Deck 11: Inferences About Population Variances

Question 1

A sample of 61 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) 40.482 and 83.298.
B) 32.357 and 71.420.
C) 34.764 and 67.505.
D) 43.188 and 79.082.

Answer 1

40.482 and 83.298.

Question 2

A sample of 31 observations yielded a sample variance of 16.If we want to test H₀: σ² = 16, the test statistic is

A) 20.
B) 21.
C) 30.
D) 7.5.

Answer 2

30.

Question 3

The symbol used for the variance of the sample is

A) σ.
B) σ².
C) s.
D) s².

Answer 3

s².

Question 4

To avoid the problem of not having access to Tables of F distribution when F values are needed for the lower tail, the numerator of the test statistic for a two-tailed test 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.

Question 5

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) H₀: σ² < 1.44.
B) H₀: s² ≥ 1.44.
C) H₀: σ < 1.20.
D) H₀: σ² ≥ 1.44.

Question 6

The bottler of a certain soft drink claims their equipment to be accurate and that the variance of all filled bottles is at most .05.The null hypothesis in a test to confirm the claim would be written as

A) H₀: σ² ≥ .05.
B) H₀: s² ≥ .05.
C) H₀: s² ≤ .05.
D) H₀: σ² ≤ .05.

Question 7

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 .2 ounces or less.A sample of 20 bottles showed a standard deviation of .12 ounces.The test statistic to test the claim is

A) 2.3.
B) 22.99.
C) 6.84.
D) 1.368.

Question 8

A sample of n observations is taken from a normal population.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.

Question 9

For an F distribution, the number of degrees of freedom for the numerator

A) must be larger than the number of degrees of freedom 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.

Question 10

Which of the following has a chi-square distribution?

A) (n - 1)s²/σ².
B) (n - 1)σ/s.
C) (n - 1)s/σ.

Question 11

The manager of the service department of a local car dealership has noted that the service times of a sample of 16 new automobiles has a standard deviation of 5 minutes.A 95% confidence interval estimate for the variance of service times for all their new automobiles is

A) 13.642 to 59.885.
B) 9.46 to 34.09.
C) 2.144 to 9.948.
D) 2.728 to 11.977.

Question 12

The producer of a certain bottling equipment claims that the variance of all their filled bottles is .027 or less.A sample of 30 bottles showed a standard deviation of .2 ounces.The p-value for the test is

A) between .025 to .05.
B) between .05 to .10.
C) .05.
D) .025.

Question 13

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.

Question 14

A sample of 20 cans of tomato juice showed a standard deviation of .4 ounces.A 95% confidence interval estimate of the variance for the population is

A) .2313 to .8533.
B) .2224 to .7924.
C) .3042 to .5843.
D) .0925 to .3413.

Question 15

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.

Question 16

The random variable for a chi-square distribution may assume

A) any value between -1 to 1.
B) any value between -∞ to +∞.
C) any negative value.
D) any value greater than zero.

Question 17

A sample of 51 observations yielded a sample standard deviation of 5.If we want to test H₀: σ² = 20, the test statistic is

A) 62.5.
B) 12.5.
C) 51.25.
D) 50.

Question 18

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 5 minutes.A 95% confidence interval estimate for the standard deviation of the service times (in minutes) for all their new automobiles is

A) 16.047 to 45.722.
B) 15.857 to 45.180.
C) 3.982 to 6.722.
D) 22.833 to 65.059.

Question 19

The symbol used for the variance of the population is

A) σ.
B) σ².
C) s.
D) s².

Question 20

Which of the following has an F distribution?

A) (n - 1)s/σ.
B) s₁/s₂.
C) (n - 1)s₁/s₂.
D) .

Question 21

The value of F_.01 with 9 numerator and 20 denominator degrees of freedom is

A) 2.39.
B) 2.94.
C) 2.91.
D) 3.46.

Question 22

A sample of 60 items from population 1 has a sample variance of 12 while a sample of 50 items from population 2 has a sample variance of 15.If we want to test whether the variances of the two populations are equal, the test statistic will have a value of

A) .8.
B) 1.56.
C) 1.5.
D) 1.25.

Question 23

The chi-square value for a one-tailed (upper tail) hypothesis test at a 5% level of significance and a sample size of 25 is

A) 33.196.
B) 36.415.
C) 39.364.
D) 37.652.

Question 24

The chi-square value for a one-tailed (lower tail) test when the level of significance is .1 and the sample size is 15 is

A) 21.064.
B) 23.685.
C) 7.790.
D) 6.571.

Question 25

The contents of a sample of 26 cans of apple juice showed a standard deviation of .06 ounces.We are interested in testing whether the variance of the population is significantly more than .003.At the .05 level of significance, the null hypothesis

A) should be rejected.
B) should not be rejected.
C) should be revised.
D) should not be tested.

Question 26

The contents of a sample of 26 cans of apple juice showed a standard deviation of .06 ounces.We are interested in testing whether the variance of the population is significantly more than .003.The null hypothesis is

A) s² > .003.
B) s² ≤ .003.
C) σ² > .003.
D) σ² ≤ .003.

Question 27

The contents of a sample of 26 cans of apple juice showed a standard deviation of .06 ounces.We are interested in testing whether the variance of the population is significantly more than .003.The test statistic is

A) 1.2.
B) 31.2.
C) 30.
D) 500.

Question 28

The sampling distribution of the quantity (n - 1)s²/σ² is the

A) chi-square distribution.
B) normal distribution.
C) F distribution.
D) t distribution.

Question 29

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) 4.589 to 62.253.
B) 46.538 to 422.171.
C) 54.941 to 746.974.
D) 62.042 to 562.895.

Question 30

We are interested in testing whether the variance of a population is significantly more than 625.The null hypothesis for this test is

A) H₀: σ² > 625.
B) H₀: σ² ≤ 625.
C) H₀: σ² ≥ 625.
D) H₀: σ² ≤ 25.

Question 31

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) F distribution.
D) t distribution.

Question 32

We are interested in testing to see if the variance of a population is less than 7.The correct null hypothesis is

A) σ < 7.
B) σ² ≥ 7.
C) σ < 49.
D) σ² ≥ 49.

Question 33

A random sample of 31 sales charge showed a sample standard deviation of $50.A 90% confidence interval estimate of the population standard deviation is

A) 1715.101 to 4055.589.
B) 1596.458 to 4466.679.
C) 39.956 to 66.833.
D) 41.393 to 63.684.

Question 34

For a sample size of 21 at 95% confidence, the chi-square values needed for interval estimation are

A) 9.591 and 34.170.
B) 2.700 and 19.023.
C) 8.260 and 37.566.
D) 10.283 and 35.479.

Question 35

A sample of 61 observations yielded a sample standard deviation of 6.If we want to test H₀: σ² = 40, the test statistic is

A) 54.
B) 9.15.
C) 54.90.
D) 9.

Question 36

The 95% confidence interval estimate of a population variance when a sample variance of 324 is obtained from a sample of 81 items is

A) 14.14 to 174.94.
B) 243.086 to 453.520.
C) 16.42 to 194.35.
D) 254.419 to 429.203.

Question 37

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) 31.410.
B) 12.443.
C) 10.851 and 31.410.
D) 12.443 and 28.412.

Question 38

The 90% confidence interval estimate of a population standard deviation when a sample variance of 50 is obtained from a sample of 15 items is

A) 26.8 to 124.356.
B) 5.177 to 11.152.
C) 5.436 to 10.321.
D) 29.555 to 106.529.

Question 39

The contents of a sample of 26 cans of apple juice showed a standard deviation of .06 ounces.We are interested in testing whether the variance of the population is significantly more than .003.The p-value for this test is

A) .05.
B) greater than .10.
C) less than .10.
D) zero.

Question 40

The critical value of F using α = .05 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.

Question 41

There is a .90 probability of obtaining a value such that ​

A) ​ < < ​.
B) ≤ ≤ ​.
C) ​ ≤ ≤ ​.
D) ​ < < ​.

Question 42

​Consider the following hypothesis problem. n = 30
H₀: σ² = 500
S² = 625
H_a: σ² ≠ 500
​
​The test statistic equals

A) ​23.20.
B) ​24.00.
C) ​36.25.
D) ​37.50.

Question 43

​Consider the following hypothesis problem. n = 14
H₀: σ² < 410
S = 20
H_a: σ² > 410
​
​The test statistic equals

A) .63.
B) ​12.68.
C) ​13.33.
D) ​13.68.

Question 44

Consider the following hypothesis problem. ​
N = 23
S² = 60
H₀: σ² ≤ 66
H_a: σ² > 66
​
If the test is to be performed at the .05 level of significance, the null hypothesis

A) should be rejected.
B) should not be rejected.
C) should be revised.
D) should not be tested.

Question 45

​Consider the following sample information from Population A and Population B. ​
​
Sample A
Sample B
N
24
16
S²
32
38
​​
We want to test the hypothesis that the population variances are equal.The test statistic for this problem equals

A) ​.67.
B) .84.
C) ​1.19.
D) ​1.50.

Question 46

To avoid the problem of not having access to tables of the F distribution when a one-tailed test is required and with F values given for the lower tail, let​ the

A) smaller sample variance be the numerator of the test statistic.
B) larger sample variance be the numerator of the test statistic.
C) sample variance from the population with the smaller hypothesized variance be the numerator of the test statistic.
D) sample variance from the population with the larger hypothesized variance be the numerator of the test statistic.

Question 47

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) 11.808 and 49.645.
B) ​14.573 and 43.195.
C) ​16.151 and 40.113.
D) ​15.308 and 44.461.

Question 48

​In practice, the most frequently encountered hypothesis test about a population variance is a

A) ​one-tailed test, with rejection region in the lower tail.
B) ​one-tailed test, with rejection region in the upper tail.
C) ​two-tailed test, with equal-size rejection regions.
D) ​two-tailed test, with unequal-size rejection regions.

Question 49

Consider the following hypothesis problem. ​
N = 23
S² = 60
H₀: σ² ≤ 66
H_a: σ² > 66
​
If the test is to be performed at the 5% level, the critical value(s) from the chi-square distribution table is(are)

A) 10.982 and 36.781.
B) 12.338 and 33.924.
C) 12.338.
D) 33.924.

Question 50

​Consider the following hypothesis problem. n = 14
H₀: σ² < 410
S = 20
H_a: σ² > 410
​
At the 5% level of significance, the null hypothesis

A) should be rejected​.
B) ​should not be rejected​.
C) ​should be revised.
D) should not be tested.

Question 51

Consider the following hypothesis problem. ​
N = 23
S² = 60
H₀: σ² ≤ 66
H_a: σ² > 66
​
The p-value is

A) less than .025.
B) between .025 and .05.
C) between .05 and .10.
D) greater than .10.

Question 52

?Which of the following rejection rules is proper?

A) ?Reject H₀ if p-value ? F_?.
B) ?Reject H₀ if F ? F_?/2.
C) ?Reject H₀ if p-value ? ?/2.
D) ?Reject H₀ if F ? F_?.

Question 53

​Consider the following hypothesis problem. n = 30
H₀: σ² = 500
S² = 625
H_a: σ² ≠ 500
​
At the 5% level of significance, the null hypothesis

A) ​should be rejected.
B) ​should not be rejected.
C) ​should be revised.
D) should not be tested.

Question 54

Consider the following hypothesis problem. n = 23
S² = 60
H₀: σ² ≤ 66
H_a: σ² > 66
​
The test statistic has a value of

A) 20.91.
B) 24.20.
C) 24.00.
D) 20.00.

Question 55

​Consider the following sample information from Population A and Population B. ​
​
Sample A
Sample B
N
24
16
S²
32
38
​​
We want to test the hypothesis that the population variances are equal.The null hypothesis is to be tested at the 10% level of significance.The critical value from the F distribution table is

A) 2.11.
B) ​2.13.
C) ​2.24.
D) ​2.29.

Question 56

Based on the sample evidence below, we want to test the hypothesis that population A has a larger variance than population B. Sample A
Sample B
N
11
10
S²
12)1
5
​
The test statistic for this problem equals

A) .4132.
B) 1.1.
C) 2.42.
D) 2.

Question 57

​Consider the following sample information from Population A and Population B. ​
​
Sample A
Sample B
N
24
16
S²
32
38
​​
We want to test the hypothesis that the population variances are equal.At the 10% level of significance, the null hypothesis

A) should be rejected.
B) ​should not be rejected.
C) ​should be revised.
D) should not be tested.

Question 58

​Consider the following hypothesis problem. n = 30
H₀: σ² = 500
S² = 625
H_a: σ² ≠ 500
​
​The null hypothesis is to be tested at the 5% level of significance.The critical value(s) from the chi-square distribution table is(are)

A) ​42.557.
B) ​43.773.
C) ​16.047 and 45.722.
D) ​16.791 and 46.979.

Question 59

​Consider the following hypothesis problem. n = 14
H₀: σ² < 410
S = 20
H_a: σ² > 410
​
​The null hypothesis is to be tested at the 5% level of significance.The critical value(s) from the chi-square distribution table is(are)

A) 22.362.
B) ​23.685.
C) ​5.009 and 24.736.
D) ​5.629 and 26.119.

Question 60

Based on the sample evidence below, we want to test the hypothesis that population A has a larger variance than population B. ​
Sample A
Sample B
N
11
10
S²
12)1
5
​
The p-value is approximately

A) .10.
B) .05.
C) .025.
D) .01.

Question 61

An egg packing company has stated that the standard deviation of the weights of their grade A large eggs is .07 ounces or less.The sample variance for 51 eggs was .0065 ounces.Can this sample result confirm the company's claim? Use α = .1.

Question 62

A sample of 30 items provided a sample mean of 28 and a sample standard deviation of 6.Test the following hypotheses using α = .05.What is your conclusion?
H₀: σ² ≤ 25
H_a: σ² > 25

Question 63

Do the following sample data indicate that the variance of the population from which it has been drawn is significantly more than 17? Use α = .05.
X
12
5
9
14
10

Question 64

A random sample of 20 observations showed a standard deviation of 8.At a .05 level of significance, test to see if the variance of the population is significantly less than 65.Use the critical value approach.

Question 65

At α = .1, test to see if the population variances from which the following samples were drawn are equal.
Sample 1
Sample 2
n₁ = 21
n₂ = 19
s₁ = 18
s₂ = 16

Question 66

The standard deviation of the ages of a sample of 16 executives from the northern states was 8.2 years; while the standard deviation of the ages of a sample of 25 executives from the southern states was 12.8 years.At α = .1, test to see if there is any difference between the variances of the ages of all the northern and all the southern executives.

Question 67

The average score of a sample of 30 students on their second statistics examination was 85 with a standard deviation of 11.Is the variance of the population significantly more than 80? Use a .05 level of significance.

Question 68

A company claims that the standard deviation in their delivery times is less than 5 days.A sample of 27 past customers was taken.The average delivery time in the sample was 14 days with a standard deviation of 4.5 days.At the 5% level of significance, test the company's claim.Use the critical value approach.

Question 69

A lumber company has claimed that the standard deviation for the lengths of their six-foot boards is .4 inches or less.To test their claim, a random sample of 16 six-foot boards is selected; and it is determined that the variance of the sample is .3.Do the results of the sample support the company's claim? Use α = .1.

Question 70

A random sample of 22 employees of a local utility firm showed that their monthly incomes had a sample standard deviation of $110.Provide a 90% confidence interval estimate for the standard deviation of the incomes of all employees of the firm.

Question 71

We are interested in determining whether or not the variances of the starting salaries of accounting majors is significantly different from management majors.The following information was gathered from two samples.
Accounting
Management
Sample Size
21
18
Average Monthly Income
$3600
$3500
Variance
$900
$400
​
At the .1 level of significance, test to determine whether or not the variances are equal.

Question 72

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.

Question 73

Student advisors are interested in determining if the variances of the scores of day students and night students are the same.The following samples are drawn:
Day
Night
n₁ = 25
n₂ = 31
s₁ = 9.8
s₂ = 14.7
​
Test the equality of the variances of the populations.Let α = .05.

Question 74

A sample of 16 students showed that the variance in the number of hours they spend studying is 25.At the 5% level of significance, test to see if the variance of the population is significantly different from 30.Use the critical value approach.

Question 75

A sample of 22 bottles of soft drink showed a variance of .64 in their contents.At the 5% level, determine whether or not the standard deviation of the population is significantly different from .7 ounces.Use the critical value approach.

Question 76

A large company has claimed that the standard deviation of the monthly incomes of their employees is less than or equal to $120.To test their claim, a random sample of 76 employees of the company was taken; and it was determined that the standard deviation of their incomes was $135.Using α = .10, test the company's claim.

Question 77

We are interested in determining whether or not the variances of the sales are equal for two small grocery stores.A sample of 16 days of sales at each store indicated the following.
Store A
Store B
n₁ = 16
n₂ = 16
s₁ = $130
s₂ = $100
​
Are the variances of the populations (from which these samples came) equal?
Use α = .05.

Question 78

A random sample of 21 checking accounts at a bank showed an average daily balance of $430 with a standard deviation of $50.
a.
Provide a 95% confidence interval estimate for the variance of the population of the checking accounts.
b.
Provide a 95% confidence interval estimate for the standard deviation of the population of the checking accounts.

Question 79

The standard deviation of the daily temperatures in Honolulu last year was 4 degrees Fahrenheit.A random sample of 25 days resulted in a standard deviation of 5.8 degrees Fahrenheit.Has there been a significant increase in the variance of the temperatures? Use α = .05.

Question 80

A sample of 14 items provides a sample mean of 20 and a sample variance of 18.Compute a 95% confidence interval estimate for the standard deviation of the population.