CHAPTER 11—INFERENCES ABOUT POPULATION VARIANCES
MULTIPLE CHOICE
1. A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the
population. The chi-square values to be used for this interval estimation are
a.
-1.96 and 1.96
b.
14.573 and 43.195
c.
16.151 and 40.113
d.
15.308 and 44.461
2. We are interested in testing whether the variance of a population is significantly less than 1.44. The
null hypothesis for this test is
a.
Ho: 2 1.44
b.
Ho: s2 1.44
c.
Ho: 1.20
d.
Ho: 2 1.44
3. A sample of 41 observations yielded a sample standard deviation of 5. If we want to test Ho: 2 = 20,
the test statistic is
a.
100
b.
10
c.
51.25
d.
50
4. The value of F.05 with 8 numerator and 19 denominator degrees of freedom is
a.
2.48
b.
2.58
c.
3.63
d.
2.96
5. To avoid the problem of not having access to Tables of F distribution with values given for the lower
tail, the numerator of the test statistic should be the one with
a.
the larger sample size
b.
the smaller sample size
c.
the larger sample variance
d.
the smaller sample variance
6. The symbol used for the variance of the population is
a.
b.
2
c.
s
d.
s2
7. The symbol used for the variance of the sample is
a.
b.
2
c.
s
d.
s2
8. The random variable for a chi-square distribution may assume
a.
any value between -1 to 1
b.
any value between – infinity to +infinity
c.
any negative value
d.
any value greater than zero
9. A sample of n observations is taken from a population. When performing statistical inference about a
population variance, the appropriate chi-square distribution has
a.
n degrees of freedom
b.
n – 1 degrees of freedom
c.
n – 2 degrees of freedom
d.
n – 3 degrees of freedom
10. For an F distribution, the number of degrees of freedom for the numerator
a.
must be larger than the number of degrees for the denominator
b.
must be smaller than the number of degrees of freedom for the denominator
c.
must be equal to the number of degrees of freedom for the denominator
d.
can be larger, smaller, or equal to the number of degrees of freedom for the denominator
11. The bottler of a certain soft drink claims their equipment to be accurate and that the variance of all
filled bottles is 0.05 or less. The null hypothesis in a test to confirm the claim would be written as
a.
Ho: 2 0.05
b.
Ho: 2 > 0.05
c.
Ho: 2 < 0.05
d.
Ho: 2 0.05
12. A sample of 20 cans of tomato juice showed a standard deviation of 0.4 ounces. A 95% confidence
interval estimate of the variance for the population is
a.
0.2313 to 0.8533
b.
0.2224 to 0.7924
c.
0.0889 to 0.3169
d.
0.0925 to 0.3413
13. The manager of the service department of a local car dealership has noted that the service times of a
sample of 15 new automobiles has a standard deviation of 4 minutes. A 95% confidence interval
estimate for the variance of service times for all their new automobiles is
a.
8.58 to 39.79
b.
4 to 16
c.
4 to 15
d.
1.64 to 1.96
14. The manager of the service department of a local car dealership has noted that the service times of a
sample of 30 new automobiles has a standard deviation of 6 minutes. A 95% confidence interval
estimate for the standard deviation of the service times for all their new automobiles is
a.
16.05 to 45.72
b.
4.78 to 8.07
c.
2.93 to 6.31
d.
22.83 to 65.06
15. The producer of a certain medicine claims that their bottling equipment is very accurate and that the
standard deviation of all their filled bottles is 0.1 ounce or less. A sample of 20 bottles showed a
standard deviation of 0.11. The test statistic to test the claim is
a.
400
b.
22.99
c.
4.85
d.
20
16. The producer of a certain bottling equipment claims that the variance of all their filled bottles is 0.027
or less. A sample of 30 bottles showed a standard deviation of 0.2. The p-value for the test is
a.
between 0.025 and 0.05
b.
between 0.05 and 0.01
c.
0.05
d.
0.025
Exhibit 11-1
Last year, the standard deviation of the ages of the students at UA was 1.81 years. Recently, a sample
of 10 students had a standard deviation of 2.1 years. We are interested in testing to see if there has
been a significant change in the standard deviation of the ages of the students at UA.
17. Refer to Exhibit 11-1. The test statistic is
a.
14.2
b.
12.1
c.
3.28
d.
2.1
18. Refer to Exhibit 11-1. At 95% confidence the null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 11-2
We are interested in determining whether or not the variances of the sales at two music stores (A and
B) are equal. A sample of 25 days of sales at store A has a sample standard deviation of 30 while a
sample of 16 days of sales from store B has a sample standard deviation of 20.
19. Refer to Exhibit 11-2. The test statistic is
a.
1.50
b.
0.67
c.
1.56
d.
2.25
20. Refer to Exhibit 11-2. At 95% confidence the null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
21. We are interested in testing to see if the variance of a population is less than 7. The correct null
hypothesis is
a.
< 7
b.
2 49
c.
s < 49
d.
s > 49
22. A random sample of 31 charge sales showed a sample standard deviation of $50. A 90% confidence
interval estimate of the population standard deviation is
a.
1,715.10 to 4,055.68
b.
1,596.45 to 4,466.73
c.
39.96 to 66.83
d.
41.39 to 63.68
23. The chi-square values (for interval estimation) for a sample size of 10 at 95% confidence are
a.
3.32511 and 16.9190
b.
2.70039 and 19.0228
c.
4.16816 and 14.6837
d.
3.24697 and 20.4831
24. The chi-square value for a one-tailed (upper tail) hypothesis test at 95% confidence and a sample size
of 25 is
a.
33.1963
b.
36.4151
c.
39.3641
d.
37.6525
25. The chi-square value for a one-tailed test (lower tail) when the level of significance is 0.1 and the
sample size is 15 is
a.
21.0642
b.
23.6848
c.
7.78453
d.
6.57063
26. The critical value of F at 95% confidence when there is a sample size of 21 for the sample with the
smaller variance, and there is a sample size of 9 for the sample with the larger sample variance is
a.
2.45
b.
2.94
c.
2.37
d.
2.10
Exhibit 11-3
The contents of a sample of 26 cans of apple juice showed a standard deviation of 0.06 ounces. We are
interested in testing to determine whether the variance of the population is significantly more than
0.003.
27. Refer to Exhibit 11-3. The null hypothesis is
a.
s2 > 0.003
b.
s2 0.003
c.
2 > 0.003
d.
2 0.003
28. Refer to Exhibit 11-3. The test statistic is
a.
1.2
b.
31.2
c.
30
d.
500
29. Refer to Exhibit 11-3. At 95% confidence, the critical value(s) from the table is(are)
a.
13.1197 and 40.6465
b.
37.6525
c.
14.6114 and 37.6525
d.
14.6114
30. Refer to Exhibit 11-3. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
31. Refer to Exhibit 11-3. The p-value for this test is
a.
0.05
b.
greater than 0.10
c.
less than 0.10
d.
1.96
32. The sampling distribution of the quantity (n-1)s2/2 is the
a.
chi-square distribution
b.
normal distribution
c.
F distribution
d.
t distribution
33. The sampling distribution of the ratio of independent sample variances extracted from two normal
populations with equal variances is the
a.
chi-square distribution
b.
normal distribution
c.
Z distribution
d.
t distribution
34. The 95% confidence interval estimate for a population variance when a sample variance of 30 is
obtained from a sample of 12 items is
a.
14.14 to 74.94
b.
15.05 to 86.48
c.
16.42 to 94.35
d.
16.77 to 72.13
35. The 99% confidence interval estimate for a population variance when a sample standard deviation of
12 is obtained from a sample of 10 items is
a.
4.58 to 62.25
b.
46.53 to 422.17
c.
54.94 to 747.01
d.
62.04 to 562.89
36. The 90% confidence interval estimate for a population standard deviation when a sample variance of
50 is obtained from a sample of 15 items is
a.
4.18 to 15.07
b.
5.18 to 11.15
c.
5.44 to 10.32
d.
29.55 to 106.53
Exhibit 11-4
n = 30
H0: 2 = 500
s2 = 625
Ha: 2 500
37. Refer to Exhibit 11-4. The test statistic for this problem equals
a.
23.2
b.
24
c.
36.25
d.
37.5
38. Refer to Exhibit 11-4. The null hypothesis is to be tested at the 5% level of significance. The critical
value(s) from the table is(are)
a.
42.5569
b.
43.7729
c.
16.0471 and 45.7222
d.
16.7908 and 46.9792
39. Refer to Exhibit 11-4. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 11-5
n = 14
H0: 2 410
s = 20
Ha: 2 > 410
40. Refer to Exhibit 11-5. The test statistic for this problem equals
a.
.63
b.
12.68
c.
13.33
d.
13.66
41. Refer to Exhibit 11-5. The null hypothesis is to be tested at the 5% level of significance. The critical
value(s) from the table is(are)
a.
22.3621
b.
23.6848
c.
5.00874 and 24.7356
d.
5.62872 and 26.119
42. Refer to Exhibit 11-5. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 11-6
Sample A
Sample B
s2
32
38
n
24
16
We want to test the hypothesis that the population variances are equal.
43. Refer to Exhibit 11-6. The test statistic for this problem equals
a.
.67
b.
.84
c.
1.19
d.
1.50
44. Refer to Exhibit 11-6. The null hypothesis is to be tested at the 10% level of significance. The critical
value from the table is
a.
2.11
b.
2.13
c.
2.24
d.
2.29
45. Refer to Exhibit 11-6. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 11-7
Sample A
Sample B
s2
22
25
n
10
8
We want to test the hypothesis that population B has a smaller variance than population A.
46. Refer to Exhibit 11-7. The test statistic for this problem equals
a.
.77
b.
.88
c.
1.14
d.
1.29
47. Refer to Exhibit 11-7. The null hypothesis is to be tested at the 5% level of significance. The critical
value from the table is
a.
3.07
b.
3.29
c.
3.35
d.
3.68
48. Refer to Exhibit 11-7. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
49. To avoid the problem of not having access to tables of the F distribution with values given for the
lower tail when a two-tailed test is required, let the smaller sample variance be
a.
the denominator of the test statistic
b.
the numerator of the test statistic
c.
at least one
d.
None of these alternatives is correct.
50. To avoid the problem of having access to tables of the F distribution with values for the lower tail
when a one-tail test is required, let
a.
the smaller sample variance be the numerator of the test statistic
b.
the larger sample variance be the numerator of the test statistic
c.
the sample variance from the population with the smaller hypothesized variance be the
numerator of the test statistic
d.
the sample variance from the population with the larger hypothesized variance be the
numerator of the test statistic
Exhibit 11-8
n = 23
H0: 2 66
s2 = 60
Ha: 2 < 66
51. Refer to Exhibit 11-8. The test statistic has a value of
a.
20.91
b.
24.20
c.
24.00
d.
20.00
52. Refer to Exhibit 11-8. If the test is to be performed at 95% confidence, the critical value(s) from the
table is(are)
a.
10.9823 and 36.7897
b.
33.9244
c.
12.3380
d.
43.7729
53. Refer to Exhibit 11-8. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
54. Refer to Exhibit 11-8. The p-value is
a.
less than 0.025
b.
less than 0.05
c.
less than 0.10
d.
greater than 0.10
55. Which of the following has a chi-square distribution?
a.
(n – 1) 2/s2
b.
(n – 2) 2/s2
c.
(n – 1)s/
d.
(n – 1)s2/2
56. Which of the following has an F distribution?
a.
(n – 1)s/
b.
s1/s2
c.
(n – 1)s1/s2
d.
57. The sampling distribution used when making inferences about a single population’s variance is
a.
an F distribution
b.
a t distribution
c.
a chi-square distribution
d.
a normal distribution
58. The sampling distribution of the ratio of two independent sample variances taken from normal
populations with equal variances is
a.
an F distribution
b.
a chi-square distribution
c.
a t distribution
d.
a normal distribution
59. In Excel, which of the following functions is used to construct a confidence interval for a population
variance?
a.
CHISQ.DIST
b.
F-Test
c.
CHI.INV
d.
None of these answers are correct.
60. In Excel, which of the following functions is used to conduct a hypothesis test (using the p-value) for a
population variance?
a.
CHISQ.DIST
b.
F-Test
c.
CHI.INV
d.
None of these answers are correct.
61. In Excel, which of the following functions is used to conduct a hypothesis test for comparing two
population variances?
a.
CHISQ.DIST
b.
F-Test
c.
CHI.INV
d.
None of these answers are correct.
Exhibit 11-9
n = 14
s = 20
H0: 500
Ha: 500
62. Refer to Exhibit 11-9. The test statistic for this problem equals
a.
.63
b.
12.68
c.
13.33
d.
13.66
63. Refer to Exhibit 11-9. The null hypothesis is to be tested at the 5% level of significance. The critical
value(s) from the table is (are)
a.
22.362
b.
23.685
c.
5.009 and 24.736
d.
5.629 and 26.119
64. Refer to Exhibit 11-9. The null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
Exhibit 11-10
65. Refer to Exhibit 11-10. The test statistic for this problem equals
a.
100
b.
101.88
c.
101.25
d.
64
66. Refer to Exhibit 11-10. The p-value is between
a.
0.025 and 0.05
b.
0.05 and 0.1
c.
0.1 and 0.2
d.
0.2 and 0.3
67. Refer to Exhibit 11-10. At 95% confidence, the null hypothesis
a.
should be rejected
b.
should not be rejected
c.
should be revised
d.
None of these alternatives is correct.
68. The value of F0.01 with 9 numerator and 20 denominator degrees of freedom is
a.
2.39
b.
2.94
c.
2.91
d.
3.46
69. A sample of 60 items from population 1 has a sample variance of 8 while a sample of 40 items from
population 2 has a sample variance of 10. If we test whether the variances of the two populations are
equal, the test statistic will have a value of
a.
0.8
b.
1.56
c.
1.5
d.
1.25
70. A sample of 21 elements is selected to estimate a 90% confidence interval for the variance of the
population. The chi-square value(s) to be used for this interval estimation is (are)
a.
-1.96 and 1.96
b.
12.443
c.
10.851 and 31.410
d.
12.443 and 28.412
71. In a hypothesis test about two population variances, the test statistic F is computed as
a. 1/(
2
1
s
/
2
2
s
)
b.
2
1
s
/
2
2
s
c. 1/(
2
1
/
2
2
)
d.
2
1
/
2
2
72. Which of the following rejection rules is proper?
a. Reject H0 if p-value > F
b. Reject H0 if p-value <
2
c. Reject H0 if p-value >
2
/2
d. Reject H0 if p-value <
73. There is a .90 probability of obtaining a
2
value such that
a.
2 2 2
.05 .95
b.
2 2 2
.10 .90
c.
2 2 2
.90 .10
d.
2 2 2
.95 .05
74.
.975
= 8.9066 indicates that
a. 97.5% of the chi-square values are greater than 8.9066.
b. 97.5% of the chi-square values are less than 8.9066.
c. 2.5% of the chi-square values are greater than 8.9066.
d. 5% of the chi-square values are more than 8.9066 from the mean.
75. In practice, the most frequently encountered hypothesis test about a population variance is a
a. one-tailed test, with rejection region in lower tail
b. one-tailed test, with rejection region in upper tail
c. two-tailed test, with equal-size rejection regions
d. two-tailed test, with unequal-size rejection regions
PROBLEM
1. A sample of 15 items provides a sample mean of 18 and a sample variance of 16. Compute a 95%
confidence interval estimate for the standard deviation of the population.
2. A sample of 30 items provided a sample mean of 28 and a sample standard deviation of 6. Test the
following hypotheses using = 0.05. What is your conclusion?
H0: 2 25
Ha: 2 > 25
3. A sample of 61 items provided a sample mean of 932, a sample mode of 900, and a sample standard
deviation of 11. Test the following hypotheses using = 0.05. What is your conclusion?
H0: 2 80
Ha: 2 > 80
4. We are interested in determining whether or not the variances of the sales at two small grocery stores
are equal. A sample of 16 days of sales at each store indicated the following.
Store A
Store B
n1 = 16
n2 = 16
s1 = $125
s2 = $105
Are the variances of the populations (from which these samples came) equal?
Use = 0.05.
5. A random sample of 25 employees of a local utility firm showed that their monthly incomes had a
sample standard deviation of $112. Provide a 90% confidence interval estimate for the standard
deviation of the incomes for all the firm’s employees.
6. A random sample of 41 scores of students taking the ACT test showed a standard deviation of 8 points.
Provide a 98% confidence interval estimate for the standard deviation of all the ACT test scores.
ANS: