Deck 11: Statistical Inference Concerning Variance

Question 1

We can use Excel's function ________ that returns the right-tailed probability of the chi-square distribution.

Answer 1

CHISQ.DIST.RT

Question 2

Statistical inference for σ² is based on the F distribution.

Answer 2

False

Question 3

The null hypothesis H₀: σ² ≤ is rejected if the value of the test statistic exceeds .

Answer 3

True

Question 4

Use the R function ________ to obtain left-tail probabilities of the chi-square distribution.

Question 5

The values of the distribution range from negative infinity to infinity.

Question 6

The population variance is one of the most widely used quantitative measures of risk in investments.

Question 7

The skewness of the chi-square distribution depends on the degrees of freedom.

Question 8

The formula for constructing the confidence interval for the ratio of two population variances is based on the assumption that the sample variances are computed from independently drawn samples from two non-normally distributed populations.

Question 9

In general, the distribution is the probability distribution of the sum of several independent squared standard ________ random variables.

Question 10

Inference for two population variances is done through their difference - .

Question 11

The estimator of / used in the inference regarding the ratio of two population variances is / .

Question 12

The distribution is negatively skewed.

Question 13

A right-tailed test for the ratio of two population variances Η₀: / ≤ 1 examines whether is greater than .

Question 14

The value of the test statistic for the hypothesis test of the population variance, σ² is computed as = .

Question 15

For a test about the ratio of two population variances, the test statistic is given by / .

Question 16

The relevant value in the ________ tail of the distribution is .

Question 17

It is preferable to place the smaller sample variance in the numerator of the statistic.

Question 18

The formula for the confidence interval of the population variance σ² is valid for the random samples drawn from any population.

Question 19

The sampling distribution of / is the χ² distribution.

Question 20

The formula for the confidence interval of the population variance σ² is valid only when the random sample is drawn from a ________ distributed population.

Question 21

The F distribution depends on ________ degrees of freedom.

Question 22

If s² is computed from a random sample of n observations drawn from an underlying normal population with a finite variance, then the variable is defined as

A) = .
B) = .
C) = .
D) = .

Question 23

The ________ is the probability distribution of the sum of several independent squared standard normal random variables.

A) F distribution
B) distribution
C) student's t distribution
D) uniform distribution

Question 24

The Excel's function ________ returns the p-value for a right-tailed test for the ratio of two population variances.

Question 25

Which of the following is a 98% confidence interval for the population variance when the sample variance is 20 for a sample of 10 items from a normal population?

A) [8.308, 86.207]
B) [7.476, 77.512]
C) [8.617, 78.125]
D) [7.755, 70.313]

Question 26

As the df grow larger, the distribution approaches the

A) F distribution
B) uniform distribution
C) student's t distribution
D) normal distribution

Question 27

Which of the following is a feature of the F distribution?

A) The F distribution depends on one degree of freedom.
B) The F distribution is bell-shaped with values ranging from negative infinity to infinity.
C) The F distribution becomes increasingly symmetric when the degrees of freedom increase.
D) The F distribution is negatively skewed.

Question 28

Which of the following is used to conduct a hypothesis test about the population variance?

A) Sample mean
B) Population mean
C) Sample proportion
D) Sample variance

Question 29

If a sample of size n is taken from a normal population with a finite variance, then the statistic = follows the distribution with degrees of freedom.

A) (n + 1)(n − 1).
B) n + 1.
C) n − 1.
D) n.

Question 30

The specification of the confidence interval for the ratio of two population variances is based on the assumption that the sample variances are computed from ________ drawn samples from two normally distributed populations.

Question 31

If P( ≥ x) = 0.05, then the value of x is

A) 14.449.
B) 10.645.
C) 12.592.
D) 1.6350.

Question 32

Which of the following is the value of for a 99% confidence level and degrees of freedom equal 6?

A) 16.812
B) 18.548
C) 0.872
D) 0.676

Question 33

All F₍
, ) distributions are ________ skewed.

Question 34

The values taken from a normally distributed population are 21 23 25 27 28 35 30 32 33.
Which of the following is a 95% confidence interval for the population variance?

A) [2.03, 16.30]
B) [10.12, 81.43]
C) [9.00, 72.41]
D) [11.39, 91.64]

Question 35

For a sample of 10 observations drawn from a normally distributed population, we obtain the sample mean and the sample variance as 50 and 75, respectively. We want to determine whether the population variance is greater than 70. The test statistic is

A) 1.645.
B) 3.325.
C) 9.642.
D) 16.919.

Question 36

Which of the following is the formula for the sample variance s² when used as an estimate of σ² for a random sample of n observations from a population?

A) s² =
B) s² =
C) s² =
D) s² =

Question 37

Statistical inference about σ² is based on which of the following distributions?

A) The F distribution
B) The student's t distribution
C) The chi-square distribution
D) The normal distribution

Question 38

Use the R function ________ to obtain a p-value for a test about the ratio of two population variances.

Question 39

Which of the following hypotheses is a right-tail test about the population variance?

A) Η₀: σ² = , Η_A: σ² ≠
B) Η₀: σ² ≤ , Η_A: σ² >
C) Η₀: σ² > , Η_A: σ² <
D) Η₀: σ² ≥ , Η_A: σ² >

Question 40

You want to test whether the population variance differs from 50. From a sample of 25 observations drawn from a normally distributed population, you calculate s² = 80. When conducting this test at the 5% significance level, the, critical value is

A) 5.625.
B) 12.401.
C) 14.400.
D) 39.364.

Question 41

Which of the following Excel functions is used to calculate the exact left-tail probability for a distribution?

A) CHISQ.DIST(x, Deg_freedom, Cumulative)
B) CHISQ.DIST(x, n−2)
C) CHISQ.DIST(x, n/2)
D) CHISQ.DIST(x/2, Deg_freedom, Cumulative)

Question 42

A random sample of 18 observations is taken from a normal population. The sample mean and sample standard deviation are 76.4 and 4.2, respectively. What is an 80% interval estimate of the population variance?

A) [12.107, 29.735]
B) [10.870, 34.581]
C) [12.819, 31.484]
D) [14.636, 23.443]

Question 43

A professor wants to compare the variances of scores between two sections of classes. The students in each section took the same test. The random samples yield sample variances of = 203.15 and = 474.42 for samples of n₁ = 13 and n₂ = 16, respectively. Which of the following is a 99% confidence interval for the ratio of the population variances?

A) [0.1540, 2.7809]
B) [0.1008, 2.0217]
C) [0.1386, 3.0895]
D) [0.0907, 1.8198]

Question 44

How does the width of the interval respond to the changes in the confidence level?

A) The width of the interval decreases with an increase in the confidence level.
B) The width of the interval increases with an increase in the confidence level.
C) The width of the interval is halved with the increase in the confidence level.
D) The width of the interval is doubled with the decrease in the confidence level.

Question 45

If independent samples of size n₁ and n₂ are drawn from normal populations with equal variances, then the value of the statistic is calculated as

A) × .
B) / .
C) (n₁ - 1) × (n₂ - 1).
D) / .

Question 46

Which of the following are the degrees of freedom df₁ and df₂ for an distribution?

A) (n₁ − 2); (n₂ − 2)
B) n₂(n₁ − 2); n₁(n₂ − 2)
C) (n₁ − 1); (n₂ − 1)
D) n(n₁ − 1); n(n₂ − 1)

Question 47

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared. He would like to add an additional billing counter if the variance exceeds the cut-off. He checks the recent sample data. For a random sample of 24 customer waiting times, he arrives at a sample variance of 3.8 minutes-squared. The manager assumes the waiting times to be normally distributed. At α = 0.05, which of the following is the critical value ?

A) 13.091
B) 32.007
C) 35.172
D) 38.076

Question 48

Which of the following characteristics is true regarding the F distribution?

A) The distribution is negatively skewed.
B) The values of the distribution range from negative infinity to infinity.
C) The distribution is the probability distribution of the ratio of two independent chi-square variables.
D) The shape of the distribution is independent of the degrees of freedom.

Question 49

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared. He would like to add an additional billing counter if the variance exceeds the cut-off. He checks the recent sample data. For a random sample of 24 customer waiting times, he arrives at a sample variance of 3.8 minutes-squared. The manager assumes the waiting times to be normally distributed. Which of the following would be null and the alternate hypothesis to test if the cut-off is surpassed?

A) Η₀: σ² ≤ 2.3, Η_A: σ² > 2.3.
B) Η₀: σ² = 2.3, Η_A: σ² ≠ 2.3.
C) Η₀: σ² ≥2.3, Η_A: σ² < 2.3.
D) Η₀: σ² < 2.3, Η_A: σ² ≥ 2.3.

Question 50

For a two-tailed test about two population variances, the null hypothesis is given by

A) = 1 / .
B) - 1 = .
C) + = 1.
D) / = 1.

Question 51

Becky owns a diner and is concerned about sustaining the business. She wants to ascertain if the standard deviation of the profits for each week is greater than $250. The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916
Assume that profits are normally distributed. Which of the following are appropriate hypotheses to test Becky's concern?

A) Η₀: σ² = 62,500, Η_A: σ² ≠ 62,500
B) Η₀: σ² < 62,500, Η_A: σ² ≥ 62,500
C) Η₀: σ² ≤ 62,500, Η_A: σ² > 62,500
D) Η₀: σ² ≥ 62,500, Η_A: σ² < 62,500

Question 52

Students of two sections of a history course took a common final examination. The course instructor examines the variance in scores between the two sections. He selects random samples of n₁ = 11 and n₂ = 16 with sample variances of = 400 and = 200, respectively. Suppose you obtain a 95% confidence for the ratio of the population variances. Which of the below allows you to conclude the first variance is smaller than the second variance?

A) The entire interval is less than 1
B) The interval captures 1
C) The entire interval is more than 1
D) None of the above

Question 53

Which of the following R functions is used to calculate the exact left-tail probability for a distribution?

A) 1-pchisq(x, df)
B) 1-pchisq(x, df, lower.tail=TRUE)
C) pchisq(x, df)
D) None of the above

Question 54

The manager of a video library would like the variance of the waiting times of the customers not to exceed 2.30 minutes-squared. He would like to add an additional billing counter if the variance exceeds the cut-off. He checks the recent sample data. For a random sample of 24 customer waiting times, he arrives at a sample variance of 3.8 minutes-squared. The manager assumes the waiting times to be normally distributed. Which of the following is the correct approximation of the p-value used to conduct this test?

A) p-value lies between 0.005 and 0.010
B) p-value lies between 0.010 and 0.025
C) p-value lies between 0.050 and 0.10
D) p-value lies between 0.025 and 0.05

Question 55

We conduct the following hypothesis test H₀: σ² = 82, H_A: σ² ≠ 82. For a random sample of 15 observations, the sample standard deviation is 12. Which of the following is the correct approximation of the p-value used to conduct this test?

A) p-value lies between 0.025 and 0.05
B) p-value lies between 0.01 and 0.025
C) p-value lies between 0.05 and 0.10
D) p-value is greater than 0.10

Question 56

Which of the following is the formula for a confidence interval for the ratio of the population variances / ?

A)
B)
C)
D)

Question 57

Becky owns a diner and is concerned about sustaining the business. She wants to ascertain if the standard deviation of the profits for each week is greater than $250. The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916
Assume that profits are normally distributed. Which of the following is the correct value of the test statistic?

A) 6.146
B) 8.604
C) 6.652
D) 7.375

Question 58

Becky owns a diner and is concerned about sustaining the business. She wants to ascertain if the standard deviation of the profits for each week is greater than $250. The details of the profits for the week are (in dollars) 1,743 1,438 1,212 1,705 1,985 1,857 1,916
Assume that profits are normally distributed. Using the critical value approach at α = 0.05, which of the following is the correct conclusion for Becky's concern?

A) We reject H₀ because the value of the test statistic is greater than = 1.635.
B) We do not reject H₀ because the value of the test statistic is greater than = 1.635.
C) We reject H₀ because the value of the test statistic is less than = 12.592.
D) We do not reject H₀ because the value of the test statistic is less than = 12.592.

Question 59

Which of the following is the value of x for which P = 0.025?

A) 4.07
B) 5.46
C) 3.22
D) 5.39

Question 60

Students of two sections of a history course took a common final examination. The course instructor examines the variance in scores between the two sections. He selects random samples of n₁ = 11 and n₂ = 16 with sample variances of and = 200, respectively. Assuming that the population distributions are normal, construct a 90% confidence interval for the ratio of the population variance.

A) [0.90, 2.41]
B) [0.50, 2.00]
C) [0.25, 4.00]
D) [0.79, 5.70]

Question 61

Construct a 95% confidence interval for the ratios of two population variances. The random samples of n₁= 9 and n₂= 11 with sample variances of = 500 and = 250, respectively. Assume that the samples were drawn from a normal population.

A) [0.50, 2.00]
B) [0.52, 8.60]
C) [0.25, 1.41]
D) [0.44, 4.30]

Question 62

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m. and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. Which of the following is the correct approximation of the p-value?

A) p-value is greater than 0.1.
B) p-value lies between 0.025 and 0.05.
C) p-value lies between 0.05 and 0.10.
D) p-value is greater than 0.2.

Question 63

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m. and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. At the 10% significance level, which of the following is the correct conclusion?

A) Do not reject H₀. We cannot conclude that the variance of wait time during morning peak hours differs from that during the evening peak hours.
B) Reject H₀. We conclude that the variance of wait time during morning peak hours differs from that during the evening peak hours.
C) Do not reject H₀. We conclude that the variance of wait time during morning peak hours differs from that during the evening peak hours.
D) Reject H₀. We cannot conclude that the variance of wait time during morning peak hours differs from that during the evening peak hours.

Question 64

A financial analyst examines the performance of two mutual funds and claims that the variances of the annual returns for the bond funds differ. To support his claim, he collects data on the annual returns (in percent) for the years 2001 through 2010. The analyst assumes that the annual returns for the two emerging market bond funds are normally distributed. Use the following summary statistics. The competing hypotheses are Η₀: / = 1, Η_A: / ≠ 1, At α = 0.10, is the analyst's claim supported by the data?

A) No, the p-value < α = 0.10.
B) Yes, the p-value > α = 0.10.
C) No, the p-value > α = 0.10.
D) Yes, the p-value < α = 0.10.

Question 65

Consider the expected returns (in percent) from the two investment options. Beth claims that the variances of the returns for the two investments differ. Use the following data to arrive at the results. Which of the following is the correct p-value?

A) 0.2873
B) 0.7127
C) 0.3564
D) 0.6436

Question 66

A financial analyst examines the performance of two mutual funds and claims that the variances of the annual returns for the bond funds differ. To support his claim, he collects data on the annual returns (in percent) for the years 2001 through 2010. The analyst assumes that the annual returns for the two emerging market bond funds are normally distributed. Use the following summary statistics. For the competing hypotheses Η₀: / = 1, Η_A: / ≠ 1, which of the following is the correct approximation of the p-value?

A) Less than 0.01
B) Between 0.01 and 0.025
C) Between 0.02 and 0.05
D) Between 0.05 and 0.10

Question 67

The following are the competing hypotheses and the relevant summary statistics Η₀: / ≤ 1, Η_A: / > 1. Which of the following statements is correct, regarding the assumptions for conducting the hypothesis test?

A) The samples are drawn from populations that are not normally distributed.
B) The values in one group are related to the values in the other group.
C) The difference of the sample variances is used to test the hypotheses.
D) The samples are independent and drawn from normally distributed populations.

Question 68

The following are the competing hypotheses and the relevant summary statistics: Η₀: / ≤ 1, Η_A: / > 1. Which of the following is the critical value at the 5% significance level?

A) 3.02
B) 3.14
C) 3.23
D) 3.39

Question 69

Consider the expected returns (in percent) from the two investment options. Beth claims that the variances of the returns for the two investments differ. Use the following data to arrive at the results. Test Beth's claim at the 5% significance level. Which of the following is the correct conclusion?

A) p-value = 0.7127 > α = 0.05; Beth's claim is correct.
B) p-value = 0.7127 > α = 0.05; Beth's claim is wrong.
C) p-value = 0.7127 < α = 0.05; Beth's claim is wrong.
D) p-value = 0.7127 < α = 0.05; Beth's claim is correct.

Question 70

The following are the competing hypotheses and the relevant summary statistics: Η₀: / ≤ 1, Η_A: / > 1. The p-value associated with the value of the test statistic is 0.3692. At the 5% significance level, which of the following conclusions is correct?

A) We reject the null hypothesis and conclude the first variance is larger than the second.
B) We do not reject the null hypothesis and conclude the first variance is larger than the second.
C) We reject the null hypothesis and cannot conclude the first variance is larger than the second.
D) We do not reject the null hypothesis and cannot conclude the first variance is larger than the second.

Question 71

Consider the expected returns (in percent) from two investment options. Beth wants to determine if investment 1 has a lower variance. Use the following summary statistics. Which of the following are the competing hypotheses for this test?

A) Η₀: / ≤ 1, Η_A: / > 1
B) Η₀: / ≤ 1, Η_A: / > 1
C) Η₀: / = 1, Η_A: / ≠ 1
D) Η₀: / ≥ 1, Η_A: / < 1

Question 72

Which of the following Excel's functions is used to calculate the right-tailed probability for a value x on the distribution?

A) F.DIST.RT(x, Deg_freedom1, Deg_freedom2, Cumulative)
B) F.DIST.RT(x, n₁, n₂)
C) F.DIST.RT(x, n₁−1, n₂−2)
D) F.DIST.RT(x, Deg_freedom1, Deg_freedom2)

Question 73

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m. and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. A 90% confidence interval is found to be [1.19, 7.36], where the morning is the first group and the evening is the second group. Which of the following is the correct conclusion?

A) We cannot conclude the variance of the wait times for the morning hours is different from the variance of the wait times for the evening hours.
B) We can conclude the variance of the wait times for the morning hours is more than the variance of the wait times for the evening hours.
C) We can conclude the variance of the wait times for the evening hours is more than the variance of the wait times for the morning hours.
D) We can conclude the variance of the wait times for the evening hours is equal to the the variance of the wait times for the morning hours.

Question 74

A financial analyst examines the performance of two mutual funds and claims that the variances of the annual returns for the bond funds differ. To support his claim, he collects data on the annual returns (in percent) for the years 2001 through 2010. The analyst assumes that the annual returns for the two emerging market bond funds are normally distributed. Use the following summary statistics. The competing hypotheses are Η₀: / = 1, Η_A: / ≠ 1. At α = 0.10, is the analyst's claim supported by the data using the critical value approach?

A) No, because the value of the test statistic is less than the critical F value.
B) Yes, because the value of the test statistic is less than the critical F value.
C) Yes, because the value of the test statistic is greater than the critical F value.
D) No, because the value of the test statistic is greater than the critical F value.

Question 75

The result of placing a larger sample variance in the numerator of the test statistic allows us to

A) focus only on the right tail of the distribution.
B) arrive at a more accurate statistic value.
C) focus only on the left tail of the distribution.
D) determine if the distribution is symmetric.

Question 76

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m. and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. Which of the following is the correct value of the test statistic?

A) 1.72
B) 2.96
C) 1.66
D) 0.34

Question 77

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m. and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. Which of the following is the 95% confidence interval for the ratio of the population variances?

A) [1.02, 8.55]
B) [1.00, 8.73]
C) [0.99, 8.83]
D) [1.19, 7.34]

Question 78

Amie Jackson, a manager at Sigma travel services, makes every effort to ensure that customers attempting to make online reservations do not have to wait too long to complete the reservation process. The travel website is open for reservations 24 hours a day, and Amie regularly checks the website for the waiting time to maintain consistency in service. She uses the following independently drawn samples of wait time during two peak hours, morning 8 a.m. to 10 a.m., and evening 7 p.m. to 9 p.m., for the analysis. Assume that wait times are normally distributed. Which of the following is the correct hypotheses to determine if the variance of wait time during morning peak hours (population 1) differs from that during the evening peak hours (population 2)?

A) Η₀: / ≤ 1, Η_A: / > 1
B) Η₀: / ≤ 1, Η_A: / > 1
C) Η₀: / = 1, Η_A: / ≠ 1
D) Η₀: / ≥ 1, Η_A: / < 1

Question 79

Which of the following R functions is used to obtain a right-tail probability for a value x of the distribution?

A) pf(x, df₁, df₂)
B) 1-pf(x, df₁, df₂)
C) pf(x, df₂, df₁)
D) 1-pf(x, df₂, df₁)

Question 80

A financial analyst examines the performance of two mutual funds and claims that the variances of the annual returns for the bond funds differ. To support his claim, he collects data on the annual returns (in percent) for the years 2001 through 2010. The analyst assumes that the annual returns for the two emerging market bond funds are normally distributed. Use the following summary statistics. The competing hypotheses are Η₀: / = 1, Η_A: / ≠ 1. Which of the following is the critical F value at the 10% significance level?

A) F_0.10,(9,9) = 2.44
B) F_0.05,(10,10) = 2.98
C) F_0.05,(9,9) = 3.18
D) F_0.10,(10,10) = 2.32