2 3
Familiar 1.63 1.42 1.52
Find the pooled estimate of the standard deviation for these data. Use the rule
for examining standard deviations in ANOVA from Chapter 12 to determine if it is
reasonable to use a pooled standard deviation for the analysis of these data.
13.26 Refer to Exercises 13.22 and 13.23. The subjects were 94 adult staff members
at a West Coast university. They watched a half-hour local news show from a
Exercises 305
different state that included the ads. The selected ads were judged to be “good”
by some experts and had been shown in regions other than where the study was
conducted. The real names of the products were replaced by either familiar or
unfamiliar brand names by a professional video editor. The ads were pretested and
no one in the pretest sample suggested that the ads were not real. Discuss each of
these facts in terms of how you would interpret the results of this study.
13.27 Refer to Exercises 13.22 and 13.23. The ratings for this study were each
measured on a seven-point scale, anchored by bad–good, low quality–high quality,
unappealing–appealing, and unpleasant–pleasant. The results presented were aver-
aged over three ads for different products: a bank, women’s clothing, and a health
care plan. Write a short report summarizing the Normality assumption for two-way
ANOVA and the extent to which it is reasonable for the analysis of these data.
13.28 One way to repair serious wounds is to insert some material as a scaffold for
the body’s repair cells to use as a template for new tissue. Scaffolds made from
extracellular material (ECM) are particularly promising for this purpose. Because
they are made from biological material, they serve as an effective scaffold and are
then resorbed. Unlike biological material that includes cells, however, they do not
trigger tissue rejection reactions in the body. One study compared 6 types of scaffold
material (S. Badylak et al., “Marrow-derived cells populate scaffolds composed of
xenogeneic extracellular matrix,” Experimental Hematology, 29 (2001), pp. 1310–
1318). Three of these were ECMs and the other three were made of inert materials.
There were three mice used per scaffold type. The response measure was the percent
of glucose phosphated isomerase (Gpi) cells in the region of the wound. A large value
is good, indicating that there are many bone marrow cells sent by the body to repair
the tissue. The experiment included additional groups of rats who received the same
types of scaffold but were measured at different times. Here are the data for 4 weeks
and 8 weeks after the repair:
Gpi (%)
Material 4 weeks 8 weeks
ECM1 55 70 70 60 65 65
ECM2 60 65 65 60 70 60
ECM3 75 70 75 70 80 70
MAT1 20 25 25 15 25 25
MAT2 5 10 5 10 5 5
MAT3 10 15 10 5 15 10
(a) Make a table giving the sample size, mean, and standard deviation for each of
the material-by-time combinations. Is it reasonable to pool the variances? Because
the sample sizes in this experiment are very small, we expect a large amount of
variability in the sample standard deviations. Although they vary more than we
would prefer, we will proceed with the ANOVA.
(b) Make a plot of the means. Describe the main features of the plot.
(c) Run the analysis of variance. Report the F statistics with degrees of freedom and
P -values for each of the main effects and the interaction. What do you conclude?
Write a short paragraph summarizing the results of your analysis.
13.29 Refer to the previous exercise. Here are the data that were collected at 2
weeks, 4 weeks, and 8 weeks:
Gpi (%)
Material 2 weeks 4 weeks 8 weeks
ECM1 70 75 65 55 70 70 60 65 65
ECM2 60 65 70 60 65 65 60 70 60
ECM3 80 60 75 75 70 75 70 80 70
MAT1 50 45 50 20 25 25 15 25 25
MAT2 5 10 15 5 10 5 10 5 5
MAT3 30 25 25 10 15 10 5 15 10
Rerun the analyses that you performed in the previous exercise. How does the ad-
dition of the data for 2 weeks change the conclusions? Write a summary comparing
these results with those in the previous exercise.
13.30 Refer to the previous exercise. Analyze the data for each time period sep-
arately using a one-way ANOVA. Use a multiple-comparisons procedure where
needed. Summarize the results.
13.31 How does the frequency that a supermarket product is promoted at a discount
affect the price that customers expect to pay for the product? Does the percent
reduction also affect this expectation? These questions were examined by researchers
in a study conducted on students enrolled in an introductory management course at
a large midwestern university. For 10 weeks 160 subjects received information about
the products. The treatment conditions corresponded to the number of promotions
(1, 3, 5, or 7) that were described during this 10-week period and the percent that the
product was discounted (10%, 20%, 30%, and 40%). Ten students were randomly
assigned to each of the 4 × 4 = 16 treatments (M. U. Kalwani and C. K. Yim,
“Consumer price and promotion expectations: an experimental study,” Journal of
Marketing Research, 29 (1992), pp. 90–100). The following table gives the data.
Exercises 307
Expected price data

Number of Percent
promotions discount Expected price ($)
1 40 4.10 4.50 4.47 4.42 4.56 4.69 4.42 4.17 4.31 4.59
1 30 3.57 3.77 3.90 4.49 4.00 4.66 4.48 4.64 4.31 4.43
1 20 4.94 4.59 4.58 4.48 4.55 4.53 4.59 4.66 4.73 5.24
1 10 5.19 4.88 4.78 4.89 4.69 4.96 5.00 4.93 5.10 4.78
3 40 4.07 4.13 4.25 4.23 4.57 4.33 4.17 4.47 4.60 4.02
3 30 4.20 3.94 4.20 3.88 4.35 3.99 4.01 4.22 3.70 4.48
3 20 4.88 4.80 4.46 4.73 3.96 4.42 4.30 4.68 4.45 4.56
3 10 4.90 5.15 4.68 4.98 4.66 4.46 4.70 4.37 4.69 4.97
5 40 3.89 4.18 3.82 4.09 3.94 4.41 4.14 4.15 4.06 3.90
5 30 3.90 3.77 3.86 4.10 4.10 3.81 3.97 3.67 4.05 3.67
5 20 4.11 4.35 4.17 4.11 4.02 4.41 4.48 3.76 4.66 4.44
5 10 4.31 4.36 4.75 4.62 3.74 4.34 4.52 4.37 4.40 4.52
7 40 3.56 3.91 4.05 3.91 4.11 3.61 3.72 3.69 3.79 3.45
7 30 3.45 4.06 3.35 3.67 3.74 3.80 3.90 4.08 3.52 4.03
7 20 3.89 4.45 3.80 4.15 4.41 3.75 3.98 4.07 4.21 4.23
7 10 4.04 4.22 4.39 3.89 4.26 4.41 4.39 4.52 3.87 4.70
(a) Summarize the means and standard deviations in a table and plot the means.
Summarize the main features of the plot.
(b) Analyze the data with a two-way ANOVA. Report the results of this analysis.
(c) Using your plot and the ANOVA results, prepare a short report explaining how
the expected price depends on the number of promotions and the percent of the
discount.
13.32 Refer to the previous exercise. Rerun the analysis as a one-way ANOVA with
4 × 4 = 16 treatments. Summarize the results of this analysis. Use a multiple-
comparisons procedure to describe combinations of number of promotions and per-
cent discounts that are similar or different.

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SUPPLEMENTARY

These exercises appeared in the third, fourth, fifth, or sixth edi-

Net assets Year-to-date Largest

defects, 430; respiratory disease, 208; AIDS, 197.

1.14 Three landmarks of baseball achievement are Ty Cobb’s batting average of

during research on ways to improve reading performance (data provided by Mari-

State Doctors State Doctors State Doctors

Salary 15–20 20–25 25–30 30–35 35–40 40–50 50–60 60–80

Year Rate Year Rate Year Rate Year Rate

Year Rate Year Rate Year Rate Year Rate

Year HRs Year HRs Year HRs

Make and Vehicle Transmission Number of City Highway

ADVANI, SURA COMM 397 B

of classes that the applet allows. Sketch the results.

Mean annual temperatures (◦ F) in two California cities

Recruitment Recruitment Recruitment Recruitment

Survival times (days) of guinea pigs

154 109 137 115 152 140 154 178 101

108 140 114 91 180 115 126 92 169 146

Year 1948 1952 1956 1960 1964 1968 1972

Beef hot dogs Meat hot dogs Poultry hot dogs

10, 001 10, 002 10, 003

100, 001 100, 002 100, 003

4,123 4,186 4,324 4,342 4,557 4,884 5,397 6,129

1.72 Outliers in percent of older residents. The stemplot in Exercise 1.42

1.75 Effects of logging in Borneo. “Conservationists have despaired over de-

1.86 Three landmarks of baseball achievement are Ty Cobb’s batting average of

Decade Mean Std. dev.

1.93 The variable Z has a standard Normal distribution.

1.102 Is the distribution of monthly returns on Philip Morris stock approximately

1.112 The variable Z has a standard Normal distribution.

1.123 Changing the mean of a Normal distribution by a moderate amount can

of years is the market down?

1.126 Heights of women. The heights of women aged 20 to 29 are approximately

1.127 Proportion of women with high cholesterol. Too much cholesterol in

Chapter 1 Review Exercises

1.130 Here is a stemplot of the percents of residents aged 25 to 34 in each of the 50

Player Salary Player Salary Player Salary

County Population County Population County Population

QB 235 QB 220 QB 215 QB 228 K 175 K 205

1.141 Distance-learning courses. The 222 students enrolled in distance-learning

Year 1966 1967 1968 1969 1970 1971 1972 1973

1.143 Damage caused by tornadoes. The average damage caused by tornadoes

Alcohol Heart disease Alcohol Heart disease

Mean NAEP Percent Percent Mean NAEP Percent Percent

Flow rate 0.31 0.85 1.26 2.47 3.75

(a) Plot the data. Which is the explanatory variable?

Board color Insects trapped

2.8 In each of the following situations, is it more reasonable to simply explore

Fuel economy (miles per gallon) for 2004 model vehicles

HAV MA HAV MA HAV MA

the form of the relationship makes sense.

2002 2003 2002 2003 2002 2003

2.19 A mutual fund company’s newsletter says, “A well-diversified portfolio includes

Year Fund EAFE Year Fund EAFE

2.24 Coffee prices in dollars or euros. Coffee is currently priced in dollars. If

Speed Fuel used Speed Fuel used

change of units affect your results?