Statistics for Business Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

Chapter 13

Experimental Design and Analysis of Variance - all with Video Answers

Educators

Chapter Questions

Problem 1

The following data are from a completely randomized design.
a. Compute the sum of squares between treatments.
b. Compute the mean square between treatments.
c. Compute the sum of squares due to error.
d. Compute the mean square due to error.
e. Set up the ANOVA table for this problem.
f. At the $\alpha=.05$ level of significance, test whether the means for the three treatments are equal.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 2

In a completely randomized design, seven experimental units were used for each of the five levels of the factor. Complete the following ANOVA table.

Sheryl Ezze

Numerade Educator

Problem 3

Refer to exercise 2.
a. What hypotheses are implied in this problem?
b. $\quad$ At the $\alpha=.05$ level of significance, can we reject the null hypothesis in part (a)? Explain.

Srikar Katta

Numerade Educator

Problem 4

In an experiment designed to test the output levels of three different treatments, the following results were obtained: $\mathrm{SST}=400, \mathrm{SSTR}=150, n_{T}=19 .$ Set up the ANOVA table and test for any significant difference between the mean output levels of the three treatments. Use $\alpha=.05$.

Sheryl Ezze

Numerade Educator

Problem 5

In a completely randomized design, 12 experimental units were used for the first treatment, 15 for the second treatment, and 20 for the third treatment. Complete the following analysis of variance. At a .05 level of significance, is there a significant difference between the treatments?

Srikar Katta

Numerade Educator

Problem 6

Develop the analysis of variance computations for the following completely randomized design. At $\alpha=.05,$ is there a significant difference between the treatment means?

Sheryl Ezze

Numerade Educator

Problem 7

Three different methods for assembling a product were proposed by an industrial engineer. To investigate the number of units assembled correctly with each method, 30 employees were randomly selected and randomly assigned to the three proposed methods in such a way that each method was used by 10 workers. The number of units assembled correctly was recorded, and the analysis of variance procedure was applied to the resulting data set. The following results were obtained: $\mathrm{SST}=10,800 ; \mathrm{SSTR}=4560$.
a. Set up the ANOVA table for this problem.
b. Use $a=.05$ to test for any significant difference in the means for the three assembly methods.

Srikar Katta

Numerade Educator

Problem 8

Refer to the NCP data in Table $13.4 .$ Set up the ANOVA table and test for any significant difference in the mean examination score for the three plants. Use $\alpha=.05$.

Sheryl Ezze

Numerade Educator

Problem 9

To study the effect of temperature on yield in a chemical process, five batches were produced at each of three temperature levels. The results follow. Construct an analysis of variance table. Use a .05 level of significance to test whether the temperature level has an effect on the mean yield of the process.

Srikar Katta

Numerade Educator

Problem 10

Auditors must make judgments about various aspects of an audit on the basis of their own direct experience, indirect experience, or a combination of the two. In a study, auditors were asked to make judgments about the frequency of errors to be found in an audit. The judgments by the auditors were then compared to the actual results. Suppose the following data were obtained from a similar study; lower scores indicate better judgments.
Use $\alpha=.05$ to test to see whether the basis for the judgment affects the quality of the judgment. What is your conclusion?

Sheryl Ezze

Numerade Educator

Problem 11

Four different paints are advertised as having the same drying time. To check the manufacturer's claims, five samples were tested for each of the paints. The time in minutes until the paint was dry enough for a second coat to be applied was recorded. The following data were obtained.
At the $a=.05$ level of significance, test to see whether the mean drying time is the same for each type of paint.

Srikar Katta

Numerade Educator

Problem 12

The Consumer Reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains (Consumer Reports website). One of the variables in the study is meal price, the average amount paid per person for dinner and drinks, minus the tip. Suppose a reporter for the Sun Coast Times thought that it would be of interest to her readers to conduct a similar study for restaurants located on the Grand Strand section in Myrtle Beach, South Carolina. The reporter selected a sample of eight seafood restaurants, eight Italian restaurants, and eight steakhouses. The following data show the meal prices (S) obtained for the 24 restaurants sampled. Use $\alpha=.05$ to test whether there is a significant difference among the mean meal price for the three types of restaurants.

Sheryl Ezze

Numerade Educator

Problem 13

The following data are from a completely randomized design.
a. At the $\alpha=.05$ level of significance, can we reject the null hypothesis that the means of the three treatments are equal?
b. Use Fisher's LSD procedure to test whether there is a significant difference between the means for treatments A and B, treatments A and C, and treatments B and C. Use $\alpha=.05$.
c. Use Fisher's LSD procedure to develop a $95 \%$ confidence interval estimate of the difference between the means of treatments A and B.

Sheryl Ezze

Numerade Educator

Problem 14

The following data are from a completely randomized design. In the following calculations, use $\alpha=.05$.
a. Use analysis of variance to test for a significant difference among the means of the three treatments.
b. Use Fisher's LSD procedure to determine which means are different.

Sheryl Ezze

Numerade Educator

Problem 15

To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, the Jacobs Chemical Company obtained the following data on the time (in minutes) needed to mix the material.
a. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use $\alpha=.05$.
b. $\quad$ At the $\alpha=.05$ level of significance, use Fisher's $L$ SD procedure to test for the equality of the means for manufacturers 1 and $3 .$ What conclusion can you draw after carrying out this test?

Sheryl Ezze

Numerade Educator

Problem 16

Refer to exercise $15 .$ Use Fisher's LSD procedure to develop a $95 \%$ confidence interval estimate of the difference between the means for manufacturer 1 and manufacturer 2.

Sheryl Ezze

Numerade Educator

Problem 17

The following data are from an experiment designed to investigate the perception of corporate ethical values among individuals specializing in marketing (higher scores indicate higher ethical values).
a. Use $\alpha=.05$ to test for significant differences in perception among the three groups.
b. At the $\alpha=.05$ level of significance, we can conclude that there are differences in the perceptions for marketing managers, marketing research specialists, and advertising specialists. Use the procedures in this section to determine where the differences occur. Use $\alpha=.05$.

Sheryl Ezze

Numerade Educator

Problem 18

To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained.
a. At the $\alpha=.05$ level of significance, what is the difference, if any, in the population mean times among the four machines?
b. Use Fisher's LSD procedure to test for the equality of the means for machines 2 and
4. Use a .05 level of significance.

Sheryl Ezze

Numerade Educator

Problem 19

Refer to exercise $18 .$ Use the Bonferroni adjustment to test for a significant difference between all pairs of means. Assume that a maximum overall experimentwise error rate of .05 is desired.

Jorge Villanueva

Jorge Villanueva

Numerade Educator

Problem 20

The International League of Triple-A minor league baseball consists of 14 teams organized into three divisions: North, South, and West. The following data show the average attendance for the 14 teams in the International League (The Biz of Baseball website, January 2009 ). Also shown are the teams' records; W denotes the number of games won, L denotes the number of games lost, and $\mathrm{PCT}$ is the proportion of games played that were won.

Sheryl Ezze

Numerade Educator

Problem 21

Consider the experimental results for the following randomized block design. Make the calculations necessary to set up the analysis of variance table.
Use $\alpha=.05$ to test for any significant differences.

Shu Naito

Numerade Educator

Problem 22

The following data were obtained for a randomized block design involving five treatments and three blocks: $\mathrm{SST}=430, \mathrm{SSTR}=310, \mathrm{SSBL}=85 .$ Set up the ANOVA table and test for any significant differences. Use $\alpha=.05$.

Sheryl Ezze

Numerade Educator

Problem 23

An experiment has been conducted for four treatments with eight blocks. Complete the following analysis of variance table.
Use $a=.05$ to test for any significant differences.

Sheryl Ezze

Numerade Educator

Problem 24

An automobile dealer conducted a test to determine if the time in minutes needed to complete a minor engine tune-up depends on whether a computerized engine analyzer or an electronic analyzer is used. Because tune-up time varies among compact, intermediate, and full-sized cars, the three types of cars were used as blocks in the experiment. The data obtained follow.

Evelyn Cunningham

Evelyn Cunningham

Numerade Educator

Problem 25

Use $\alpha=.05$ to test for any significant differences. Prices for vitamins and other health supplements increased over the past several years, and the prices charged by different retail outlets often vary a great deal. The following data show the prices for 13 products at four retail outlets in Rochester, New York (Democrat and Chronicle, February 13,2005 ).
Use $a=.05$ to test for any significant difference in the mean price for the four retail outlets.

Rashmi Sinha

Numerade Educator

Problem 26

The Scholastic Aptitude Test (SAT) contains three parts: critical reading, mathematics, and writing. Each part is scored on an 800 -point scale. Information on test scores for the 2009 version of the SAT is available at the College Board website. A sample of SAT scores for six students follows.
a. Using a .05 level of significance, do students perform differently on the three portions of the SAT?
b. Which portion of the test seems to give the students the most trouble? Explain.

Shu Naito

Numerade Educator

Problem 27

A study reported in the Journal of the American Medical Association investigated the cardiac demands of heavy snow shoveling. Ten healthy men underwent exercise testing with a treadmill and a cycle ergometer modified for arm cranking. The men then cleared two tracts of heavy, wet snow by using a lightweight plastic snow shovel and an electric snow thrower. Each subject's heart rate, blood pressure, oxygen uptake, and perceived exertion during snow removal were compared with the values obtained during treadmill and arm-crank ergometer testing. Suppose the following table gives the heart rates in beats per minute for each of the 10 subjects.
At the .05 level of significance, test for any significant differences.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 28

A factorial experiment involving two levels of factor A and three levels of factor $\mathrm{B}$ resulted in the following data.
Test for any significant main effects and any interaction. Use $\alpha=.05$.

Dominador Tan

Numerade Educator

Problem 29

The calculations for a factorial experiment involving four levels of factor $A$, three levels of factor $\mathbf{B},$ and three replications resulted in the following data: $\mathrm{SST}=280, \mathrm{SSA}=26$, $\mathrm{SSB}=23, \mathrm{SSAB}=175 .$ Set up the ANOVA table and test for any significant main effects and any interaction effect. Use $a=.05$.

Shu Naito

Numerade Educator

Problem 30

A mail-order catalog firm designed a factorial experiment to test the effect of the size of a magazine advertisement and the advertisement design on the number of catalog requests received (data in thousands). Three advertising designs and two different size advertisements were considered. The data obtained follow. Use the ANOVA procedure for factorial designs to test for any significant effects due to type of design, size of advertisement, or interaction. Use $\alpha=.05$.

Shu Naito

Numerade Educator

Problem 31

An amusement park studied methods for decreasing the waiting time (minutes) for rides by loading and unloading riders more efficiently. Two alternative loading/unloading methods have been proposed. To account for potential differences due to the type of ride and the possible interaction between the method of loading and unloading and the type of ride, a factorial experiment was designed. Use the following data to test for any significant effect due to the loading and unloading method, the type of ride, and interaction. Use $\alpha=.05$.

Dominador Tan

Numerade Educator

Problem 32

As part of a study designed to compare hybrid and similarly equipped conventional vehicles, Consumer Reports tested a varicty of classes of hybrid and all-gas model cars and sport utility vehicles (SUVs). The following data show the miles-per-gallon rating Consumer Reports obtained for two hybrid small cars, two hybrid midsize cars, two hybrid small SUVs, and two hybrid midsized SUVs; also shown are the miles per gallon obtained for eight similarly equipped conventional models (Consumer Reports, October 2008 ). At the $a=.05$ level of significance, test for significant effects due to class, type, and interaction.

Shu Naito

Numerade Educator

Problem 33

A study reported in The Accounting Review examined the separate and joint effects of two levels of time pressure (low and moderate) and three levels of knowledge (naive, declarative, and procedural) on key word selection behavior in tax research. Subjects were given a tax case containing a set of facts, a tax issue, and a key word index consisting of 1336 key words. They were asked to select the key words they believed would refer them to a tax authority relevant to resolving the tax case. Prior to the experiment, a group of tax experts determined that the text contained 19 relevant key words. Subjects in the naive group had little or no declarative or procedural knowledge, subjects in the declarative group had significant declarative knowledge but little or no procedural knowledge, and subjects in the procedural group had significant declarative knowledge and procedural knowledge. Declarative knowledge consists of knowledge of both the applicable tax rules and the technical terms used to describe such rules. Procedural knowledge is knowledge of the rules that guide the tax researcher's search for relevant key words. Subjects in the low time pressure situation were told they had 25 minutes to complete the problem, an amount of time which should be "more than adequate" to complete the case; subjects in the moderate time pressure situation were told they would have "only" 11 minutes to complete the case. Suppose 25 sub jects were selected for each of the six treatment combinations and the sample means for each treatment combination are as follows (standard deviations are in parentheses). Use the ANOVA procedure to test for any significant differences due to time pressure, knowledge, and interaction. Use a .05 level of significance. Assume that the total sum of squares for this experiment is 327.50.

Shu Naito

Numerade Educator

Problem 34

In a completely randomized experimental design, three brands of paper towels were tested for their ability to absorb water. Equal-size towels were used, with four sections of towels tested per brand. The absorbency rating data follow. At a .05 level of significance, does there appear to be a difference in the ability of the brands to absorb water?

Rashmi Sinha

Numerade Educator

Problem 35

A study reported in the Jourmal of Small Business Management concluded that selfemployed individuals do not experience higher job satisfaction than individuals who are not self-employed. In this study, job satisfaction is measured using 18 items, each of which is rated using a Likert-type scale with $1-5$ response options ranging from strong agreement to strong disagreement. A higher score on this scale indicates a higher degree of job satisfaction. The sum of the ratings for the 18 items, ranging from $18-90$, is used as the measure of job satisfaction. Suppose that this approach was used to measure the job satisfaction for lawyers, physical therapists, cabinetmakers, and systems analysts. The results obtained for a sample of 10 individuals from each profession follow.
At the $\alpha=.05$ level of significance, test for any difference in the job satisfaction among the four professions.

Srikar Katta

Numerade Educator

Problem 36

Money magazine reports percentage returns and expense ratios for stock and bond funds. The following data are the expense ratios for 10 midcap stock funds, 10 small-cap stock funds, 10 hybrid stock funds, and 10 specialty stock funds (Money, March 2003 ).
Use $\alpha=.05$ to test for any significant difference in the mean expense ratio among the four types of stock funds.

Sheryl Ezze

Numerade Educator

Problem 37

The U.S. Census Bureau computes quarterly vacancy and homeownership rates by state and metropolitan statistical area. Each metropolitan statistical area (MSA) has at least one urbanized area of 50,000 or more inhabitants. The following data are the rental vacancy rates
$(\%)$ for MSAs in four geographic regions of the United States for the first quarter of 2008
(U.S. Census Bureau website, January 2009 ).
Use $a=.05$ to test whether there the mean vacancy rate is the same for each geographic region.

Srikar Katta

Numerade Educator

Problem 38

Three different assembly methods have been proposed for a new product. A completely randomized experimental design was chosen to determine which assembly method results in the greatest number of parts produced per hour, and 30 workers were randomly selected and assigned to use one of the proposed methods. The number of units produced by each worker follows.
Use these data and test to see whether the mean number of parts produced is the same with each method. Use $\alpha=.05$.

Sheryl Ezze

Numerade Educator

Problem 39

In a study conducted to investigate browsing activity by shoppers, each shopper was initially classified as a nonbrowser, light browser, or heavy browser. For each shopper, the study obtained a measure to determine how comfortable the shopper was in a store. Higher scores indicated greater comfort. Suppose the following data were collected.
a. Use $\alpha=.05$ to test for differences among comfort levels for the three types of browsers.
b. Use Fisher's LSD procedure to compare the comfort levels of nonbrowsers and light browsers. Use $\alpha=.05 .$ What is your conclusion?

Dominador Tan

Numerade Educator

Problem 40

A research firm tests the miles-per-gallon characteristics of three brands of gasoline. Because of different gasoline performance characteristics in different brands of automobiles, five brands of automobiles are selected and treated as blocks in the experiment; that is, each brand of automobile is tested with each type of gasoline. The results of the experiment (in miles per gallon) follow.
a. $\quad$ At $\alpha=.05,$ is there a significant difference in the mean miles-per-gallon characteristics of the three brands of gasoline?
b. Analyze the experimental data using the ANOVA procedure for completely randomized designs. Compare your findings with those obtained in part
(a). What is the advantage of attempting to remove the block effect?

Shu Naito

Numerade Educator

Problem 41

Wegmans Food Markets and Tops Friendly Markets are the major grocery chains in the Rochester, New York, area. When Wal-Mart opened a Supercenter in one of the Rochester suburbs, experts predicted that Wal-Mart would undersell both local stores. The Democrat and Chronicle obtained the price data in the following table for a 15 -item market basket (Democrat and Chronicle, March 17,2002 ). At the .05 level of significance, test for any significant difference in the mean price for the 15 -item shopping basket for the three stores.

Jeremiah Mbaria

Jeremiah Mbaria

Numerade Educator

Problem 42

The U.S. Department of Housing and Urban Development provides data that show the fair market monthly rent for metropolitan areas. The following data show the fair market monthly rent $(\$)$ in 2005 for 1 -bedroom, 2 -bedroom, and 3 -bedroom apartments for five metropolitan areas (The New York Times Almanac, 2006).
At the .05 level of significance, test whether the mean fair market monthly rent is the same for each metropolitan area.

Srikar Katta

Numerade Educator

Problem 43

A factorial experiment was designed to test for any significant differences in the time needed to perform English to foreign language translations with two computerized language translators. Because the type of language translated was also considered a significant factor, translations were made with both systems for three different languages: Spanish, French, and German. Use the following data for translation time in hours. Test for any significant differences due to language translator, type of language, and interaction. Use $\alpha=.05$.

Shu Naito

Numerade Educator

Problem 44

A manufacturing company designed a factorial experiment to determine whether the number of defective parts produced by two machines differed and if the number of defective parts produced also depended on whether the raw material needed by each machine was loaded manually or by an automatic feed system. The following data give the numbers of defective parts produced. Use $\alpha=.05$ to test for any significant effect due to machine, loading system, and interaction.

Shu Naito

Numerade Educator