FA21_ISOM2500 Final_ Solutions

1. Where should the control limits for an X-bar chart be placed if the design of the
process sets α=0.0027 with the following parameters: μ = 10, σ = 5 and n =
18 cases per batch. [Hint: Please use z-test]
A) [3.846, 10.535]
B) [6.476, 13.523]
C) [7.642, 15.357]
D) [8.821, 11.178]

2. The probability that house sales will increase in the next 6 months is
estimated to be 0.32. The probability that the interest rates on housing loans
will go up in the same period is estimated to be 0.63. The probability that
house sales or interest rates will go up during the next 6 months is estimated
to be 0.78. The probability that neither house sales nor interest rates will
increase during the next 6 months is
A) 0.11.
B) 0.19.
C) 0.22.
D) 0.90.

3. Which of following statements is NOT TRUE?

A) Var(X+Y) = Var(X-Y), if X and Y are independent.
B) Cov(X, Y) = E(XY)-E(X)E(Y)
C) If X and Y are strictly positively correlated, then Var(X+Y) > Var(X-Y).
D) The covariance between a random variable and a constant is NOT always zero.

4. In order to determine the average amount spent in November on Amazon.com, a

random sample of 13 Amazon accounts were selected. The sample mean amount spent
in November was $253.78 with a standard deviation of $23.45. What is a 90%
confidence interval for the population mean amount spent on Amazon.com in
November? (Correct your answer to 2 decimal place.)
A) [$213.13, $284.54]
B) [$242.19, $265.37]
C) [$345.03, $362.65]
D) [$125.44, $262.12]
E) Cannot be determined by the given information

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5. Suppose that you own an auto parts warehouse and you have recently started selling a
certain type of tire. According to the manufacturer, the tire lifetimes (measured in
kilometers (km)) are normally distributed with a mean of 75,000 (km) and a standard
deviation of 2,500 (km). You would like to offer a guarantee for free replacement of
any tire that does not last a specified minimum number of miles. Which of the
following should be the guaranteed lifetime if you desire that about 1% of this type of
tires will fail to last the guaranteed number of miles?
[Hint: Please use z-test]
A) 70,100
B) 68,550
C) 52,000
D) 69,175
E) 70,900

6. A sales manager selects a simple random sample of 400 customers who shopped at
one of the company’s hundred chain grocery stores in Sai Kung last week. From this
sample, she determines the average amount spent by these customers. The manager
returns to the grocer’s headquarters and tells the management team that she has an
estimate of the average amount spent per visit by customers at all of their stores in
Hong Kong. Do you agree or disagree that the average amount obtained by the
manager should be used as an estimate in this manner? Briefly explain why you agree
or disagree.
A) Agree – an estimate is any sample statistic calculated from a subset of the
population.
B) Agree – because a random sample was selected, her sample statistic can be
considered an estimate of the population parameter.
C) Disagree – the sample statistic is only from the customers last visit; in order to get
a true estimate, the manager would need to take more than 1 measurement from
each customer.
D) Disagree – the sample statistic is a measurement of only a store in Sai Kung while
the manager is claiming to have an estimate of all stores in Hong Kong.
E) Both (A) and (B) are correct.

7. The weights of bags of potting soil, X, filled by a distributor of garden products have
an expected value of 20 pounds and a standard deviation of 1.6 pounds. The weight
of any bag is independent of the weight of any other bag. The shipping department
secures 16 bags to a pallet and sends them to retail outlets.
Determine the standard deviation of the total weight of the potting soil on a pallet.
A) 6.4 lbs.
B) 25.6 lbs.
C) 1.6 lbs.
D) 409.6 lbs.
E) The standard deviation cannot be determined without being given the
covariance of the weights between individual bags.

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 [Q8-Q9]
Consider the following test of whether a coin is fair.Toss the coin three times. If the coin
lands either all heads or all tails, reject H0 : p=1/2. (The p denotes the chance for the coin
to land on heads.)
[Hint: Remember that you cannot use the normal approximation because the sample size
is three”]

8. What is the probability of a Type I error for this procedure?

A) 1/3
B) 1/4
C) 1/5
D) 1/8

9. If p=3/4, what is the probability of a Type II error for this procedure?

A) 9/16
B) 8/25
C) 3/16
D) 5/18

[Q10-Q11]
A software distributor has opened a new customer call center to assist customers with the
installation and use of their software. The manager is assuming that the amount of time
required to service a customer, X, is a normally distributed random variable. In an attempt
to get some information on the population mean service time, μ, the manager has selected
a random sample of 16 calls and found the average service time is 24 minutes with s = 4.2
minutes.

10. If a 95% confidence interval for the population mean service time is to be computed,
what is the margin of error? (Round your answer to 3 decimal places.)
[Hint: Please use t-test]
A) 2.058
B) 1.050
C) 2.237
D) No error actually committed sample was selected randomly

11. The manager would like to take another sample and determine a 95% confidence
interval with a margin of error of 1.5 minutes. What sample size would be needed to
achieve this goal? (Find the closest number.)
[Hint: Please use z-test]
A) 24 calls
B) 32 calls
C) 6 calls
D) 64 calls

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[Q12-Q13]
A manufacturer of flu medicine claims in its advertising that “90% of people with the
flu get relief from their symptoms” when using this medicine. A consumer group
selects a random sample of 400 people suffering from flu symptoms and provides
them with this medicine. After using the medicine, 348 of the people indicate that they
did get relief from the flu symptoms.

12. Determine a 98% confidence interval for the population proportion of people
with the flu who will get relief from their symptoms using this medicine.
(Round your answer to 3 decimal places.)
[Hint: Please use z-test]
A) [0.805, 0.885]
B) [0.811, 0.939]
C) [0.648, 0.852]
D) [0.831, 0.909]
E) [0.663, 0.787]

13. A member of the consumer group independently selects her own random
sample of 400 people with the flu and computes a 95% confidence interval for
the population proportion of people with the flu who will get relief from their
symptoms using this medicine. She provides the following interval: [.841,
.899]. Which of the following interpretations of this result is appropriate?

A) The company’s claim is not true because the interval does not contain
the value of .90 (90%).
B) There is a 95% chance that the population proportion is less than .90.
C) If they obtain a new sample and follow the same procedure to construct the
interval, it covers the population proportion with a probability of 95%.
D) 95% of the sample proportions based on a sample of this size will be in this
interval.
E) The population proportion will be in this interval 95% of the time.

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 14. Answer “True” or “False” for these two following statements.
(I) The size of the sample for a survey should be a fixed percentage of the
population size in order to produce representative results.
(II) Randomization produces samples that mimic the various characteristics of the
population without systematic bias.
A) (I) True ; (II) True
B) (I) False ; (II) False
C) (I) False ; (II) True
D) (I) True ; (II) False

[Q15-Q16]
15. In order to determine the average amount spent in October on hktvmall.com, a
random sample of 168 hktvmall accounts were selected. The sample mean amount
spent in October was $245.68 with a standard deviation of $24.75. Assuming
conditions are met, what is a 90% confidence interval for the population mean
amount spent on hktvmall.com in October?
[Hint: Please use z-test]
A) ($250.66, $256.90)
B) ($242.53, $248.83)
C) ($253.64, $259.92)
D) ($242.53, $256.22)

16. The confidence interval in previous question was calculated for October. If the amount
spent in November increased by 55%, then what is the 90% interval for the increase in
spending from October to November? (Assume standard deviation is $24.75 and
n=168)
[Hint: Please use z-test]
A) ($375.92, $385.69)
B) ($37, 592.80, $54,016.85)
C) ($364.44, $371.52)
D) ($36.34, $37.25)

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 [Q17-Q21]
A manufacturer of household products is considering a proposal by its research
department to invest in a development program for a new “environmentally safe” laundry
detergent to add to their product line. The sales department believes that the population
proportion of consumers who will buy such a product with their brand name is .20 (20%).
However, due to the cost of the development program, management believes that the
product will only be profitable if the proportion of consumers buying this product is
greater than .20. It is decided that a random sample of 400 consumers will be selected and
the sample proportion who indicate they will buy such a product will be computed. This
result will be used to reach a conclusion concerning what they believe to be true about the
value of the population proportion.

17. State an appropriate null and alternative hypothesis for the test described above.

A) H0: p ≥ 0.20 Ha: p < 0.20

B) H0: p ≤ 0.20 Ha: p > 0.20
C) H0: µ ≥ 0.20 Ha: µ < 0.20
D) H0: µ ≤ 0.20 Ha: µ = 0.20

18. Describe the consequences to the manufacturer of making a Type I error for this test.
A) They will invest in developing a product that will not be profitable.
B) They will invest in developing a product that will be profitable.
C) They will not invest in developing a product that will not be profitable.
D) They will not invest in developing a product that will be profitable.

19. The sales manager indicates that she will not be comfortable concluding that the
population proportion is greater than .20 unless the p-value for the test statistic is .01 (or
less). What must the value of the sample proportion be in order to observe a p-value this
small? (Round your answer to 3 decimal places.)
[Hint: Please use z-test]
A) 2.330
B) 0.247
C) 0.153
D) 0.201

20. Despite the feeling of the sales manager, it is decided that the test will be done using
α=.05. The survey results are that 92 out of 400 of the consumers indicated they will
buy such a product. What can be the p-value of this hypothesis test? (Round your
answer to 4 decimal places.)
[Hint: Please use z-test]
A) 0.2300
B) 0.5230
C) 0.9332
D) 0.0668

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21. If the survey results are still 92 out of 400, what conclusion should be reached based
on this data?
A) Do not reject H0, the product should be developed.
B) Reject H0, the product should not be developed.
C) Do no reject H0, the product should not be developed.
D) Reject H0, the product should be developed.

[Q22-Q24]

A large metropolitan bank has analyzed the amount of time required to process
home loans and determined that the times follow a normal distribution with
mean time μ=45 hours. The bank’s operations manager has developed a new
procedure for processing the loans that involves extensive use of new computer
software. He believes that the new procedure will decrease the population mean
amount of time required to process home loans. After training a group of loan
officers, a random sample of 25 loan applications will be selected and the
average amount of time required to process the loans will be determined. If the
switch is made to the new procedure, the cost of the additional software will be
more than offset by the savings in manpower required to process the loans. Use
the hypotheses Ho: μ≥45 hours and Ha: μ<45 hours.

22. Determine the p-value of the test statistic if the sample mean amount of time is x =
43.118 hours with the sample standard deviation s = 5.5 hours.
[Hint: Please use t-test]
A) 0.005
B) 0.01
C) 0.025
D) 0.05

23. If the sample mean amount of time is x =43.118 hours with the sample standard
deviation s = 5.5 hours, give the appropriate conclusion, for α=.025.
A) Do not reject H0, do not switch to the new procedure.
B) Reject H0, do not switch to the new procedure.
C) Do not reject H0, switch to the new procedure.
D) Reject H0, switch to the new procedure.

24. If you prefer to use a confidence interval to perform the same test as Q23, what
should the confidence level be set to?

A) 95%
B) 97.5%
C) 2.5%
D) 5%

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[Q25-Q28]
An insurance agent has selected a sample of drivers that she insures whose ages are in the
range from 16-42 years old. For each driver, she records the age of the driver and the
dollar amount of claims that the driver filed in the previous 12 months. A scatterplot
showing the dollar amount of claims as the response variable and the age as the predictor
shows linearity. The least squares regression line is determined to be: yˆ = 3715 − 75.4 x. A
plot of the residuals versus age of the drivers showed no pattern, and the following were
reported:

𝑟 2 = 0.822 Standard deviation of the residuals 𝑠𝑒 = 312.1

25. If the age of a driver increases by one year, by how much and in what direction
would the dollar amount of claims be predicted to change for the driver?
A) Increase by 75.4 dollars
B) Decrease by 75.4 dollars
C) Increase by 3715 dollars
D) Increase by 312.1 dollars

26. Using the fitted line given above to estimate the dollar amount of claims for a driver
whose age is 55 years would provide a prediction that is unreliable because it is an
_________.
A) unsolvable problem
B) extended result
C) extrapolation
D) extorted point

27. What percentage of the variation in the dollar amount of claims is due to factors
other than age?
A) 17.8%
B) 0.178%
C) 75.4%
D) 31.21%

28. For the response variable and the predictor described above, is the correlation
r between the variables positive or negative? Explain how you reached your
conclusion.
A) Positive – because the slope is positive.
B) Positive – because the y-intercept is positive.
C) Negative – because the slope is negative.
D) Negative – because the y-intercept is negative.

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29. A kindergarten teacher developed a least squares regression equation
predicting the height of her kindergarteners (in inches) from their age (in
months). The resulting equation is yˆ = 26.68 + .293x with an r2 = .263. Which
variable is the explanatory variable?
A) Height
B) Age
C) Months
D) Inches

30. A cosmetic company has selected a sample of its female customers whose age are in
the range from 16 to 42 years. For each customer, the company records her age and
the amount (in dollars) that she spends on cosmetics this year. A scatterplot showing
the dollar amount of expense as the response variable and the age as the predictor
exhibits linearity. The least squares regression line is determined to be: 𝑦̂ = 2,600 +
250𝑥 . Other summary results are also reported: 𝑟2 = 0.6, estimate of the standard
deviation of the residual s=212.

Which of the following is correct ?

A) The dollar amount of expense for a female customer of 50 years old is

accurately predicted to be $ 15,100.
B) The correlation coefficient between the response variable and the predictor is
negative 0.775
C) A customer in the data set whose age is 25 years had a residual of -$150 using
the fitted line above; this means her dollar amount of claims is $8700.
D) If the age of a female customer increases by one year, the dollar amount of
expense is predicted to be increased by $2,600

31. (True or False) Two variables that have a least square regression line fit of
r2 = 0 have no association.
A) True
B) False

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[Q32-Q34]
Each worker at an assembly plant that produces clock radios is responsible for
the entire assembly of each unit they work on. The plant manager has collected
data from a sample of workers: the number of years (YRS) of experience at the
plant, and the number of hours per unit (TIME) required for assembly. The
scatterplot of TIME versus YRS is shown below.

32. Which of the following is an appropriate reason why a regression line

should not be used to make predictions based on this data?
A) The magnitude of the slope of the line is too large.
B) The intercept of the fitted line has no practical interpretation in this context.
C) The linear condition for simple regression appears to be met.
D) The association between TIME and YRS appears to be negative and nonlinear.

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The manager has decided to transform the response variable from TIME (hours/unit) to
1/TIME (units/hour).
Note: This is a reciprocal transformation. The scatterplot of 1/TIME versus YRS is
shown below.

33. Which of the following would be appropriate interpretations of these results?

A) The units on se are hours per unit.
B) More experienced workers are predicted to produce more units per hour
on average than less-experienced workers.
C) Workers with no experience are predicted to require an average of 0.015
hours per unit (rounded to one decimal place).
D) The slope b1 measures the elasticity between 1/TIME and YRS.

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34. The fitted line given above question 32 predicts that workers with 3.8 years of
experience would require an average TIME of .46 hours per unit. What value
would be predicted for the average hours per unit required for a worker with 3.8
years of experience using the fitted line given above question 33 after the
reciprocal transformation? (Round your answer to 2 decimal places.)
A) 2.64 hours per unit
B) 1.27 hours per unit
C) 2.25 hours per unit
D) 3.98 hours per unit
E) 0.25 hours per unit

35. Two production plants, A and B, make wire cables that are sent to a common
distributor. 40% of the cables sent to the distributor come from Plant A, and the
remaining 60% come from Plant B. Among the cables produced at Plant A, 95%
meet the strength specifications; among the cables produced at Plant B, 98% meet
the strength specifications.
The distributor selects one cable at random from among all the cables in stock. If
the cable selected is found to meet the strength specifications, what is the
probability that the cable was produced at Plant A? (Round your answer to 4
decimal places.)
A) 0.2800
B) 0.5623
C) 0.3926
D) 0.4706
E) 0.3714

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[Q36-Q40]
A manufacturer of automobile spark plugs has several production facilities spread across
a wide geographical area. The quality control manager at each plant budgets a certain
amount of money (QC $, measured in thousands of dollars) for quality control efforts
each week (maintenance and calibration of equipment, time spent on process evaluation,
etc.). Each manager also determines the percentage of defective spark plugs (% DEF)
produced each week. The company’s vice president of operations has collected a sample
of data to determine the association between these variables. A scatterplot, fitted line, and
summary output for a simple regression model are provided:

Regression Analysis

r2 0.876 n 85
r -0.936
se 0.967 Dep. Var. % DEF

Regression output
variables coefficients std. error t (df=83) p-value
Intercept 12.0554 0.2767 43.564 5.93E-59
QC $ (000) -1.2045 0.0498 -24.169 2.55E-39

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 36. Which of the following conclusions is an appropriate interpretation of these
results?
A) The simple regression model should not be used because the residual plot
displays no linear pattern.
B) The simple regression model should not be used because the sign of r 2 does not
match the sign of the slope b1 .
C) The similar variances condition for a simple regression model appears to be
satisfied.
D) The model does not appear to explain statistically significant variation in the
percentage of defectives being produced.
E) On average, a $1000 increase in the amount of money spent on quality control is
estimated to result in a .936% decrease in the percentage of defectives.

37. Which of the following represents a 95% confidence interval for the slope 1 ?
(Round your answer to two decimal places.)
[Hint: Please use z-test]
A) (-1.30, -1.11)
B) (-3.14, 0.73)
C) (-1.22, -1.19)
D) (-1.25, -1.15)
E) (-1.93, 1.93)

38. Which of the following can be used as justification to reject the hypothesis
H 0 :  1 = 0 at a level of significance of α=.05?
A) A 95% prediction interval for %DEF does not contain the value of zero.
B) A 95% confidence interval for the slope 1 does not contain the value of zero.
C) The standard error of the residuals is less than 1.
D) The value of se(b1 ) is not equal to zero.
E) None of these results would allow us to conclude to reject the hypothesis .

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39. What is the value of the standard deviation of the residuals?
A) 0.2767
B) 0.0498
C) 0.9670
D) 0.4592
.

40. What is the value of the test statistic for the hypothesis test for slope different from
zero?
A) -24.169
B) 43.564
C) -1.205
D) 2.550

-----Then End---

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