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Essentials of Statistics for Business & Economics

David R. Anderson; Dennis J. Sweeney; Thomas A. Williams; Jeffrey D. Camm; James J. Cochran

Chapter 11

Inferences About Population Variances - all with Video Answers

Educators

Chapter Questions

01:18

Problem 1

Find the following chi-square distribution values from Table 11.1 or Table 3 of Appendix B.
a. $\chi_{\cos }^2$ with $d f=5$
b. $\chi_{\cos }^2$ with $d f=15$
c. $\chi_{\text {ors }}$ with $d f=20$
d. $\chi_{\text {Dol }}^2$ with $d f=10$
e. $\chi_{\text {sos }}^2$ with $d f=18$

Nick Johnson
Nick Johnson
Numerade Educator
02:38

Problem 2

A sample of 20 items provides a sample standard deviation of 5 .
a. Compute the $90 \%$ confidence interval estimate of the population variance.
b. Compute the $95 \%$ confidence interval estimate of the population variance.
c. Compute the $95 \%$ confidence interval estimate of the population standard deviation.

Nick Johnson
Nick Johnson
Numerade Educator
02:04

Problem 3

A sample of 16 items provides a sample standard deviation of 9.5 . Test the following hypotheses using $\alpha=.05$. What is your conclusion? Use both the $p$-value approach and the critical value approach.
$$
\begin{aligned}
& H_0: \sigma^2 \leq 50 \\
& H_{\mathrm{x}}: \sigma^2>50
\end{aligned}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:33

Problem 4

Package Delivery by Drones. Amazon.com is testing the use of drones to deliver packages for same-day delivery. In order to quote narrow time windows, the variability in delivery times must be sufficiently small. Consider a sample of 24 drone deliveries with a sample variance of $s^2=.81$.
a. Construct a $90 \%$ confidence interval estimate of the population variance for the drone delivery time.
b. Construct a $90 \%$ confidence interval estimate of the population standard deviation.

Nick Johnson
Nick Johnson
Numerade Educator
07:33

Problem 5

College Basketball Coaches' Salaries. In 2018, Mike Krzyewski and John Calipari topped the list of highest-paid college basketball coaches (Sports Illustrated website, https:/www.si.com/college-basketball/2018/03/01/highest-paid-college-basketball -coaches-salaries-mike-krzyewski-john-calipari). The sample below shows the head basketball coach's salary for a sample of 10 schools playing NCAA Division I basketball. Salary data are in millions of dollars.$$
\begin{array}{lclc}
\text { University } & \text { Coach's Salary } & \text { University } & \text { Coach's Salary } \\
\text { North Carolina State } & 2.2 & \text { Miami (FL) } & 1.5 \\
\text { lona } & .5 & \text { Creighton } & 1.3 \\
\text { Texas A\&M } & 2.4 & \text { Texas Tech } & 1.5 \\
\text { Oregon } & 2.7 & \text { South Dakota State } & .3 \\
\text { lowa State } & 2.0 & \text { New Mexico State } & .3
\end{array}
$$
a. Use the sample mean for the 10 schools to estimate the population mean annual salary for head basketball coaches at colleges and universities playing NCAA Division I basketball.
b. Use the data to estimate the population standard deviation for the annual salary for head basketball coaches.
c. What is the $95 \%$ confidence interval for the population variance?
d. What is the $95 \%$ confidence interval for the population standard deviation?

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
03:26

Problem 6

Volatility of General Electric Stock. To analyze the risk, or volatility, associated with investing in General Electric common stock, consider a sample of the eight quarterly percent total returns. The percent total return includes the stock price change plus the dividend payment for the quarter.
$20.0 \quad-20.5$
12.2
12.6
10.5
$-5.8$
$-18.7$
a. What is the value of the sample mean? What is its interpretation?
b. Compute the sample variance and sample standard deviation as measures of volatility for the quarterly return for General Electric.
c. Construct a $95 \%$ confidence interval for the population variance.
d. Construct a $95 \%$ confidence interval for the population standard deviation.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:34

Problem 7

Halloween Spending. In 2017, Americans spent a record-high $$\$ 9.1$$ billion on Halloween-related purchases (the balance website, https://www.thebalance.com halloween-spending-statistics-facts-and-trends-3305716). Sample data showing the amount, in dollars, 16 adults spent on a Halloween costume are as follows.
$$
\begin{array}{llll}
12 & 69 & 22 & 64 \\
33 & 36 & 31 & 44 \\
52 & 16 & 13 & 98 \\
45 & 32 & 63 & 26
\end{array}
$$
a. What is the estimate of the population mean amount adults spend on a Halloween costume?
b. What is the sample standard deviation?
c. Provide a $95 \%$ confidence interval estimate of the population standard deviation for the amount adults spend on a Halloween costume.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:05

Problem 8

Variability in Daily Change in Stock Price. Consider a day when the Dow Jones Industrial Average went up 149.82 points. The following table shows the stock price changes for a sample of 12 companies on that day.
a. Compute the sample variance for the daily price change.
b. Compute the sample standard deviation for the price change.$$
\begin{aligned}
&\text { Price Change }\\
&\begin{array}{lc}
\text { Company } & \text { (\$) } \\
\text { Aflac } & .81 \\
\text { Altice USA } & .41 \\
\text { Bank of America } & -.05 \\
\text { Diageo plc } & 1.32 \\
\text { Fluor Corporation } & 2.37 \\
\text { Goodrich Petroleum } & .3
\end{array}
\end{aligned}
$$
$$
\begin{aligned}
&\text { Price Change }\\
&\begin{array}{lr}
\text { Company } & \text { (\$) } \\
\text { Johnson \& Johnson } & 1.46 \\
\text { Lows Corporation } & .92 \\
\text { Nokia Corporation } & .21 \\
\text { Sempra Energy } & .97 \\
\text { Sunoco LP } & .52 \\
\text { Tyson Foods, Inc. } & .12
\end{array}
\end{aligned}
$$
c. Provide $95 \%$ confidence interval estimates of the population variance and the population standard deviation.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:35

Problem 9

Aerospace Part Manufacturing. The competitive advantage of small American factories such as Tolerance Contract Manufacturing lies in their ability to produce parts with highly narrow requirements, or tolerances, that are typical in the aerospace industry. Consider a product with specifications that call for a maximum variance in the lengths of the parts of $\mathbf{0} 0004$. Suppose the sample variance for 30 parts turns out to be $s^2=.0005$. Use $\alpha=.05$ to test whether the population variance specification is being violated.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:50

Problem 10

Costco Customer Satisfaction. Consumer Reports uses a 100 -point customer satisfaction score to rate the nation's major chain stores. Assume that from past experience with the satisfaction rating score, a population standard deviation of $\sigma=12$ is expected. In 2012 , Costco, with its 432 warehouses in 40 states, was the only chain store to earn an outstanding rating for overall quality. A sample of 15 Costco customer satisfaction scores follows.
$$
\begin{array}{lllll}
95 & 90 & 83 & 75 & 95 \\
98 & 80 & 83 & 82 & 93 \\
86 & 80 & 94 & 64 & 62
\end{array}
$$
a. What is the sample mean customer satisfaction score for Costco?
b. What is the sample variance?
c. What is the sample standard deviation?
d. Construct a hypothesis test to determine whether the population standard deviation of $\sigma=12$ should be rejected for Costco. With a .05 level of significance, what is your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:08

Problem 11

Variability in GMAT Scores. In 2016, the Graduate Management Admission Council reported that the variance in GMAT scores was 14,660 . At a recent summit, a group of economics professors met to discuss the GMAT performance of undergraduate students majoring in economics. Some expected the variability in GMAT scores achieved by undergraduate economics students to be greater than the variability in GMAT scores of the general population of GMAT takers. However, others took the opposite view. The file EconGMAT contains GMAT scores for 51 randomly selected undergraduate students majoring in economics.
a. Compute the mean, variance, and standard deviation of the GMAT scores for the 51 observations.
b. Develop hypotheses to test whether the sample data indicate that the variance in GMAT scores for undergraduate students majoring in economics differs from the general population of GMAT takers.
c. Use $\alpha=.05$ to conduct the hypothesis test formulated in part (b). What is your conclusion?

Dominador Tan
Dominador Tan
Numerade Educator
02:21

Problem 12

Vehicle Ownership by Fortune Magazine Subscribers. A Forrune study found that the variance in the number of vehicles owned or leased by subscribers to Forrune magazine is 94 . Assume a sample of 12 subscribers to another magazine provided the following data on the number of vehicles owned or leased: $2,1,2,0,3,2,2,1,2,1,0$, and 1 .
a. Compute the sample variance in the number of vehicles owned or leased by the 12 subscribers.
b. Test the hypothesis $H_0: \sigma^2=.94$ to determine whether the variance in the number of vehicles owned or leased by subscribers of the other magazine differs from $\sigma^2=.94$ for Fortune. At a .05 level of significance, what is your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:23

Problem 13

Find the following $F$ distribution values from Table 4 of Appendix B.
a. $F_{05}$ with degrees of freedom 5 and 10
b. $F_{\text {J25 }}$ with degrees of freedom 20 and 15
c. $F_{01}$ with degrees of freedom 8 and 12
d. $F_{10}$ with degrees of freedom 10 and 20

Nick Johnson
Nick Johnson
Numerade Educator
01:09

Problem 14

A sample of 16 items from population I has a sample variance $s_1^2=5.8$ and a sample of 21 items from population 2 has a sample variance $s_2^2=2.4$. Test the following hypotheses at the .05 level of significance.
$$
\begin{aligned}
& H_0: \sigma_1^2 \leq \sigma_2^2 \\
& H_{\mathrm{a}}: \sigma_1^2>\sigma_2^2
\end{aligned}
$$
a. What is your conclusion using the p-value approach?
b. Repeat the test using the critical value approach.

Dominador Tan
Dominador Tan
Numerade Educator
02:45

Problem 15

Consider the following hypothesis test.
$$
\begin{aligned}
& H_0: \sigma_1^2=\sigma_2^2 \\
& H_{\mathrm{a}}: \sigma_1^2 \neq \sigma_2^2
\end{aligned}
$$
a. What is your conclusion if $n_1=21, s_1^2=8.2, n_2=26$, and $s_2^2=4.0$ ? Use $\alpha=.05$ and the $p$-value approach.
b. Repeat the test using the critical value approach.

Nick Johnson
Nick Johnson
Numerade Educator
02:58

Problem 16

Comparing Risk of Mutual Funds. Investors commonly use the standard deviation of the monthly percentage return for a mutual fund as a measure of the risk for the fund; in such cases, a fund that has a larger standard deviation is considered more risky than a fund with a lower standard deviation. The standard deviation for the American Century Equity Growth fund and the standard deviation for the Fidelity Growth Discovery fund were recently reported to be $15.0 \%$ and $18.9 \%$, respectively. Assume that each of these standard deviations is based on a sample of 60 months of returns. Do the sample results support the conclusion that the Fidelity fund has a larger population variance than the American Century fund? Which fund is more risky?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:38

Problem 17

Repair Costs as Automobiles Age. In its 2016 Auto Reliability Survey, Consumer Reports asked subscribers to report their maintenance and repair costs. Most individuals are aware of the fact that the average annual repair cost for an automobile depends on the age of the automobile. A researcher is interested in finding out whether the variance of the annual repair costs also increases with the age of the automobile. A sample of 26 automobiles 4 years old showed a sample standard deviation for annual repair costs of $$\$ 170$$ and a sample of 25 automobiles 2 years old showed a sample standard deviation for annual repair costs of $$\$ 100$$.
a. State the null and alternative versions of the research hypothesis that the variance in annual repair costs is larger for the older automobiles.
b. At a . 01 level of significance, what is your conclusion? What is the $p$-value? Discuss the reasonableness of your findings.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:49

Problem 18

Variance in Fund Amounts: Merrill Lynch versus Morgan Stanley. Barron's has collected data on the top 1000 financial advisers. Merrill Lynch and Morgan Stanley have many of their advisers on this list. A sample of 16 of the Merrill Lynch advisers and 10 of the Morgan Stanley advisers showed that the advisers managed many very large accounts with a large variance in the total amount of funds managed. The standard deviation of the amount managed by the Merrill Lynch advisers was $$s_1=\$ 587$$ million. The standard deviation of the amount managed by the Morgan Stanley advisers was $$s_2=\$ 489$$ million. Conduct a hypothesis test at $\alpha=.10$ to determine if there is a significant difference in the population variances for the amounts managed by the two companies. What is your conclusion about the variability in the amount of funds managed by advisers from the two firms?

Nick Johnson
Nick Johnson
Numerade Educator
01:52

Problem 19

Bag-Filling Machines at Jelly Belly. The variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance. Jelly Belly Candy Company is testing two machines that use different technologies to fill three pound bags of jelly beans. The file Bags contains a sample of data on the weights of bags (in pounds) filled by each machine. Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for two machines. Use a .05 level of significance. What is your conclusion? Which machine, if either, provides the greater opportunity for quality improvements?

Nick Johnson
Nick Johnson
Numerade Educator
02:53

Problem 20

Salaries at Public Accounting Firms. On the basis of data provided by a Romac salary survey, the variance in annual salaries for senior partners in public accounting firms is approximately 2.1 and the variance in annual salaries for managers in public accounting firms is approximately 11.1. The salary data were provided in thousands of dollars. Assuming that the salary data were based on samples of 25 senior partners and 26 managers, test the hypothesis that the population variances in the salaries are equal. At a .05 level of significance, what is your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:35

Problem 21

Smartphone Battery Life. Battery life is an important issue for many smartphone owners. Public health studies have examined "low-battery anxiety" and acute anxiety called nomophobia that results when a smartphone user's phone battery charge runs low and then dies (Wall Sireet Journal, https//www.wsj.com/articles/your-phone-is-almost -out-of-battery-remain-calm-call-a-doctor-1525449283). Battery life between charges for the Samsung Galaxy S9 averages 31 hours when the primary use is talk time and 10 hours when the primary use is Internet applications. Because the mean hours for talk time usage is greater than the mean hours for Internet usage, the question was raised as to whether the variance in hours of usage is also greater when the primary use is talk time. Sample data showing battery life between charges for the two applications follows.$$
\begin{array}{|cccccc}
\text { Primary Use: Talking } & & & & & \\
35.8 & 22.2 & 24.0 & 32.6 & 18.5 & 42.5 \\
28.0 & 23.8 & 30.0 & 22.8 & 20.3 & 35.5 \\
\text { Primary Use: Internet } & & & & & \\
14.0 & 12.5 & 16.4 & 11.9 & 9.9 & 3.1 \\
5.4 & 11.0 & 15.2 & 4.0 & 4.7 &
\end{array}
$$
a. Formulate hypotheses about the two population variances that can be used to determine if the population variance in battery life is greater for the talk time application.
b. What are the standard deviations of battery life for the two samples?
c. Conduct the hypothesis test and compute the p-value. Using a . 05 level of significance, what is your conclusion?

Nick Johnson
Nick Johnson
Numerade Educator
03:12

Problem 22

Stopping Distances of Automobiles. A research hypothesis is that the variance of stopping distances of automobiles on wet pavement is substantially greater than the variance of stopping distances of automobiles on dry pavement. In the research study, 16 automobiles traveling at the same speeds are tested for stopping distances on wet pavement and then tested for stopping distances on dry pavement. On wet pavement, the standard deviation of stopping distances is 32 feet. On dry pavement, the standard deviation is 16 feet.
a. At a . 05 level of significance, do the sample data justify the conclusion that the variance in stopping distances on wet pavement is greater than the variance in stopping distances on dry pavement? What is the p-value?
b. What are the implications of your statistical conclusions in terms of driving safety recommendations?

Nick Johnson
Nick Johnson
Numerade Educator
02:28

Problem 23

Daily Hotel Room Occupancy. Because of staffing decisions, managers of the Gibson-Marimont Hotel are interested in the variability in the number of rooms occupied per day during a particular season of the year. A sample of 20 days of operation shows a sample mean of 290 rooms occupied per day and a sample standard deviation of 30 rooms.
a. What is the point estimate of the population variance?
b. Provide a $90 \%$ confidence interval estimate of the population variance.
c. Provide a $90 \%$ confidence interval estimate of the population standard deviation.

Nick Johnson
Nick Johnson
Numerade Educator
01:58

Problem 24

Pricing of Initial Public Offerings. Initial public offerings (IPOs) of stocks are on average underpriced. The standard deviation measures the dispersion, or variation, in the underpricing-overpricing indicator. A sample of 13 Canadian IPOs that were subsequently traded on the Toronto Stock Exchange had a standard deviation of 14.95. Develop a $95 \%$ confidence interval estimate of the population standard deviation for the underpricing-overpricing indicator.

Nick Johnson
Nick Johnson
Numerade Educator
02:35

Problem 25

Business Travel Costs. According to the 2017 Corporate Travel Index compiled by Business Travel News, the average daily cost for business travel in the United States rose to $$\$ 321$$ per day (Executhe Travel website, https://executivetravel.com/new -business-travel-study-says-average-per-diem-is-now-321day/). The file Travel contains sample data for an analogous study on the estimated daily living costs for an executive traveling to various international cities. The estimates include a single room at a four-star hotel, beverages, breakfast, taxi fares, and incidental costs.
$$
\begin{array}{lclc}
\text { City } & \text { Daily Living Cost (\$) } & \text { City } & \text { Daily Living Cost (\$) } \\
\text { Bangkok } & 242.87 & \text { Mexico City } & 212.00 \\
\text { Bogota } & 260.93 & \text { Milan } & 284.08 \\
\text { Cairo } & 194.19 & \text { Mumbai } & 139.16 \\
\text { Dublin } & 260.76 & \text { Paris } & 436.72 \\
\text { Frankfurt } & 355.36 & \text { Rio de Janeiro } & 240.87 \\
\text { Hong Kong } & 346.32 & \text { Seoul } & 310.41 \\
\text { Johannesburg } & 165.37 & \text { Tel Aviv } & 223.73 \\
\text { Lima } & 250.08 & \text { Toronto } & 181.25 \\
\text { London } & 326.76 & \text { Warsaw } & 238.20 \\
\text { Madrid } & 283.56 & \text { Washington, D.C. } & 250.61
\end{array}
$$
a. Compute the sample mean.
b. Compute the sample standard deviation.
c. Compute a $95 \%$ confidence interval for the population standard deviation.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:27

Problem 26

Manufacture of Ball Bearings. Ball bearing manufacturing is a highly precise business in which minimal part variability is critical. Large variances in the size of the ball bearings cause bearing failure and rapid wearout. Production standards call for a maximum variance of .0001 inches $^2$. Gerry Liddy has gathered a sample of 15 bearings that shows a sample standard deviation of .014 inches.
a. Use $\alpha=.10$ to determine whether the sample indicates that the maximum acceptable variance is being exceeded.
b. Compute the $90 \%$ confidence interval estimate of the variance of the ball bearings in the population.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:51

Problem 27

Count Chocula Cereal. Filling boxes with consistent amounts of its cereals is critical to General Mills's success. The filling variance for boxes of Count Chocula cereal is designed to be .02 ounces $^2$ or less. A sample of 41 boxes of Count Chocula shows a sample standard deviation of .16 ounces. Use $\alpha=.05$ to determine whether the variance in the cereal box fillings is exceeding the design specification.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:35

Problem 28

OrderUp Food Delivery. OrderUp is a service that delivers food that its customers order online from participating restaurants. OrderUp claims consistent delivery times for its deliveries. A sample of 22 meal deliveries shows a sample variance of 1.5 . Test to determine whether $H_0: \sigma^2 \leq 1$ can be rejected. Use $\alpha=.10$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:09

Problem 29

Daily Patient Volume at Dental Clinic. A sample of 9 days over the past six months showed that Philip Sherman, DDS, treated the following numbers of patients at his dental clinic: $22,25,20,18,15,22,24,19$, and 26 . If the number of patients seen per day is normally distributed, would an analysis of these sample data reject the hypothesis that the variance in the number of patients seen per day is equal to 10 ? Use a . 10 level of significance. What is your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:44

Problem 30

Passenger Volume on Allegiant Airlines. A sample standard deviation for the number of passengers taking a particular Allegiant Airlines flight is $8 . \mathrm{A} 95 \%$ confidence interval estimate of the population standard deviation is 5.86 passengers to 12.62 passengers.
a. Was a sample size of 10 or 15 used in the statistical analysis?
b. Suppose the sample standard deviation of $s=8$ was based on a sample of 25 flights. What change would you expect in the confidence interval for the population standard deviation? Compute a $95 \%$ confidence interval estimate of $\sigma$ with a sample size of 25 .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:01

Problem 31

Golf Scores. Is there any difference in the variability in golf scores for players on the LPGA Tour (the women's professional golf tour) and players on the PGA Tour (the men's professional golf tour)? A sample of 20 tournament scores from LPGA events showed a standard deviation of 2.4623 strokes, and a sample of 30 tournament scores from PGA events showed a standard deviation of 2.2118 . Conduct a hypothesis test for equal population variances to determine if there is any statistically significant difference in the variability of golf scores for male and female professional golfers. Use $\alpha=.10$. What is your conclusion?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:34

Problem 32

Grade Point Average Comparison. The grade point averages of 352 students who completed a college course in financial accounting have a standard deviation of .940 . The grade point averages of 73 students who dropped out of the same course have a standard deviation of .797 . Do the data indicate a difference between the variances of grade point averages for students who completed a financial accounting course and students who dropped out? Use a .05 level of significance. Note: $F_{\text {. }}$ with 351 and 72 degrees of freedom is 1.466 .

Nick Johnson
Nick Johnson
Numerade Educator
02:24

Problem 33

Weekly Cost Reporting. Stable cost reporting in a manufacturing setting is typically a sign that operations are running smoothly. The accounting department at Rockwell Collins, an avionics manufacturer, analyzes the variance of the weekly costs reported by two of its production departments. A sample of 16 cost reports for each of the two departments shows cost variances of 2.3 and 5.4 , respectively. Is this sample sufficient to conclude that the two production departments differ in terms of weekly cost variance? Use $\alpha=.10$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:50

Problem 34

Lean Process Improvement at the New York City Food Bank. In an effort to make better use of its resources, the New York City Food Bank engaged in lean process improvement. This employee-driven kaizen effort resulted in a new method for packing meals for distribution to needy families. One goal of the process improvement effort was to reduce the variability in the meal-packing time. The following table summarizes information from a sample of data using the current method and the new method. Did the kaizen event successfully reduce the population variation? Use $\alpha=.10$ and formulate the appropriate hypothesis test.
Current Method
Sample Size
Sample Variance
$$
n_1=31
$$
$$
5_1^2=25
$$

New Method
$$
\begin{aligned}
& n_2=25 \\
& s_2^2=12
\end{aligned}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator