Sociology 592 - Research Statistics I Exam 1 Answer Key September 28, 2001
Sociology 592 - Research Statistics I Exam 1 Answer Key September 28, 2001
Sociology 592 - Research Statistics I Exam 1 Answer Key September 28, 2001
Where appropriate, show your work - partial credit may be given. (On the other hand, don't waste a lot of time on
excess verbiage.) Do not spend too much time on any one problem. It is legitimate (and probably essential) to refer
to results that have previously been proven in class or homework, without re-proving them - for example, you
wouldn't need to prove that P(-1.96 Z 1.96) = .95, since we have already shown that in class. Likewise, you are
free to refer to anything that was demonstrated in the homework or handouts.
1. (5 points each, 20 points total). Indicate whether the following statements are true or false. If you think the
statement is false, indicate how the statement could be corrected. For false statements, do not just say that you could
substitute not equals for equals. For example, the statement P(Z 0) = .7 is false. To make it correct, don't just say
P(Z 0) <> .7, instead say P(Z 0) = .5 or P(Z .525) = .7.
TRUE. Reverse the definitions of success and failure. Then turn to Appendix E, Table
II, and look up N = 17, p = .30, r = 4. Or, if you prefer, compute
N r N r 17 13 4 17!
p q = .7 .3 = .00007848 = 2380 * .00007848 = .1868
r 13 4!13!
B. V(3X + 7) = 9 * V(X) + 49
FALSE. Adding a constant to all cases does not affect the variance. V(3X + 7) = 9 *
V(X).
C. If A and B are mutually exclusive events, then P(A | B) = P(B | A).
2. (10 points each, 30 points total) Answer three of the following. The answers to most of these are fairly
straightforward, so do not spend a great deal of time on any one problem. NOTE: I will give up to 5 points extra
credit for each additional problem you do correctly.
A. A long, hard-fought season is finally about to end. To no ones surprise, the national
championship game features a rematch between the two teams that have dominated the sport all season long: The
Nebraska Cornhuskers and The Notre Dame Fighting Irish (this is Womens College Soccer we are talking about, of
course). The Huskers are eager to avenge their only loss of the season, a 1-0 shutout at Notre Dame on September
25. However, the Irish, fresh from a spectacular 7-0 semifinal win over hated archrival North Carolina, are
determined to claim their second-ever national title.
This is the championship game so it cannot end in a tie; overtime will be played if necessary. Defense will
be a key. There is a 60% chance that Notre Dames stubborn defense will hold Nebraska to 2 goals or less. But, if
Nebraskas high-powered offense does score 3 goals or more, there is a 70% chance the Huskers will win. Nebraska
has a good defense too, so there is a 21% chance that the Huskers will score 2 goals or less and win the game.
We are told:
P(2 goals or less) = .60, which implies P(3 goals or more) = .40
P(NU Winning | 3 goals or more) = .70
P(NU Winning Scoring 2 goals or less) = .21
P(NU Winning) =
P(NU Winning scoring 3 goals or more) + P(NU Winning scoring 2 goals or less) =
.28 + .21 = .49
P(Winning team scoring more goals than losing team) = 1.0 (otherwise it wouldnt be the
winning team.)
=4
s=4
For = 4:
x ( z/2 * ), i.e.,
N
32 - (1.96 * 4 / 4 ) 32 + (1.96 * 4 / 4 ), i.e.,
30.04 33.96
x ( t/2,v * s/ N ), i.e.
32 - ( 2.131* 4 / 4 ) 32 + ( 2.131* 4 / 4 ), i.e.
29.869 34.131
C. Here are the results from a previous cohorts first exam in statistics. Compute the mean and
variance of the scores. There were 9 Students in the class.
41 1
72 1
83 1
90 1
94 1
97 1
102 1
108 2
D. A company wants to cut its payroll. It has decided to offer early retirement incentives to those
managers whose salaries put them in the top 10% of the pay scale. If Salary N($65000, 160002), how high does
your salary have to be to qualify for the early retirement plan?
Turning to Appx E, Table 1, we see the critical value for Z is about 1.28 (1.29 is also
ok). So,
The True Standard Error of the Mean = N . Ergo, if the True SE = 5 and = 90,
N = 90 N = 5, Thus
18 = N and hence N = 324
If the True SE = 1,
N = 90 N = 1, Thus
90 = N and hence N = 8100
So, the more precise you want your estimates to be, the larger your sample will have to
be and hence the greater your data collection costs will be. The researcher would have
to collect a sample 25 times as large if she wanted the True SE to be 1 rather than 5.
Both the researcher and the funding agency have to carefully consider whether the
added precision is really necessary or not given the substantially greater costs.
3. (25 points) A Sociologist and a Biologist are both concerned about the increasing prevalence of health
problems such as obesity and diabetes. They believe that these problems are related to how much one exercises as a
child, and that exercise may in turn be related to the race and income of individuals. They have decided to team up
and do a study of racial and economic differences in childrens exercise. They have gathered data on the exercise
habits and income of 1000 black and 1000 white children . Their study reveals that 70% of the black children have
low incomes and the remainder have high incomes. For whites, 20% have low incomes and the rest have high
incomes. Among blacks, 40% of those with low incomes regularly exercise, compared to 60% of the high-income
blacks. For whites, 35% of those with low incomes regularly exercise, compared to 55% of the high-income whites.
a. Finish filling in the numbers for the following table. Remember that, as is already noted in the
table, there are a total of 1000 blacks and 1000 whites. [HINT: If you dont find that more whites exercise than do
blacks, youve done something wrong.]
Exercises regularly
For blacks, we are told P(Low Income) = .70, implying 700 low income blacks and 300
high income blacks. Also, P(Exercise | Low Income) = .40, which means that 40% of
the 700 low income blacks, or 280, exercise regularly, while the other 420 do not. Also,
P(Exercise | High Income) =.60, which means that 60% of the 300 high income blacks,
or 180, exercise regularly, while the other 120 do not.
For whites, we are told P(Low Income) = .20, implying 200 low income whites and 800
high income whites. Also, P(Exercise | Low Income) = .35, which means that 35% of
the 200 low income whites, or 70, exercise regularly, while the other 130 do not. Also,
P(Exercise | High Income) =.55, which means that 55% of the 800 high income whites,
or 440, exercise regularly, while the other 360 do not. We therefore get
Black White
b. Looking at the above table, the Biologist argued that blacks are less likely to exercise than are
whites. The Sociologist, however, argued that blacks are more likely to exercise than are comparable whites.
Briefly explain the evidence that supports each of their positions.
The biologist is correct, in that only 46% of all Blacks exercise regularly, compared to
51% of all whites. However, the Sociologist is also correct: 40% of all low income
blacks exercise regularly, compared to only 35% of all low income whites. And, 60% of
all high income blacks exercise regularly, compared to only 55% of all high income
whites. That is, for blacks and whites of comparable incomes, blacks are somewhat
more likely to exercise. However, blacks are much more likely to be low income, and
low income people are less likely to exercise, hence blacks as a whole exercise less
than whites do.
If blacks had the same income distribution as whites (20% low income, 80% high
income) while maintaining their income-specific rates of exercise (40% for low income,
60% for high income), then the probability of a black exercising would be
So, if blacks had the same income distribution as whites, they would actually exercise more
than whites currently do. This implies that it is their lower incomes, rather than their race,
that causes blacks to exercise less overall than whites do.
4. (25 points) An association has reassured a hotel that, despite recent events, no more than 15% of its
members plan to cancel their reservations for an upcoming national convention. The hotel, of course, fears
otherwise. A random sample of 81 association members reveals that 20 are going to cancel their reservations. Test
the associations claim at the .01 level of significance. Be sure to indicate:
(a) The null and alternative hypotheses - and whether a one-tailed or two-tailed test is called for.
(b) The appropriate test statistic
(c) The critical region
(d) The computed value of the test statistic
(e) Your decision - should the null hypothesis be rejected or not be rejected? Why?
A one-tailed alternative is appropriate. The hotel wont be upset if the cancellation rate
turns out to be less than 15%, but it will be upset if it is more than that.
For the correction for continuity, we will subtract .5 if there are more than 12.15
cancellations, we will add .5 if there are less than 12.15 cancellations.
Or, equivalently,
(e) Decision: Do not reject the null. The computed test statistic falls just barely within
the acceptance region. (I imagine the hotel is probably still going to feel a bit nervous
though.)