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CHAPTER 13 SECTION 1-2: INFERENCE ABOUT COMPARING TWO
POPULATIONS
MULTIPLE CHOICE
1. The expected value of the difference of two sample means equals the difference of the corresponding
population means when:
a. the populations are normally distributed.
b. the samples are independent.
c. the populations are approximately normal and the sample sizes are large.
d. All of these choices are true.
ANS: D
PTS: 1
REF: SECTION 13.1-13.2
2. In testing the difference between the means of two normally distributed populations, the number of
degrees of freedom associated with the unequal-variances t-test statistic usually results in a non-integer
number. It is recommended that you:
a. round to the nearest integer.
b. change the sample sizes until the number of degrees of freedom becomes an integer.
c. assume that the population variances are equal, and then use df = n1 + n2  2.
d. None of these choices.
ANS: A
3. The quantity
PTS: 1
REF: SECTION 13.1-13.2
is called the pooled variance estimate of the common variance of two unknown but
equal population variances. It is the weighted average of the two sample variances, where the weights
represent the:
a. sample variances.
b. sample standard deviations.
c. degrees of freedom for each sample.
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
4. Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed
populations. Assume that the population variances are unknown but equal. In order to test the
difference between the population means,
, the sampling distribution of the sample mean
difference,
, is:
a. normal.
b. Student-t with 50 degrees of freedom.
c. Student-t with 48 degrees of freedom.
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
5. Two independent samples of sizes 40 and 50 are randomly selected from two populations to test the
difference between the population means
. Assume the population variances are known. The
sampling distribution of the sample mean difference
is:
a. normally distributed.
b. approximately normal.
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c. Student t-distributed with 88 degrees of freedom.
d. None of these choices.
ANS: B
PTS: 1
REF: SECTION 13.1-13.2
6. Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with
equal variances (assumed to be unknown). In order to test the difference between the population
means, the test statistic is:
a. a standard normal random variable.
b. approximately standard normal random variable.
c. Student t-distributed with 58 degrees of freedom.
d. Student t-distributed with 33 degrees of freedom.
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
7. In testing the difference between two population means using two independent samples, we use the
pooled variance in estimating the standard error of the sampling distribution of the sample mean
difference
if:
a. the sample sizes are both large.
b. the populations are normal with equal variances.
c. the populations are non-normal with unequal variances.
d. All of these choices are true.
ANS: B
PTS: 1
REF: SECTION 13.1-13.2
8. In testing the difference between two population means for which the population variances are
unknown and not assumed to be equal, two independent samples are drawn from the populations.
Which of the following tests is appropriate?
a. z-test
b. pooled-variances t-test
c. unequal variances t-test
d. None of these choices.
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
9. In testing the difference between the means of two normal populations using two independent samples
when the population variances are unequal, the sampling distribution of the resulting statistic is:
a. normal.
b. Student-t.
c. approximately normal.
d. approximately Student-t.
ANS: D
PTS: 1
REF: SECTION 13.1-13.2
10. In constructing a confidence interval estimate for the difference between the means of two independent
normally distributed populations, we:
a. pool the sample variances when the unknown population variances are equal.
b. pool the sample variances when the population variances are known and equal.
c. pool the sample variances when the population means are equal.
d. never pool the sample variances.
ANS: A
PTS: 1
REF: SECTION 13.1-13.2
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11. The t-test for the difference between the means of two independent populations assumes that the
respective:
a. sample sizes are equal.
b. means are equal.
c. populations are normal.
d. All of these choices are true.
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
12. If we are testing for the difference between the means of two independent populations with equal
variances, samples of n1 = 15 and n2 = 15 are taken, then the number of degrees of freedom is equal to
a. 29
b. 28
c. 14
d. 13
ANS: B
PTS: 1
REF: SECTION 13.1-13.2
13. In testing for the differences between the means of two independent populations where the variances in
each population are unknown but assumed equal, the degrees of freedom is:
a. n1 + n2
b. n1 + n2  1
c. n1 + n2  2
d. None of these choices
ANS: C
PTS: 1
REF: SECTION 13.1-13.2
14. In testing for differences between the means of two independent populations the null hypothesis is:
a.
b.
c.
d.
ANS: B
PTS: 1
REF: SECTION 13.1-13.2
15. Given the information:
be used in the pooled variance t-test is:
a. 40
b. 38
c. 15
d. 25
ANS: B
PTS: 1
the number of degrees of freedom that should
REF: SECTION 13.1-13.2
16. When testing
vs.
2.15. Then, the p-value for this test would be
a. .0158
b. .0316
c. .9842
d. .9684
ANS: B
PTS: 1
, the observed value of the z-score was found to be
REF: SECTION 13.1-13.2
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17. A political analyst in Texas surveys a random sample of registered Democrats and compares the
results with those obtained from a random sample of registered Republicans. This would be an
example of:
a. independent samples.
b. dependent samples.
c. independent samples only if the sample sizes are equal.
d. dependent samples only if the sample sizes are equal.
ANS: A
PTS: 1
REF: SECTION 13.1-13.2
18. Suppose we randomly selected 200 people, and on the basis of their responses to a survey we assigned
them to one of two groups: high-risk group and low-risk group. We then recorded the blood pressure
for the members of each group. Such data are called:
a. observational.
b. experimental.
c. matched.
d. None of these choices.
ANS: A
PTS: 1
REF: SECTION 13.1-13.2
TRUE/FALSE
19. Independent samples are those for which the selection process for one is not related to the selection
process for the other.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
20. The pooled-variances t-test requires that the two population variances need not be the same.
ANS: F
PTS: 1
REF: SECTION 13.1-13.2
21. In testing the difference between two population means using two independent samples, we use the
pooled variance in estimating the standard error of the sampling distribution of the sample mean
difference
if the populations are normal with equal variances.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
22. In testing the difference between two population means using two independent samples, the sampling
distribution of the sample mean difference
is normal if the sample sizes are both greater than
30.
ANS: F
PTS: 1
REF: SECTION 13.1-13.2
23. A political analyst in Iowa surveys a random sample of registered Democrats and compares the results
with those obtained from a random sample of registered Republicans. This would be an example of
two independent samples.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
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24. Two samples of sizes 25 and 20 are independently drawn from two normal populations, where the
unknown population variances are assumed to be equal. The number of degrees of freedom of the
equal-variances t-test statistic is 44.
ANS: F
PTS: 1
REF: SECTION 13.1-13.2
25. The sampling distribution of
is normal if the sampled populations are normal, and
approximately normal if the populations are nonnormal and the sample sizes n1 and n2 are large.
ANS: T
PTS: 1
26. The expected value of
ANS: T
27. The variance of
ANS: F
REF: SECTION 13.1-13.2
is
PTS: 1
is
PTS: 1
.
REF: SECTION 13.1-13.2
.
REF: SECTION 13.1-13.2
28. The equal-variances test statistic of
is Student t-distributed with n1 + n2 degrees of freedom,
provided that the two populations are normal.
ANS: F
PTS: 1
REF: SECTION 13.1-13.2
29. When the population variances are unequal, we estimate each population variance with its sample
variance. Hence, the unequal-variances test statistic of
is approximately Student t-distributed
with n1 + n2  2 degrees of freedom.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
30. Both the equal-variances and unequal variances test statistic and confidence interval estimator of
require that the two populations be normally distributed.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
31. The best estimator of the difference between two population means
two sample means
.
ANS: T
PTS: 1
is the difference between
REF: SECTION 13.1-13.2
32. When we test for differences between the means of two independent populations, we can only use a
two-tailed test.
ANS: F
PTS: 1
REF: SECTION 13.1-13.2
33. Unless we can conclude that the population variances are equal, we cannot use the pooled variance
estimate.
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ANS: T
PTS: 1
REF: SECTION 13.1-13.2
34. When the sample sizes are equal, the pooled variance of the two samples is the average of the two
sample variances.
ANS: T
PTS: 1
REF: SECTION 13.1-13.2
COMPLETION
35. When the sample sizes are equal, the pooled variance of the two samples is the
____________________ of the two sample variances.
ANS: average
PTS: 1
REF: SECTION 13.1-13.2
36. The equal-variances test statistic of
is Student t-distributed with n1 + n2  2 degrees of freedom
provided that the two populations are ____________________.
ANS: normal
PTS: 1
REF: SECTION 13.1-13.2
37. The unequal-variances test statistic of
has an approximate ____________________
distribution with n1 + n2  2 degrees of freedom.
ANS:
Student t
Student-t
PTS: 1
REF: SECTION 13.1-13.2
38. ____________________ samples are those for which the selection process for one is not related to the
selection process for the other.
ANS: Independent
PTS: 1
REF: SECTION 13.1-13.2
39. When two population variances are ____________________ we estimate each population variance
with its sample variance. The test statistic of
is approximately Student t-distributed with n1 +
n2  2 degrees of freedom.
ANS: unequal
PTS: 1
REF: SECTION 13.1-13.2
40. A political analyst in Iowa surveys a random sample of registered Democrats and compares the results
with those obtained from a random sample of registered Republicans. This would be an example of
____________________ samples.
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copied, or distributed without the prior consent of the publisher.
ANS:
two independent
independent
PTS: 1
REF: SECTION 13.1-13.2
41. The pooled-variances t-test is used when the two population variances are ____________________.
ANS: equal
PTS: 1
REF: SECTION 13.1-13.2
42. When the population variances are unknown and unequal, we estimate each population variance with
its ____________________ variance.
ANS: sample
PTS: 1
REF: SECTION 13.1-13.2
43. The pooled variance estimator is the ____________________ average of the two sample variances.
ANS: weighted
PTS: 1
REF: SECTION 13.1-13.2
SHORT ANSWER
Final Exam Scores
Two random samples of 40 students were drawn independently from two populations of students.
Assume their final exams are normally distributed (total points = 100). The following statistics
regarding their scores in a final exam were obtained:
.
44. {Final Exam Scores Narrative} Test at the 5% significance level to determine whether we can infer
that the two population means differ.
ANS:
,
Rejection region: | t | > t0.025,78 = 1.99
Test Statistic: t = 2.45
Note: Cannot assume population variances are equal. Use all formulas assuming unequal variances.
Conclusion: Reject the null hypothesis. Yes, the two student populations differ in their mean final
exam scores. Population 1 has a higher mean, according to this data.
PTS: 1
REF: SECTION 13.1-13.2
45. {Final Exam Scores Narrative} Estimate with 95% confidence the difference between the two
population means.
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ANS:
4  3.25. Thus, LCL = 0.75, and UCL = 7.25 is our 95% confidence interval for the difference in mean
final exam scores (Population 1  Population 2).
PTS: 1
REF: SECTION 13.1-13.2
46. {Final Exam Scores Narrative} Explain how to use the 95% confidence interval to test the hypotheses
at  = .05.
ANS:
Since the hypothesized value 0 is not included in the 95% confidence level for
 = .05.
PTS: 1
, we reject H0 at
REF: SECTION 13.1-13.2
Starting Salary
In testing the hypotheses
vs.
, two random samples from two
populations of college of business graduates majoring in finance and international business produced
the following statistics regarding their starting salaries (in $1000s):
,
,
,
,
, and
. (Assume the salaries have normal distributions.)
47. {Starting Salary Narrative} What conclusion can we draw at the 5% significance level?
ANS:
Rejection region: | t | > t0.025,98 = 1.98
Test statistic: t = 1.19
Note: Can't assume population variances are equal. Use all formulas assuming unequal variances.
Conclusion: Don't reject the null hypotheses. Not enough evidence to say the average starting salaries
differ, according to this data.
PTS: 1
REF: SECTION 13.1-13.2
48. {Starting Salary Narrative} Estimate with 95% confidence the difference between the two population
means.
ANS:
5.0  8.41. Thus, a 95% confidence interval for the difference in the starting salaries (finance 
international business) has LCL = 3.41, and UCL = 13.41 (in thousands of dollars).
PTS: 1
REF: SECTION 13.1-13.2
49. {Starting Salary Narrative} Explain how to use the 95% confidence interval to test the hypotheses at 
= .05.
ANS:
Since the hypothesized value 0 = 0 is included in the 95% confidence interval, we fail to reject the
null hypothesis at  = 0.05.
PTS: 1
REF: SECTION 13.1-13.2
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copied, or distributed without the prior consent of the publisher.
50. The owner of a service station wants to determine if owners of new cars (two years old or less) change
their cars' oil more frequently than owners of older cars (more than two years old). From his records he
takes a random sample of ten new cars and ten older cars and determines the number of times the oil
was changed in the last 12 months. The data follow. Do these data allow the service station owner to
infer at the 10% significance level that new car owners change their cars' oil more frequently than
older car owners?
Frequency of Oil Changes in Past 12 Months
New Car Owners
Old Cars Owners
6
4
3
2
3
1
3
2
4
3
3
2
6
2
5
3
5
2
4
1
ANS:
,
Rejection region: t > t0.10,18 = 1.33
Test statistic: t = 4.12
Conclusion: Reject the null hypothesis. Yes, new car owners change their cars' oil more frequently
than older car owners, according to this data.
Note: OK to say population variances are equal, since sample variances are so close. Use equal
variance version of all formulas.
PTS: 1
REF: SECTION 13.1-13.2
Employees Test Scores
Thirty-five employees who completed two years of college were asked to take a basic mathematics
test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random
sample of 50 employees who only completed high school, the mean and standard deviation of the test
scores were 72.1 and 14.6, respectively.
51. {Employees Test Scores Narrative} Can we infer at the 10% significance level that a difference exists
between the two groups?
ANS:
,
Rejection region: | t | > t0.05,83  1.664
Test statistic: t = 1.00
Conclusion: Don't reject the null hypothesis. No, we can't infer at the 10% significance level that a
difference exists between the two groups, according to this data.
Note: Cannot assume population variances are equal. Use the unequal version of all formulas.
PTS: 1
REF: SECTION 13.1-13.2
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copied, or distributed without the prior consent of the publisher.
52. {Employees Test Scores Narrative} Estimate with 90% confidence the difference in mean scores
between the two groups of employees.
ANS:
3.0  4.98. Thus, LCL = 1.98, and UCL = 7.98.
PTS: 1
REF: SECTION 13.1-13.2
53. {Employees Test Scores Narrative} Explain how to use the interval estimate to test the hypotheses.
ANS:
Since the hypothesized value 0 is included in the 90% confidence interval, we fail to reject the null
hypothesis at  = 0.10.
PTS: 1
REF: SECTION 13.1-13.2
Preservatives
A food processor wants to compare two preservatives for their effects on retarding spoilage. Suppose
16 cuts of fresh meat are treated with preservative A and 16 are treated with preservative B, and the
number of hours until spoilage begins is recorded for each of the 32 cuts of meat. The results are
summarized in the table below
Sample Mean
Sample Standard Deviation
Preservative A
108.7 hours
10.5 hours
Preservative B
98.7 hours
13.6 hours
54. {Preservatives Narrative} State the null and alternative hypotheses to determine if the average number
of hours until spoilage begins differs for the preservatives A and B.
ANS:
vs.
PTS: 1
REF: SECTION 13.1-13.2
55. {Preservatives Narrative} Assume population variances are equal. Calculate the pooled variance and
the value of the test statistic.
ANS:
= 147.61, t = 2.33
PTS: 1
REF: SECTION 13.1-13.2
56. {Preservatives Narrative} Determine the rejection region at  = .05 and write the proper conclusion.
ANS:
t < t.025,30 = 2.042 or t > t.025,30 = 2.042. Since t = 2.33 > 2.042, we reject the null hypothesis at  =
.05 and conclude that the average number of hours until spoilage begins differs for preservatives A and
B. Preservative A helps the product last longer than Preservative B, on average, according to this data.
PTS: 1
REF: SECTION 13.1-13.2
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