Deck 11: Inferences About Population Variances

Question 1

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 ≥ 49
C) s < 49
D) s > 49

Answer 1

σ² ≥ 49

Question 2

A sample of 20 cans of tomato juice showed a standard deviation of 0.4 ounce. A 95% confidence interval estimate of the variance for the population is _____.

A) .2313 to .8533
B) .2224 to .7924
C) .0889 to .3169
D) .0925 to .3413

Answer 2

.0925 to .3413

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

The χ² 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

Question 5

The χ² 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

Question 6

The symbol used for the variance of the population is _____.

A) σ
B) σ²
C) s
D) s²

Question 7

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

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

Question 8

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

A) between .025 and .05
B) between .05 and .01
C) .05
D) .025

Question 9

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 10

A sample of 41 observations yielded a sample standard deviation of 5. If we want to test H₀: σ² = 20, the test statistic is _____.

A) 100
B) 10
C) 51.25
D) 50

Question 11

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

Question 12

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 its new automobiles is _____.

A) 8.58 to 39.79
B) 4 to 16
C) 4 to 15
D) 1.64 to 1.96

Question 13

A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The χ² 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

Question 14

A sample of n observations is taken from a population. When performing statistical inference about a population variance, the appropriate χ² 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 15

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 its 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

Question 16

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 17

The random variable for a chi-square distribution may assume any _____.

A) value between -1 and 1
B) value between -infinity and +infinity
C) negative value
D) value greater than 0

Question 18

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

Question 19

The producer of a certain medicine claims that its bottling equipment is very accurate and that the standard deviation of all its filled bottles is 0.1 ounce or less. A sample of 20 bottles showed a standard deviation of .11. The test statistic to test the claim is _____.

A) 400
B) 22.99
C) 4.85
D) 20

Question 20

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 the _____.

A) larger sample size
B) smaller sample size
C) larger sample variance
D) smaller sample variance

Question 21

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

Question 22

The sampling distribution of the ratio of independent sample variances extracted from two normal populations with equal variances is the _____.

A) χ² distribution
B) normal distribution
C) Z distribution
D) t distribution

Question 23

The χ² value for a one-tailed test (lower tail) when the level of significance is .1 and the sample size is 15 is _____.

A) 21.0642
B) 23.6848
C) 7.78453
D) 6.57063

Question 24

A sample of 21 elements is selected to estimate a 90% confidence interval for the variance of the population. The χ² 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

Question 25

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

Question 26

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 the answers is correct.

Question 27

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

Question 28

Which of the following has a χ² distribution?

A) (n - 1)σ²/s²
B) (n - 2)σ²/s²
C) (n - 1)s/σ
D) (n - 1)s²/σ²

Question 29

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

Question 30

The sampling distribution of the $\frac { ( n - 1 ) s ^ { 2 } } { \sigma ^ { 2 } }$ quantity is the _____.

A) ?² distribution
B) normal distribution
C) F distribution
D) t distribution

Question 31

The value of F_0.01 with 9 numerator and 20 denominator degrees of freedom is _____.

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

Question 32

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 the answers is correct.

Question 33

To avoid the problem of having access to tables of the F distribution with values for the lower tail when a one-tailed test is required, let the _____ variance be the numerator of the test statistic.

A) smaller sample
B) larger sample
C) sample variance from the population with the smaller hypothesized
D) sample variance from the population with the larger hypothesized

Question 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

Question 35

Which of the following has an F distribution?

A) $\frac { ( n - 1 ) s } { \sigma }$
B) $\frac { s _ { 1 } } { s _ { 2 } }$
C) $\frac { ( n - 1 ) s _ { 1 } } { s _ { 2 } }$
D) $\frac { s _ { 1 } ^ { 2 } } { s _ { 2 } ^ { 2 } } n$

Question 36

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 the answers is correct.

Question 37

The sampling distribution of the ratio of two independent sample variances taken from normal populations with equal variances is a(n) _____ distribution.

A) F
B) χ²
C) t
D) normal

Question 38

In a hypothesis test about two population variances, the test statistic F is computed as _____.

A) $\frac { 1 } { \frac { s _ { 1 } ^ { 2 } } { s _ { 2 } ^ { 2 } } }$
B) $\frac { s _ { 1 } ^ { 2 } } { s _ { 2 } ^ { 2 } }$
C) $\frac { 1 } { \frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } } }$
D) $\frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } }$

Question 39

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 1
D) None of the answers is correct.

Question 40

The sampling distribution used when making inferences about a single population's variance is a(n) _____ distribution.

A) F
B) t
C) χ²
D) normal

Question 41

A company has claimed that the standard deviation of the monthly incomes of its employees is less than or equal to $120. To test its claim, a random sample of 15 employees of the company was taken, and it was determined that the standard deviation of their incomes was $135. At a 5% level of significance, test the company's claim.

Question 42

A lumber company has claimed that the standard deviation for the lengths of its six-foot boards is 0.5 inch or less. To test its claim, a random sample of 17 six-foot boards is selected, and it is determined that the standard deviation of the sample is .43. Do the results of the sample support the company's claim? Use α = .1.

Question 43

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 44

There is a .90 probability of obtaining a ?² value such that _____.

A) $\chi _ { .05 } ^ { 2 } \leq \chi ^ { 2 } \leq \chi _ { .95 } ^ { 2 }$
B) $\chi _ { .10 } ^ { 2 } \leq \chi ^ { 2 } \leq \chi _ { .90 } ^ { 2 }$
C) $\chi _ { .90 } ^ { 2 } \leq \chi ^ { 2 } \leq \chi _ { .10 } ^ { 2 }$
D) $\chi _ { .95 } ^ { 2 } \leq \chi ^ { 2 } \leq \chi _ { .05 } ^ { 2 }$

Question 45

A coach is deciding on whether to buy stopwatches from company A or company B. A test was set up to see how many seconds each stopwatch was off in a precise 10-minute test period. For samples of watches from companies A and B, the following information on stopwatch errors was found. Test the hypothesis of equal variances using a .05 level of significance. Be sure to state the null and alternative hypotheses being tested and the final conclusions of the test.

Question 46

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

Question 47

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 α = .05. What is your conclusion?
​
H₀: α² ≤ 80
H_a: α² > 80

Question 48

The average grade of a sample of 25 students on their second statistics examination was 85 with a standard deviation of 9. Is the variance of the population significantly more than 80? Use a .05 level of significance.

Question 49

At α = .1, test to see if the population variances from which the following samples were drawn are equal. ​

Question 50

Do the following data indicate that the variance of the population from which this sample has been drawn is 17? Use α = .05.

Question 51

Which of the following rejection rules is proper?

A) Reject H₀ if p-value $\geq$ F $\alpha$
B) Reject H₀ if p-value $\leq$ $x _ { a } ^ { 2 }$
C) Reject H₀ if p-value $\geq$ $x _ { a / 2 } ^ { 2 }$
D) Reject H₀ if p-value $\leq$ $\alpha$

Question 52

Two classes in business statistics showed the following results on a recent test. Carry out a test to determine whether the difference in the variance of the scores on this test is due to chance variation or is statistically significant with a .02 level of significance.

Question 53

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: Are the variances of the populations (from which these samples came) equal?
Use α = .05.

Question 54

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

Question 55

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 in the standard deviations of the ages of all the northern and southern executives.

Question 56

We are interested in determining whether or not the variances of the starting salaries of accounting majors are significantly different from the starting salaries of management majors. The following information was gathered from two samples: At a 5% level of significance, test to determine whether or not the variances are equal.
​

Question 57

The standard deviation of the daily temperatures in Honolulu last year was 3.2 degrees Fahrenheit. A random sample of 19 days resulted in a standard deviation of 4 degrees Fahrenheit. Has there been a significant change in the variance of the temperatures? Use α = .02.

Question 58

An egg-packing company has stated that the standard deviation of the weights of its grade A large eggs is 0.07 ounce or less. The sample variance for 51 eggs was 0.0065 ounce. Can this sample result confirm the company's claim? Use α = .1.

Question 59

Student advisors are interested in determining if the variances of the grades of day students and night students are the same. The following samples are drawn: Test the equality of the variances of the populations at 95% confidence.

Question 60

Test scores of two independent samples of students from UA and UB on a national examination are given below. At a .05 level of significance, test to determine if there is a significant difference between the variances of the two populations.

Question 61

The time it takes to complete a test was recorded for a sample of 15 students. The results follow. Use Excel to estimate the variance with a 90% level of confidence.

Question 62

The State Highway Patrol (SHP) periodically samples vehicle speeds at Milepost 92 on Interstate 17. The SHP is concerned about the dispersion of speeds of vehicles sharing the same highway because significant difference in speed is a known cause of accidents. The speeds of a random sample of 16 vehicles are shown below.
​ The SHP's policy is to position a patrol car at Milepost 92 on Interstate 17 if the vehicle speed variance at that location is believed to be greater than 6 mph. Conduct a hypothesis test (with a = .10) to determine whether the speed variance for all vehicles passing Milepost 92 exceeds 6 mph.

Question 63

The specifications for the filling of soft drink bottles is a variance of .05 (ounces)². A sample of 20 bottles of soft drink showed the following results: At a 5% level of significance, use Excel to determine whether or not the variance of the population is significantly different from .05.

Question 64

In a manufacturing production process, two machines are being compared. The data below show the diameters, measured in millimeters, of a sample of ball bearings produced by each machine. Use Excel to determine whether the variance in diameters differs between the two machines. Use a .05 level of significance.

Question 65

A sample of 10 earnings per share estimates is shown below. Use Excel to estimate the variance with a 95% level of confidence.

Question 66

The data below shows samples of annual salaries (in $1000s) for accountants in two different firms. Use Excel to determine whether the variance in salaries in Company A differs from that of Company B. Use a .10 level of significance.

Question 67

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

Question 68

A random sample of 20 observations showed a standard deviation of 8. At a 5% level of significance, test to see if the variance of the population is significantly less than 65.

Question 69

A manufacturing company is considering changing suppliers of a particular raw material. The standard deviation of days until delivery for the current supplier is 3. A sample of five delivery times is taken from the new supplier. The data follow:
2
2
1
5
3
The manufacturing company will use the new supplier if the variance in delivery time is less than that of the current supplier. Use Excel to determine whether the company should use the new supplier. Use a 10% level of significance.

Question 70

Do the following data indicate that the variance of the population from which this sample has been drawn is significantly more than 12? Use α = .05.

Question 71

A professor believes the variability in time to complete tests has increased. In the past, the variance in minutes to complete a particular test was 25. The time it takes to complete a test was recorded for a sample of 15 students. The results follow. Use Excel to determine if the variance in times has increased. Use a .02 level of significance.

Question 72

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