Business Statistics

David F. Groebner, Patrick W. Shannon, Phillip C. Fry, Kent D. Smith

Chapter 12 Analysis of Variance - all with Video Answers

Educators

Chapter Questions

Problem 1

The Lottaburger restaurant chain in central New Mexico is conducting an analysis of its restaurants, which take pride in serving burgers and fries to go faster than the competition. As a part of its analysis, Lottaburger wants to determine if its speed of service is different across its four outlets. Orders at Lottaburger restaurants are tracked electronically, and the chain is able to determine the speed with which every order is filled. The chain decided to randomly sample 20 orders from each of the four restaurants it operates. The speed of service for each randomly sampled order was noted and is contained in the file Lottaburger.
a. At the $\alpha=0.05$ level of service, can Lottaburger conclude that the speed of service is different across the four restaurants in the chain?
b. If the chain concludes that there is a difference in speed of service, is there a particular restaurant the chain should focus its attention on? Use the appropriate test for multiple comparisons to support your decision. Use $\alpha=0.05$.

Sheryl Ezze

Numerade Educator

02:34

Problem 1

A start-up cell phone applications company is interested in determining whether houschold incomes are different for subscribers to three different service providers. A random sample of 25 subscribers to each of the three service prowiders was taken, and the annual household income for each subscriber was reconded. The partially completed ANOVA table for the analysis is shown here:
a. Complete the ANOVA table by filling in the missing sums of squares, the degrees of freedom for each source, the mean square, and the calculated $F$-test statistic,
b. Based on the sample results, can the start-up firm conclude that there is a difference in houschold incomes for subscribers to the three service providers? You may assume normal distributions and equal variances. Conduct your test at the $\alpha=$ 0.10 level of significance. Be sure to state a critical F-statistic, a decision rule, and a conclusion.

Sheryl Ezze

Numerade Educator

01:40

Problem 2

An analyst is interested in testing whether four populations have equal means. The following sample data have been collected from populations that are assumed to be normally distributed with equal variances:
Conduct the appropriate hypothesis test using a significance level equal to 0.05 .

Shu Naito

Numerade Educator

Problem 3

A manager is interested in testing whether three populations of interest have equal population means. Simple random samples of size 10 were selected from each population. The following ANOVA table and related statistics were computed:
a. State the appropriate null and alternative hypotheses.
b. Conduct the appropriate test of the null hypothesis assuming that the populations have equal variances and the populations are normally distributed. Use a 0.05 level of significance.
c. If warranted, use the Tukey-Kramer procedure for multiple comparisons to determine which populations have different means. (Assume $\alpha=0.05$.)

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Problem 4

Respond to each of the following questions using this partially completed one-way ANOVA table:
a. How many different popolations are being considered in this analysis?
b. Fill in the ANOVA table with the missing values.
c. State the appropriate null and alternative hypotheses.
d. Based on the analysis of variance F-test, what conclusion should be reached regarding the null hypothesis? Test using a significance level of 0.01 .

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Problem 5

Respond to each of the following questions using this partially completed one-way ANOVA table:
a. How many different populations are being considered in this analysis?
b. Fill in the ANOVA table with the missing values.
c. State the appropriate null and alternative hypotheses.
d. Based on the analysis of variance $F$-test, what conclusion should be reached regarding the null hypothesis? Test using $\alpha=0.05$.

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Problem 6

Given the following sample data
a. Based on the computations for the within- and between-sample variation, develop the ANOVA table and test the appropriate null hypothesis using $\alpha=0.05$. Use the $p$-value approach.
b. If warranted, use the Tukey-Kramer procedure to determine which populations have different means. Use $\alpha=0.05$.

Victor Salazar

Numerade Educator

03:05

Problem 7

Examine the three samples obtained independently from three populations:

John Piaszynski

Numerade Educator

Problem 8

In conjunction with the housing foreclosure crisis of 2009, many economists expressed increasing concern about the level of credit card debt and efforts of banks to raise interest rates on these cards. The banks claimed the increases were justified. A Senate sub-committee decided to determine if the average credit card balance depends on the type of credit card used. Under consideration are Visa, MasterCard, Discower, and American Express. The sample sizes to be used for each level are $25,25,26$, and 23 , respectively.
a. Describe the parameter of interest for this analysis.
b. Determine the factor associated with this experiment.
c. Describe the levels of the factor associated with this analysis.
d. State the number of degrees of freedom available for determining the between-samples variation.
e. State the number of degrees of freedom available for determining the within-samples variation.
f. State the number of degrees of freedom available for determining the total variation.

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03:58

Problem 9

EverRun Incorporated produces treadmills for use in exercise clubs and recreation centers. EverRun assembles, sells, and services its treadmills, but it does not manufacture the treadmill motors. Rather, treadmill motors are purchased from an outside vendor. Currently, EverRun is considering which motor to include in its new ERI500 series. Three potential suppliers have been identified: Venetti, Madison, and Edison; however, only one supplier will be used. The motors produced by these three suppliers are identical in terms of noise and cost. Consequently, EverRun has decided to make its decision based on how long a motor operates at a high level of speed and incline before it fails. A random sample of 10 motors of each type is selected, and each motor is tested to determine how many minutes (rounded to the nearest minute) it operates before it needs to be repaired. The sample information for each motor is as follows:
a. At the $\alpha=0.01$ level of significance, is there a difference in the average time before failure for the three different supplier motors?
b. Is it possible for EverRun to decide on a single motor supplier based on the analysis of the sample results? Support your answer by conducting the appropriate post-test analysis.

Victor Salazar

Numerade Educator

Problem 10

ESSROC Cement Corporation is a leading North American cement producer, with over 6.5 million metric tons of annual capacity. With headquarters in Nazareth, Pennsylvania, ESSROC operates production facilities strategically located throughout the United States, Canada, and Puerto Rico. One of its products is Portland cement. Portland cement's properties and performance standards are defined by its type designation. Each type is designated by a Roman numeral. Ninety-two percent of the Portland cement produced in North America is Type I, II, or I/II.
One characteristic of the type of cement is its compressive strength. Sample data for the compressive strength (psi) are shown as follows:
a. Develop the appropriate ANOVA table to determine if there is a difference in the average compressive strength among the three types of Portland cement. Use a significance level of 0.01 .
b. If warranted, use the Tukey-Kramer procedure to determine which populations have different mean compressive strengths. Use an experiment-wide error rate of 0.01 .

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Problem 11

The Weidmann Group Companies, with headquarters in Rapperswil, Switzerland, are worldwide leaders in insulation systems technology for power and distribution transformers. One facet of its expertise is the development of dielectric fluids in electrical equipment. Mineral oil-based dielectric fluids have been used more extensively than other dielectric fluids. Their only shortcomings are their relatively low flash and fire point. One study examined the fire point of mineral oil, high-molecular-weight hydrocarbon (HMWH), and silicone. The fire points for each of these fluids were as follows:
a. Develop the appropriate ANOVA table to determine if there is a difference in the average fire points among the types of dielectric fluids. Use a significance level of 0.05 .
b. If warranted, use the Tukey-Kramer procedure to determine which populations have different mean fire points. Use an experiment-wide error rate of 0.05 .

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02:14

Problem 12

The manager at the Hillsberg Savings and Loan is interested in determining whether there is a difference in the mean time that customers spend completing their transactions depending on which of four tellers they use. To conduct the test, the manager has selected simple random samples of 15 customers for each of the tellers and has timed them (in seconds) from the moment they start their transaction to the time the transaction is completed and they leave the teller station. The manager then asked one of her assistants to perform the appropriate statistical test. The assistant returned with the following partially completed ANOVA table.
a. State the appropriate null and alternative hypotheses.
b. Test to determine whether the population variances are equal. Use a significance level equal to 0.05 .
c. Fill in the missing parts of the ANOVA table and perform the statistical hypothesis test using $\alpha=0.05$.
d. Based on the result of the test in part c , if warranted, use the Tukey-Kramer method with $\alpha=0.05$ to determine which teller require the most time on average to complete a customer's transaction.

James Kiss

Numerade Educator

Problem 12

Suppose as part of your job you are responsible for installing emergency lighting in a series of state office buildings. Bids have been received from four manufacturers of battery-operated emergency lights. The costs are about equal, so the decision will be based on the length of time the lights last before failing. A sample of four lights from each manufacturer has been tested with the following values (time in hours) recorded for each manufacturer:
a. State the appropriate null and alternative hypotheses.
b. Test to determine whether the population variances are equal. Use a significance level equal to 0.05 .
c. Fill in the missing parts of the ANOVA table and perform the statistical hypothesis test using $\alpha=0.05$.
d. Based on the result of the test in part c , if warranted, use the Tukey-Kramer method with $\alpha=0.05$ to determine which teller require the most time on average to complete a customer's transaction.

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02:48

Problem 13

Pratyush Raitan

Numerade Educator

Problem 14

Damage to homes caused by burst piping can be expensive to repair. By the time the leak is discovered, hundreds of gallons of water may have already flooded the home. Automatic shutoff valves can prevent extensive water damage from plumbing failures. The valves contain sensors that cut off water flow in the event of a leak, thereby preventing flooding. One important characteristic is the time (in milliseconds) required for the sensor to detect the water leak. Sample data obtained for four different shutoff valves are contained in the file entitled Waterflow.
a. Produce the relevant ANOVA table and conduct a hypothesis test to determine if the mean detection time differs among the four shutoff valve models. Use a significance level of 0,05 .
b. Use the Tukey-Kramer multiple comparison technique to discover any differences in the average detection time. Use a significance level of 0.05 .
c. Which of the four shutoff valves would you recommend? State your criterion for your selection.

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03:11

Problem 15

A regional package delivery company is considering changing from full-size vans to minivans. The company sampled minivans from each of three manufacturers. The number sampled represents the number the manufacturer was able to provide for the test. Each minivan was driven for 5,000 miles, and the operating cost per mile was computed. The operating costs, in cents per mile, for the 12 are provided in the data file called Delivery:
a. Perform an analysis of variance on these data. Assume a significance level of 0.05 . Do the experimental data provide evidence that the average operating costs per mile for the three types of minivans are different? Use a $p$-value approach.
b. Referring to part a, based on the sample data and the appropriate test for multiple comparisons, what conclusions should be reached concerning which type of car the delivery company should adopt? Discuss and prepare a report to the company CEO. Use $\alpha=0.05$.
c. Provide an estimate of the maximum and minimum difference in average savings per year if the CEO chooses the "best" versus the "worst" minivan using operating costs as a criterion. Assume that minivans are driven 30,000 miles a year. Use a $90 \%$ confidence interval.

Sheryl Ezze

Numerade Educator

08:33

Problem 17

Most auto batteries are made by just three manufacturers-Delphi, Exide, and Johnson Controls Industries. Each makes batteries sold under several different brand names. Delphi makes ACDelco and some EverStart (Wal-Mart) models. Exide makes Champion, Exide, Napa, and some EverStart batteries. Johnson Controls makes Diehard (Sears), Duralast (AutoZone), Interstate, Kirkland (Costco), Motoreraft (Ford), and some EverStarts. To determine if who makes the auto batteries affects the average length of life of the battery, the samples in the file entitled Start were obtained. The data represent the length of life (months) for batteries of the same specifications for each of the three manufacturers.
a. Determine if the average length of battery life is different among the batteries produced by the three manufacturers. Use a significance level of 0.05 .
b. Which manufacturer produces the battery with the longest average length of life? If warranted, conduct the Tukey-Kramer procedure to determine this. Use a significance level of 0.05 . (Note: You will need to manipulate the data columns to obtain the appropriate factor levels).

Sonam Khatri

Numerade Educator

02:03

Problem 18

A study was conducted to determine if differences in new textbook prices exist between on-campus bookstores, off-campus bookstores, and Internet bookstores. To control for differences in textbook prices that might exist across disciplines, the study randomly selected 12 textbooks and recorded the price of each of the 12 books at each of the three retailers. You may assume normality and equal-variance assumptions have been met. The partially completed ANOVA table based on the study's findings is shown here:

Sheryl Ezze

Numerade Educator

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Problem 19

The following data were collected for a randomized block analysis of variance design with four populations and cight blocks:
a. State the appropriate null and alternative hypotheses for the treatments and determine whether blocking is necessary.
b. Construct the appropriate ANOVA table.
c. Using a significance level equal to 0.05 , can you conclude that blocking was necessary in this case? Use a test-statistic approach.
d. Based on the data and a significance level equal to 0.05 , is there a difference in population means for the four groups? Use a p-value approach.
c. If you found that a difference exists in part d, use the $L S D$ approach to determine which populations have different means.

Rashmi Sinha

Numerade Educator

02:17

Problem 19

The Cordage Institute, based in Wayne, Pennsylvania, is an international association of manufacturers, producers, and resellers of cordage, rope, and twine. It is a not-for-profit corporation that reports on research concerning these products. Although natural fibers like manila, sisal, and cotton were once the predominant rope materials, industrial synthetic fibers dominate the marketplace today. with most ropes made of nylon, polyester, or polypropylene. One of the principal traits of rope material is its breaking strength. A research project generated data given in the file entitled Knots. The data listed were gathered on 10 different days from K/2-diameter ropes.

a. Test to determine if inserting the day on which the testing was done was necessary. Use a significance level of 0.05 .
b. Based on the data gathered by the Cordage Institute, can it be concluded that there is a difference in the average breaking strength of nylon, polyester, and polypropylene?
c. If you concluded that there was a difference in the average breaking strength of the rope material, use Fisher's LSD approach to determine which material has the highest breaking strength.

Adriano Chikande

Numerade Educator

Problem 20

The following ANOVA table and accompanying information are the result of a randomized block ANOVA test.
a. How many blocks were used in this study?
b. How many populations are involved in this test?
c. Test to determine whether blocking is effective using an alpha level equal to 0.05 .
d. Test the main hypothesis of interest using $\alpha=0.05$.
e. If warranted, conduct an LSD test with $\alpha=0.05$ to determine which population means are different.

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07:11

Problem 20

When the world's largest retailer, Wal-Mart, decided to enter the grocery marketplace in a big way with its "Super Stores," it changed the retail grocery landscape in a major way. The other major chains such as Albertsons have struggled to stay competitive. In addition, regional discounters such as WINCO in the western United States have made it difficult for the traditional grocery chains. Recently, a study was conducted in which a "market basket" of products was selected at random from those items offered in three stores in Boise, Idaho: Wal-Mart, Winco, and Albertsons. At issue was whether the mean prices at the three stores are equal or whether there is a difference in prices. The sample data are in the data file called Food Price Comparisons. Using an alpha level equal to 0.05, test to determine whether the three stores have equal population mean prices. If you conclude that there are differences in the mean prices, perform the appropriate posttest to determine which stores have different means.

Jeremiah Mbaria

Numerade Educator

Problem 21

The following sample data were recently collected in the course of conducting a randomized block analysis of variance. Based on these sample data, what conclusions should be reached about blocking effectiveness and about the means of the three populations involved? Test using a significance level equal to 0.05 .

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Problem 22

A randomized complete block design is carried out, resulting in the following statistics:
a. Determine if blocking was effective for this design.
b. Using a significance level of 0.05 , produce the relevant ANOVA and determine if the average responses of the factor levels are equal to each other.
c. If you discovered that there were differences among the average responses of the factor levels, use the LSD approach to determine which populations have different means.

Rashmi Sinha

Numerade Educator

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Problem 23

Frasier and Company manufactures four different products that it ships to customers throughout the United States. Delivery times are not a driving factor in the decision as to which type of carrier to use (rail, plane, or truck) to deliver the product. However, breakage cost is very expensive, and Frasier would like to select a mode of delivery that reduces the amount of product breakage. To help it reach a decision, the managers have decided to exarmine the dollar amount of breakage incurred by the three alternative modes of transportation under consideration. Because each product's fragility is different, the executives conducting the study wish to control for differences due to type of product. The company randomly assigns each product to each carrier and monitors the dollar breakage that occurs over the course of 100 shipments. The dollar breakage per shipment (to the nearest dollar) is as follows:

a. Was Frasier and Company correct in its decision to block for type of product? Conduct the appropriate hypothesis test using a level of significance of 0.01 .
b. Is there a difference due to carrier type? Conduct the appropriate hypothesis test using a level of significance of 0.01 .

Victor Salazar

Numerade Educator

06:51

Problem 24

The California Lettuce Research Board was originally formed as the Iceberg Lettuce Advisory Board in 1973. The primary function of the board is to fund research on iceberg and leaf lettuce. The California Lettuce Research Board published research (M. Cahn and H. Ajwa. "Salinity Effects on Quality and Yield of Drip Irrigated Lettuce") concerning the effect of varying levels of sodium absorption ratios (SAR) on the yield of head lettuce. The trials followed a randomized complete block design where variety of lettuce (Salinas and Sniper) was the main factor and salinity levels were the blocks. The measurements (the number of lettuce heads from each plot) of the kind observed were

a. Determine if blocking was effective for this design.
b. Using a significance level of 0.05 , produce the relevant ANOVA and determine if the average number of lettuce heads among the SARs are equal to each other.
c. If you discovered that there were differences among the average number of lettuce heads among the SARs, use the LSD approach to determine which populations have different means.

Raymond Matshanda

Numerade Educator

Problem 25

CB Industries operates three shifts every day of the week. Each shift includes full-time hourly workers, nonsupervisory salaried employees, and supervisors/ managers. CB Industries would like to know if there is a difference among the shifts in terms of the number of hours of work missed due to employee illness. To control for differences that might exist across employee groups, CB Industries randomly selects one employee from each employee group and shift and records the number of hours missed for one year. The results of the study are shown here:

a. Develop the appropriate test to determine whether blocking is effective or not. Conduct the test at the $\alpha=0.05$ level of significance.
b. Develop the appropriate test to determine whether there are differences in the average number of hours missed due to illness across the three shifts. Conduct the test at the $\alpha=0.05$ level of significance.
c. If it is determined that a difference in the average hours of work missed due to illness is not the same for the three shifts, use the $L S D$ approach to determine which shifts have different means.

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Problem 26

Grant Thomton LLP is the U.S. member firm of Grant Thomton International, one of the six global accounting, tax, and business advisory organizations. It provides firmwide auditing training for its employees in three different auditing methods. Auditors were grouped into four blocks according to the education they had received: (1) high school, (2) bachelor's, (3) master's, (4) doctorate. Three auditors at each education level were used-one assigned to each method. They were given a posttraining examination consisting of complicated auditing scenarios. The scores for the 12 auditors were as follows:

a. Indicate why blocking was employed in this design.
b. Determine if blocking was effective for this design by producing the relevant ANOVA.
c. Using a significance level of 0.05 , determine if the average posttraining examination scores among the auditing methods are equal to each other.
d. If you discovered that there were differences among the average posttraining examination scores among the auditing methods, use the $L S D$ approach to determine which populations have different means.

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07:00

Problem 27

Applebee's International, Inc., is a U.S. company that develops, franchises, and operates the Applebee's Neighborhood Grill and Bar restaurant chain. It is the largest chain of casual dining restaurants in the country, with over 1,500 restaurants across the United States. The headquarters is located in Overland Park, Kansas. The company is interested in determining if mean weekly revenue differs among three restaurants in a particular city. The file entitled Applebees contains revenue data for a sample of weeks for each of the three locations.
a. Test to determine if blocking the week on which the testing was done was necessary. Use a significance level of 0.05 .
b. Based on the data gathered by Applebee's, can it be concluded that there is a difference in the average revenue among the three restaurants?
c. If you did conclude that there was a difference in the average revenue, use Fisher's LSD approach to determine which restaurant has the lowest mean sales.

Sheryl Ezze

Numerade Educator

07:11

Problem 28

In a local community there are three grocery chain stores. The three have been carrying out a spirited advertising campaign in which each claims to have the lowest prices. A local news station recently sent a reponter to the three stores to check prices on several items. She found that for certain items each store had the lowest price. This survey didn't really answer the question for consumers. Thus, the station set up a test in which 20 shoppers were given different lists of grocery items and were sent to each of the three chain stores. The sales receipts from each of the three stores are recorded in the data file Groceries.

a. Why should this price test be conducted using the design that the television station used? What was it attempting to achieve by having the same shopping lists used at each of the three grocery stores?
b. Based on a significance level of 0.05 and these sample data, test to determine whether blocking was necessary in this example. State the null and alternative hypotheses. Use a test-statistic approach.
c. Based on these sample data, can you conclude the three grocery stores have different sample means? Test using a significance level of 0.05 . State the appropriate null and alternative hypotheses. Use a p-value approach.
d. Based on the sample data, which store has the highest average prices? Use Fisher's $L S D$ test if appropriate.

Jeremiah Mbaria

Numerade Educator