Deck 11: Statistical Inference Concerning Variance

Question 1

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

Answer 1

n - 1

Question 2

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

Answer 2

False

Question 3

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

Answer 3

True

Question 4

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

Question 5

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

Question 6

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 7

The sampling distribution of is the χ² distribution.

Question 8

The distribution is positively skewed.

Question 9

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 10

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

Question 11

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

Question 12

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 nonnormal populations.

Question 13

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 14

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 15

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

Question 16

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

Question 17

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

Question 18

Statistical inferences regarding σ² are based on the F distribution.

Question 19

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

Question 20

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

Question 21

Exhibit 11-1.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 listed below (in dollars. )Assume that profits are normally distributed. Refer to Exhibit 11-1.The appropriate null and alternate hypotheses are:

A)
B)
C)
D)

Question 22

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 23

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 24

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 25

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 26

Exhibit 11-2.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. Refer to Exhibit 11-2.Approximate the p-value.

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 27

Find the value of x for which = 0.05.

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

Question 28

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 29

Exhibit 11-1.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 listed below (in dollars. )Assume that profits are normally distributed. Refer to Exhibit 11-1.What is the value of the test statistic?

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

Question 30

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

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

Question 31

We conduct the following hypothesis test: versus .For a random sample of 15 observations,the sample standard deviation is 12.Approximate 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 32

Identify the hypothesis that depicts a right-tailed test for the population variance from the list of hypotheses given below.

A)
B)
C)
D)

Question 33

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 34

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 35

Exhibit 11-1.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 listed below (in dollars. )Assume that profits are normally distributed. Refer to Exhibit 11-1.Test Becky's concern using the critical value approach at α = 0.05.What is the result?

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

Question 36

Compute 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 to 86.207
B)7.476 to 77.512
C)8.617 to 78.125
D)7.755 to 70.313

Question 37

Given below are the values taken from a normal population. A 95% confidence interval for the population variance is closest to:

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

Question 38

Exhibit 11-2.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. Refer to Exhibit 11-2.Which of the following would be null and the alternate hypothesis to test if the cut-off is surpassed?

A)
B)
C)
D)

Question 39

For a 99% confidence level find with 6 degrees of freedom.

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

Question 40

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 41

A professor analyzes 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.Construct 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 42

Exhibit 11-2.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. Refer to Exhibit 11-2.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 43

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.Identify the 95% interval estimate 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 44

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.State the null and the alternate 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 45

Exhibit 11-3.The following are the competing hypotheses and the relevant summary statistics. Sample 1: n₁ = 10 Sample 2: n₂ = 9
Refer to Exhibit 11-3.Which of the following statements is true with regard to the assumptions for conducting the hypothesis test?

A)The samples are drawn from a nonnormal population.
B)The samples are dependent and taken from a normal population.
C)The difference of the sample variances is used to test the hypotheses.
D)The samples are independent and taken from a normal population.

Question 46

Which of the following Excel functions is used to calculate the exact probability for a left-tailed 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 47

Exhibit 11-3.The following are the competing hypotheses and the relevant summary statistics. Sample 1: n₁ = 10 Sample 2: n₂ = 9
Refer to Exhibit 11-3.What is the critical value at the 5% significance level?

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

Question 48

The result of placing a larger sample variance in the numerator of the F_(df1,df2) statistic allows us to

A)focus only on the right tail of the distribution.
B)arrive at a more accurate F_(df1,df2) statistic value.
C)focus only on the left tail of the distribution.
D)determine if the distribution is symmetric.

Question 49

Exhibit 11-3.The following are the competing hypotheses and the relevant summary statistics. Sample 1: n₁ = 10 Sample 2: n₂ = 9
Refer to Exhibit 11-3.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 50

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 to arrive at the results. Investment 1: = 33.53;n₁ = 8
Investment 2: = 44.76;n₂ = 8
Identify the relevant null and alternate hypotheses for this test.

A)
B)
C)
D)

Question 51

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 52

If independent samples of size n₁ and n₂ are drawn from normal populations with equal variances,then the value of the F_(df1,df2) statistic is calculated as ____.

A)
B)
C)
D)

Question 53

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 and ,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 54

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

A)
B)
C)
D)

Question 55

The degrees of freedom df₁ and df₂ for an F(df₁,df₂) distribution is given by _____ and ____.

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 56

Find the value of x given the equation P(F_(6,10) ≥ x)= 0.025.

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

Question 57

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 58

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.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.

Question 59

Which of the following characteristics is true with regard to the F distribution?

A)The F_(df1,df2) distribution is negatively skewed.
B)The values of the F_(df1,df2) distribution range from negative infinity to infinity.
C)The F_(df1,df2) distribution is the probability distribution of the ratio of two independent chi-square variables.
D)The shape of the F_(df1,df2) distribution is independent of the degrees of freedom.

Question 60

Exhibit 11-4.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. Refer to Exhibit 11-4.Test Beth's claim at the 5% significance level.What is the 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 61

A random sample of 21 pages is used to estimate the population variance of the number of typographical errors in a book.The sample mean and sample standard deviation are calculated as 7.34 and 5.11,respectively.Assume that the population is normally distributed.

A)Construct a 95% interval estimate of the population variance.
B)Construct a 99% interval estimate of the population variance.
C)Use your answers to discuss the impact of the confidence level on the width of the interval.

Question 62

Find the value x for which:

A) b. c. d.

Question 63

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.Compute the value of the test statistic.

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

Question 64

Consider the following hypotheses: Approximate the p-value based on the following sample information,where the sample is drawn from a normally distributed population.

A)s² = 500;n = 15
B)s² = 600;n = 10
C)Which of the above sample information enables us to reject the null hypothesis at α = 0.10?

Question 65

Find and under the following scenarios:

A)A 90% confidence level with n = 10.
B)A 95% confidence level with n = 15.
C)A 99% confidence level with n = 20.

Question 66

The supervisor of an automobile sales and service industry would like the variance of the number of automobiles serviced not to exceed 5 units-squared.He would recruit one more service executive if the variance exceeds this threshold.In a recent random sample of 14 services,he computes the sample standard deviation as 3.3.He believes that the automobile services are normally distributed.

A)State the null and the alternative hypotheses to test if the threshold has been crossed.
B)Use the critical value approach to conduct the test at α = 0.01.
C)What should the supervisor do?

Question 67

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.Calculate 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 68

Exhibit 11-6.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.Here are some relevant summary statistics. Refer to Exhibit 4-6.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 69

A random sample of 10 homes sold during last month in the city of Fontana had the following number of bedrooms in each house: 4,2,3,5,4,3,3,4,5,3.At α = 0.01,test the claim that the variance of a normally distributed population is less than 1 using (a)the p-value approach and (b)the critical value approach.

Question 70

The sample standard deviation of the monthly sales (in thousands of dollars)of a telecommunications firm in U.S.for two years,2010 and 2011,is computed as 6.7.Assuming that the sales data are drawn from a normally distributed population,conduct the following hypothesis tests for the population variance.Use the critical value approach at α = 0.05.

A) b.

Question 71

Exhibit 11-5.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 am to 10 am and evening 7 pm to 9 pm,for the analysis.Assume that wait times are normally distributed. Refer to Exhibit 11-5.At the 10% significance level,what is the 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 72

Consider the following hypotheses: A sample variance of 98 is obtained from a random sample of 10 observations drawn from a normally distributed population.At 1% significance level,conduct a test for the population variance using the p-value approach.

Question 73

Exhibit 11-6.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.Here are some relevant summary statistics. Refer to Exhibit 11-6.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 74

The following table shows the annual returns (in percent)for the Fidelity Utilities fund from 2006 through 2010.

A)Calculate the point estimate of σ.
B)Construct a 95% confidence interval of σ.

Question 75

Find the value x for which:

A) b. c. d.

Question 76

Exhibit 11-6.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.Here are some relevant summary statistics. Refer to Exhibit 11-6.For the competing hypotheses: since ,approximate the p-value for the test.

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 77

Exhibit 11-6.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.Here are some relevant summary statistics. Refer to Exhibit 11-6.Repeat the hypothesis test, since .What is the critical F value at the 10% significance level?

A)
B)
C)
D)

Question 78

The sample mean and the sample standard deviation of annual returns on a stock over a 12-year period as computed by a stock analyst were 18% and 12%,respectively.The analyst wants to know if the risk,as measured by the standard deviation,differs from 15%.

A)Construct a 90% confidence interval of the population variance and the population standard deviation.
B)What assumption is required in constructing the confidence interval?
C)Based on the above confidence interval,does the risk differ from 15%?

Question 79

The following data,drawn from a normal population,is the waiting time (in minutes)of 12 randomly selected customers at a travel desk.
1.8,2.4,2.7,2.1,1.3,1.7,1.5,1.5,2.3,2.8,2.5,1.9

A)Calculate the point estimates of the population variance and the population standard deviation.
B)Compute a 90% confidence interval for the population variance and the population standard deviation.

Question 80

Consider the following hypotheses: A sample variance of 98 is obtained from a random sample of 10 observations drawn from a normally distributed population.At 1% significance level,conduct a test for the population variance using the critical value approach.