Deck 11: Statistical Inference Concerning Variance

Question 1

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

Answer 1

normal

Question 2

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

Answer 2

True

Question 3

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.

Answer 3

False

Question 4

The value of the test statistic to test the ratio of two population variances is .

Question 5

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

Question 6

Statistical inferences regarding σ² are based on the F distribution.

Question 7

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

Question 8

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

Question 9

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

Question 10

A right-tailed test for the ratio of two population variances examines whether is greater than .
.

Question 11

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

Question 12

The parameter of interest for inferences regarding the ratio of two population variances is their sum .

Question 13

The null hypothesis is rejected if the value of the test statistic exceeds .

Question 14

We can use the confidence intervals to implement _________ hypothesis tests regarding the population variance.

Question 15

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

Question 16

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

Question 17

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

Question 18

The sampling distribution of is the χ² distribution.

Question 19

The relevant value on the ____ tail of the distribution is .

Question 20

The distribution is negatively skewed.

Question 21

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)
B)
C)
D)

Question 22

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

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

Question 23

Statistical inferences pertaining to σ² are 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 24

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 25

We conduct the following hypothesis test .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 26

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 27

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 28

The F distribution depends on _____ degrees of freedom.

Question 29

Placing the larger sample variance in the numerator of the statistic allows one to focus only on the _____ tail of the distribution.

Question 30

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 31

All distributions are __________ skewed.

Question 32

Which of the following hypotheses depicts a right-tailed test for the population variance?

A)
B)
C)
D)

Question 33

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 34

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 critical value at a 5% significance level is

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

Question 35

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 36

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 37

A distribution tends to the ____,as the degrees of freedom increase.

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

Question 38

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 value of test statistic, is

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

Question 39

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 40

Which of the following factors is used to conduct hypothesis tests regarding the population variance?

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

Question 41

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) .
D)

Question 42

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

A)
B)
C)
D)

Question 43

Which of the following Excel functions is used to calculate the exact probability for a left-tailed is 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 44

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

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

Question 45

Which of the following is the value of x for which

A) 20.483
B) 18.307
C) 15.987
D) 4.1680

Question 46

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 47

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 ,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 48

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 49

Which of the following is the value of x for which ?

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

Question 50

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 an appropriate hypotheses to test Becky's concern?

A)
B)
C)
D)

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.Using the critical value approach at α = 0.05,which of the following is the correct conclusion for Becky's concern?

A) We reject H₀becausethe value of the test statistic is greater than .
B) We do not reject H₀ becausethe value of the test statistic is greater than .
C) We reject H₀becausethe value of the test statistic is less than
D) We do not reject H0becausethe value of the test statistic is less than

Question 52

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)
B)
C)
D)

Question 53

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 54

Which of the following Excel functions is used to determine the left-tailed value given any probability?

A) CHISQ.INV(Probability,n)
B) CHISQ.INV(Probability,n−2)
C) CHISQ.INV(Probability,Deg_freedom)
D) CHISQ.INV(Probability,n/2)

Question 55

A professor applies the variance in scores between two sections that he teaches.The students of each section took the same test.The random samples drawn from the observations 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 56

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 57

A two-tailed test is used to determine if two population variances differ.The null hypothesis takes the form H₀

A) .
B) .
C) .
D)

Question 58

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

Question 59

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 60

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 61

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.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 62

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 63

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)
B)
C)
D)

Question 64

A financial analyst maintains that the risk,measured by the variance,of investing in emerging markets is more than 280(%)².Data on 20 stocks from emerging markets revealed the following sample results: = 12.1% and s² = 361(%)².Assume that the returns are normally distributed.Which of the following is the correct conclusion of the financial analyst's claim?

A) Do not reject H₀.The financial analyst's claim is supported by the sample data at 1% significance level.
B) Reject H₀.The financial analyst's claim is not supported by the sample data at 1% significance level.
C) Do not reject H₀.The financial analyst's claim is not supported by the sample data at 1% significance level.
D) Reject H₀.The financial analyst's claim is supported by the sample data at 1% significance level.

Question 65

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 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 66

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 67

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.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 68

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.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)
B)
C)
D)

Question 69

A financial analyst maintains that the risk,measured by the variance,of investing in emerging markets is more than 280(%)².Data on 20 stocks from emerging markets revealed the following sample results: = 12.1% and s² = 361(%)² .Assume that the returns are normally distributed.Which of the following is the correct value of the test statistic?

A) = 36.19
B) = 24.50
C) = 24.50
D)

Question 70

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.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 71

The following are the competing hypotheses and the relevant summary statistics: 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 that the variances differ.
B) We do not reject the null hypothesis and conclude that the variances differ.
C) We reject the null hypothesis and cannot conclude that the variances differ.
D) We do not reject the null hypothesis and cannot conclude that the variances differ.

Question 72

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 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 73

The following are the competing hypotheses and the relevant summary statistics: . 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 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 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 75

The following are the competing hypotheses and the relevant summary statistics . . Which of the following statements is correct,regarding the assumptions for conducting the hypothesis test?

A) The samples are drawn from not normally distributed population.
B) The samples are dependent and taken from normally distributed population.
C) The difference of the sample variances is used to test the hypotheses.
D) The samples are independent and taken from normally distributed population.

Question 76

A financial analyst maintains that the risk,measured by the variance,of investing in emerging markets is more than 280(%)².Data on 20 stocks from emerging markets revealed the following sample results: = 12.1% and s² = 361(%)².Assume that the returns are normally distributed.Which of the following are appropriate hypotheses to test the analyst's claim?

A)
B)
C)
D)

Question 77

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 78

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 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

Question 79

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.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 80

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 8a.m.to 10 a.m.and evening 7p.m.to 9p.m. ,for the analysis.Assume that wait times are normally distributed. Construct a 90% confidence interval and test if the ratio of the population variances differs from one at the 10% significance level.Which of the following outcomes supports the test?

A) [1.47,5.99].The ratio of the population variances differs from one at the 10% significance level.
B) [1.19,7.35].The ratio of the population variances differs from one at the 10% significance level.
C) [1.47,5.99].The ratio of the population variances does not differ from one at the 10% significance level.
D) [1.19,7.35].The ratio of the population variances does not differ from one at the 10% significance level.