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Statistics for Business and Economics

Paul Newbold, William Carlson, Betty Thorne

Chapter 15

Analysis of Variance - all with Video Answers

Educators

Chapter Questions

03:19

Problem 1

Given the following analysis of variance table, compute mean squares for between groups and within groups. Compute the $F$ ratio and test the hypothesis that the group means are equal.
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 1,000 & 4 \\
\text { Within groups } & 750 & 15 \\
\text { Total } & 1,750 & 19 \\
\hline
\end{array}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:28

Problem 2

Given the following analysis of variance table, compute mean squares for between groups and within groups. Compute the $F$ ratio and test the hypothesis that the group means are equal.
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 879 & 3 \\
\text { Within groups } & 798 & 16 \\
\text { Total } & 1,677 & 19 \\
\hline
\end{array}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:57

Problem 3

Given the following analysis of variance table, compute mean squares for between groups and within groups. Compute the $F$ ratio and test the hypothesis that the group means are equal.
$$
\begin{array}{lcc}
\begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 1,000 & 2 \\
\text { Within groups } & 743 & 15 \\
\text { Total } & 1,743 & 17 \\
\hline
\end{array}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:54

Problem 4

A manufacturer of cereal is considering three alternative box colors-red, yellow, and blue. To check whether such a consideration has any effect on sales,
16 stores of approximately equal size are chosen. Red boxes are sent to 6 of these stores, yellow boxes to 5 others, and blue boxes to the remaining 5. After a few days a check is made on the number of sales in each store. The results (in tens of boxes) shown in the following table were obtained.
$$
\begin{array}{ccc}
\hline \text { Red } & \text { Yellow } & \text { Blue } \\
\hline 43 & 52 & 61 \\
52 & 37 & 29 \\
59 & 38 & 38 \\
76 & 64 & 53 \\
61 & 74 & 79 \\
81 & & \\
\hline
\end{array}
$$
a. Calculate the within-groups, between-groups, and total sum of squares.
b. Complete the analysis of variance table, and test the null hypothesis that the population mean sales levels are the same for all three box colors.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:15

Problem 5

An instructor has a class of 23 students. At the beginning of the semester, each student is randomly assigned to one of four teaching assistants-Smiley, Haydon, Alleline, or Bland. The students are encouraged to meet with their assigned teaching assistant to discuss difficult course material. At the end of the semester, a common examination is administered. The scores obtained by students working with these teaching assistants are shown in the accompanying table.
$$
\begin{array}{cccc}
\hline \text { Smiley } & \text { Haydon } & \text { Alleline } & \text { Bland } \\
\hline 72 & 78 & 80 & 79 \\
69 & 93 & 68 & 70 \\
84 & 79 & 59 & 61 \\
76 & 97 & 75 & 74 \\
64 & 88 & 82 & 85 \\
& 81 & 68 & 63 \\
\hline
\end{array}
$$
a. Calculate the within-groups, between-groups, and total sum of squares.
b. Complete the analysis of variance table and test the null hypothesis of equality of population mean scores for the teaching assistants.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:05

Problem 6

Three suppliers provide parts in shipments of 500 units. Random samples of six shipments from each of the three suppliers were carefully checked, and the numbers of parts not conforming to standards were recorded. These. numbers are listed in the following table:
$$
\begin{array}{ccc}
\hline \text { Supplier A } & \text { Supplier B } & \text { Supplier C } \\
\hline 28 & 22 & 33 \\
37 & 27 & 29 \\
34 & 29 & 39 \\
29 & 20 & 33 \\
31 & 18 & 37 \\
33 & 30 & 38 \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table for these data.
b. Test the null hypothesis that the population mean numbers of parts per shipments not conforming to standards are the same for all three suppliers.
c. Compute the minimum significant difference and indicate which subgroups have different means.

John Piaszynski
John Piaszynski
Numerade Educator
03:11

Problem 7

A corporation is trying to decide which of three makes of automobile to order for its fleet-domestic, Japanese, or European. Five cars of each type were ordered, and, after 10,000 miles of driving, the operating cost per mile of each was assessed. The accompanying results in cents per mile were obtained.
$$
\begin{array}{ccc}
\hline \text { Domestic } & \text { Japanese } & \text { European } \\
\hline 18.0 & 20.1 & 19.3 \\
15.6 & 15.6 & 15.4 \\
15.4 & 16.1 & 15.1 \\
19.1 & 15.3 & 18.6 \\
16.9 & 15.4 & 16.1 \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table for these data.
b. Test the null hypothesis that the population mean operating costs per mile are the same for these three types of car.
c. Compute the minimum significant difference and indicate which subgroups have different means.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:21

Problem 8

Random samples of seven freshmen, seven sophomores, and seven juniors taking a business statistics class were drawn. The accompanying table shows scores on the final examination.
$$
\begin{array}{ccc}
\hline \text { Freshmen } & \text { Sophomores } & \text { Juniors } \\
\hline 82 & 71 & 64 \\
93 & 62 & 73 \\
61 & 85 & 87 \\
74 & 94 & 91 \\
69 & 78 & 56 \\
70 & 66 & 78 \\
53 & 71 & 87 \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table.
b. Test the null hypothesis that the three population mean scores are equal.
c. Compute the minimum significant difference and indicate which subgroups have different means.

Jon Southam
Jon Southam
Numerade Educator
02:23

Problem 9

Samples of four salespeople from each of four regions were asked to predict percentage increases in sales volume for their territories in the next 12 months. The predictions are shown in the accompanying table.
$$
\begin{array}{cccc}
\hline \text { West } & \text { Midwest } & \text { South } & \text { East } \\
\hline 6.8 & 7.2 & 4.2 & 9.0 \\
4.2 & 6.6 & 4.8 & 8.0 \\
5.4 & 5.8 & 5.8 & 7.2 \\
5.0 & 7.0 & 4.6 & 7.6 \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table.
b. Test the null hypothesis that the four population mean sales growth predictions are equal.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:26

Problem 10

Independent random samples of six assistant professors, four associate professors, and five full professors were asked to estimate the amount of time outside the classroom spent on teaching responsibilities in the last week. Results, in hours, are shown in the accompanying table.
$$
\begin{array}{rcc}
\hline \text { Assistant } & \text { Associate } & \text { Full } \\
\hline 7 & 15 & 11 \\
12 & 12 & 7 \\
11 & 15 & 6 \\
15 & 8 & 9 \\
9 & & 7 \\
14 & & \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table.
b. Test the null hypothesis that the three population mean times are equal.

Dominador Tan
Dominador Tan
Numerade Educator
05:47

Problem 11

Two tutoring services offer crash courses in preparation for the CPA exam. To check on the effectiveness of these services, 15 students were chosen. Five students. were randomly assigned to service $\mathrm{A}, 5$ were assigned to service $\mathrm{B}$, and the remaining 5 did not take a crash course. Their scores on the examination, expressed as percentages, are given in the table.
$$
\begin{array}{ccc}
\hline \begin{array}{c}
\text { Service A } \\
\text { Course }
\end{array} & \begin{array}{c}
\text { Service B } \\
\text { Course }
\end{array} & \begin{array}{c}
\text { No } \\
\text { Course }
\end{array} \\
\hline 79 & 74 & 72 \\
74 & 69 & 71 \\
92 & 87 & 81 \\
67 & 81 & 61 \\
85 & 64 & 63 \\
\hline
\end{array}
$$
a. Prepare the analysis of variance table.
b. Test the null hypothesis that the three population mean scores are the same.
c. Compute the minimum significant difference and indicate which subgroups have different means.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:42

Problem 12

In the study of Example 15.1, independent random samples of six advertisements from True Confessions, People Weekly, and Newsweek were taken. The fog indices for these advertisements are given in the accompanying table. Test the null hypothesis that the population mean fog indices are the same for advertisements in these three magazines and compute the minimum significant difference and indicate which subgroups have different means.
$$
\begin{array}{ccc}
\hline \text { True Confessions } & \text { People Weekly } & \text { Newstoeek } \\
\hline 12.89 & 9.50 & 10.21 \\
12.69 & 8.60 & 9.66 \\
11.15 & 8.59 & 7.67 \\
9.52 & 6.50 & 5.12 \\
9.12 & 4.79 & 4.88 \\
7.04 & 4.29 & 3.12 \\
\hline \hline
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:04

Problem 13

For the one-way analysis of variance model, we write the jth observation from the $i$ th group as
$$
X_{i j}=\mu+G_{i}+\varepsilon_{i j}
$$
where $\mu$ is the overall mean, $G_{i}$ is the effect specific to the, ith group, and $\varepsilon_{i j}$ is a random error for the jth observation from the ith group. Consider the data of Example 15.1.
a. Estimate $\mu$.
b. Estimate $G_{i}$ for each of the three magazines.
c. Estimate $\varepsilon_{32}$, the error term corresponding to the second observation $(8.28)$ for the New Yorker.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:04

Problem 14

Use the model for the oneway analysis of variance for the data of Exercise $15.12$.
a. Estimate $\mu$
b. Estimate $G_{i}$ for each of the three magazines.
c. Estimate $\varepsilon_{13}$, the error term corresponding to the third observation (11.15) for True Confessions.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:27

Problem 15

Consider a problem with three subgroups with the sum of ranks in each of the subgroups equal to 45,98, and 88 and with subgroup sizes equal to 6,6, and $7 .$ Complete the Kruskal-Wallis test and test the null hypothesis of equal subgroup ranks.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:16

Problem 16

Consider a problem with four subgroups with the sum of ranks in each of the subgroups equal to 49,84 , 76 , and 81 and with subgroup sizes equal to $4,6,7$, and $6 .$ Complete the Kruskal-Wallis test and test the null hypothesis of equal subgroup ranks.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:18

Problem 17

Consider a problem with four subgroups with the sum of ranks in each of the subgroups equal to 71,88, 82, and 79 and with subgroup sizes equal to $5,6,6$, and 7. Complete the Kruskal-Wallis test and test the null hypothesis of equal subgroup ranks.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:34

Problem 18

For the data of Exercise $15.4$, use the Kruskal-Wallis test of the null hypothesis that the population mean sales levels are identical for three box colors.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:50

Problem 19

Using the data of Exercise 15.5, perform a KruskalWallis test of the null hypothesis that the population mean test scores are the same for students assigned to the four teaching assistants.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:24

Problem 20

Using the data of Exercise 15.6, carry out a test of the null hypothesis of equality of the three population mean numbers of parts per shipment not conforming to standards without assuming normality of population distributions.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:15

Problem 21

For the data of Exercise 15.7, test the null hypothesis that the population mean operating costs per mile are the same for all three types of automobiles without assuming normal population distributions.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:10

Problem 22

Using the data of Exercise $15.8$, carry out a nonparametric test of the null hypothesis of equality of population mean examination scores for freshmen, sophomores, and juniors.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:13

Problem 23

Based on the data of Exercise 15.9, use the KruskalWallis method to test the null hypothesis of equality of growth predictions for population mean sales for the four regions.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:48

Problem 24

Refer to Exercise 15.10. Without assuming normal population distributions, test the null hypothesis that the population mean times spent outside the classroom on teaching responsibilities are the same for assistant, associate, and full professors.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:24

Problem 25

Based on the data of Exercise 15.11, perform the KruskalWallis test of the null hypothesis of equal population mean scores on the CPA exam for students using no tutoring services and using services $\mathrm{A}$ and $\mathrm{B}$.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:47

Problem 26

Independent random samples of 101 college sophomores, 112 college juniors, and 96 college seniors were asked to rate, on a scale of 1 to 7, the importance attached to brand name when purchasing a car. The obtained value of the Kruskal-Wallis statistic was $0.15$.
a. What null hypothesis can be tested using this information?
b. Carry out this test.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:28

Problem 27

Consider a two-way analysis of variance with one observation per cell and randomized blocks with the following results:
$$
\begin{array}{lcc}
\begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 231 & 4 \\
\text { Between blocks } & 348 & 5 \\
\text { Error } & 550 & 20 \\
\text { Total } & 1,129 & 29 \\
\hline
\end{array}
$$
Compute the mean squares and test the hypotheses that between-group means are equal and betweenblock means are eoual.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:28

Problem 28

Consider a two-way analysis of variance with one observation per cell and randomized blocks with the following results:
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 380 & 6 \\
\text { Between blocks } & 232 & 5 \\
\text { Error } & 387 & 30 \\
\text { Total } & 989 & 41 \\
\hline
\end{array}
$$
Compute the mean squares and test the hypotheses that between-group means are equal and betweenblock means are equal.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:28

Problem 29

Consider a two-way analysis of variance with one observation per cell and randomized blocks with the following results:
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 131 & 3 \\
\text { Between blocks } & 287 & 6 \\
\text { Error } & 360 & 18 \\
\text { Total } & 778 & 27 \\
\hline
\end{array}
$$
Compute the mean squares and test the hypotheses that between-group means are equal and betweenblock means are equal.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:19

Problem 30

Four financial analysts were asked to predict earnings growth over the coming year for five oil companies. Their forecasts, as projected percentage
increases in earnings, are given in the accompanying table.
a. Prepare the two-way analysis of variance table.
b. Test the null hypothesis that the population mean growth forecasts are the same for all oil companies.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
06:33

Problem 31

An agricultural experiment designed to assess differences in yields of corn for four different varieties, using three different fertilizers, produced the results (in bushels per acre) shown in the following table:
a. Prepare the two-way analysis of variance table.
b. Test the null hypothesis that the population mean yields are identical for all four varieties of corn.
c. Test the null hypothesis that population mean yields are the same for all three brands of fertilizer.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
02:00

Problem 32

A company has test-marketed three new types of soup in selected stores over a period of 1 year. The following table records sales achieved (in thousands of dollars) for each of the three soups in each quarter of the year.
a. Prepare the two-way analysis of variance table.
b. Test the null hypothesis that population mean sales are the same for all three types of soup.

Jameson Kuper
Jameson Kuper
Numerade Educator
02:00

Problem 33

A company has test-marketed three new types of soup in selected stores over a period of 1 year. The following table records sales achieved (in thousands of dollars) for each of the three soups in each quarter of the year.
a. Prepare the two-way analysis of variance table.
b. Test the null hypothesis that population mean sales are the same for all three types of soup.

Jameson Kuper
Jameson Kuper
Numerade Educator
02:54

Problem 33

A diet soda manufacturer wants to compare the effects on sales of three can colors-red, yellow, and blue. Four regions are selected for the test, and three stores are randomly chosen from each region, each to display one color of cans. The accompanying table shows sales (in tens of cans) at the end of the experimental period.
a. Prepare the appropriate analysis of variance table.
b. Test the null hypothesis that population mean sales are the same for each can color.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:54

Problem 34

An instructor in an economics class is considering three different texts. He is also considering three types of examinations - multiple choice, essay, and a mix of multiple choice and essay questions. During the year he teaches nine sections of the course and randomly assigns a text-examination type combination of each section. At the end of the course he obtained students' evaluations for each section. These ratings are shown in the accompanying table.
a. Prepare the analysis of variance table.
b. Test the null hypothesis of equality of population mean ratings for the three texts.
c. Test the null hypothesis of equality of population mean ratings for the three examination types.

James Kiss
James Kiss
Numerade Educator
02:10

Problem 35

We introduced for the two-way analysis of variance the population model
$$
X_{i j}-\mu=G_{i}+\beta_{i}+\varepsilon_{i j}
$$
For the data of Exercise 15.33, obtain sample estimates for each term on the right-hand side of this equation for the east region-red can combination.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:16

Problem 36

For the data of Exercise $15.34$, obtain sample estimates for each term on the right-hand side of the equation used in the previous exercise for the text C-multiple choice combination.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:08

Problem 37

Four real estate agents were asked to appraise the values of 10 houses in a particular neighborhood. The appraisals were expressed in thousands of dollars, with the results shown in the following table.
a. Complete the analysis of variance table.
b. Test the null hypothesis that population mean assessments are the same for these four real estate agents.

Adriano Chikande
Adriano Chikande
Numerade Educator
10:01

Problem 38

Four brands of fertilizer were evaluated. Each brand was applied to each six plots of land containing soils of different types. Percentage increases in corn yields were then measured for the 24 brand-soil-type combinations. The results obtained are summarized in the accompanying table.
a. Complete the analysis of variance table.
b. Test the null hypothesis that population mean yield increases are the same for the four fertilizers.
c. Test the null hypothesis that population mean yield increases are the same for the six soil types.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
02:07

Problem 39

Three television pilots for potential situation-comedy series were shown to audiences in four regions of the country-the East, the South, the Midwest, and the West Coast. Based on audience reactions, a score (on a scale from 0 to 100 ) was obtained for each show. The sums of squares between groups (shows) and between blocks (regions) were found to be
$$
\mathrm{SSG}=95.2 \text { and } \mathrm{SSB}=69.5
$$
and the error sum of squares was as follows:
$$
\mathrm{SSE}=79.3
$$
Prepare the analysis of variance table, and test the null hypothesis that the population mean scores for audience reactions are the same for all three shows.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:57

Problem 40

Suppose that, in the two-way analysis of variance setup with one observation per cell, there are just two groups. Show in this case that the $F$ ratio for testing the equality of the group population means is precisely the square of the test-statistic discussed in Section $10.1$ for testing equality of population means, given a sample of matched pairs. Hence, deduce that the two tests are equivalent in this particular case.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:09

Problem 41

Consider an experiment with treatment factors A and B, with factor A having four levels and factor B having three levels. The results of the experiment are summarized in the following analysis of variance table.
Compute the mean squares and test the null hypotheses of no effect from either treatment and no interaction effect.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
05:00

Problem 42

Consider an experiment with treatment factors A and B, with factor A having five levels and factor B having six levels. The results of the experiment are summarized in the following analysis of variance table:
$$
\begin{array}{lcc}
\hline \text { Source of Variation } & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Treatment A groups } & 86 & 4 \\
\text { Treatment B groups } & 75 & 5 \\
\text { Interaction } & 75 & 20 \\
\text { Error } & 300 & 90 \\
\text { Total } & 536 & 119 \\
\hline
\end{array}
$$
Compute the mean squares and test the null hypotheses of no effect from either treatment and no interaction effect

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:17

Problem 43

Consider an experiment with treatment factors A and B, with factor A having three levels and factor B having seven levels. The results of the experiment are summarized in the following analysis of variance table:
$$
\begin{array}{lcc}
\hline \text { Source of Variation } & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Treatment A groups } & 37 & 2 \\
\text { Treatment B groups } & 58 & 6 \\
\text { Interaction } & 57 & 12 \\
\text { Error } & 273 & 84 \\
\text { Total } & 425 & 104 \\
\hline
\end{array}
$$
Compute the mean squares and test the null hypotheses of no effect from either treatment and no interaction effect.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
02:07

Problem 44

Suppose that scores given by judges to competitors in the ski-jumping events of the Winter Olympics were analyzed. For the men's ski-jumping competition, suppose there were 22 contestants and
9 judges. Each judge in seven subevents assessed each contestant. The scores given can, thus, be treated in the framework of a two-way analysis of variance with 198 contestant-judge cells, seven observations per cell. The sums of squares are given in the following table:
$$
\begin{array}{lr}
\hline \text { Source of Variation } & \text { Sum of Squares } \\
\hline \text { Between contestants } & 364.50 \\
\text { Between judges } & 0.81 \\
\text { Interaction } & 4.94 \\
\text { Error } & 1,069.94 \\
\hline
\end{array}
$$
a. Complete the analysis of variance table.
b. Carry out the associated $F$ tests and interpret your findings.

Adriano Chikande
Adriano Chikande
Numerade Educator
View

Problem 45

Refer to Exercise 15.44. Twelve pairs were entered in the ice-dancing competition. Once again, there were 9 judges, and contestants were assessed in seven subevents. The sums of squares between groups (pairs of contestants) and between blocks (judges) were found to be
$$
\mathrm{SSG}=60.10 \text { and } \mathrm{SSB}=1.65
$$
while the interaction and error sums of squares were as follows:
$$
\mathrm{SSI}=3.35 \text { and } \mathrm{SSE}=31.61
$$
Analyze these results and verbally interpret the conclusions.

Victor Salazar
Victor Salazar
Numerade Educator
01:54

Problem 46

A psychologist is working with three types of aptitude tests that may be given to prospective management trainees. In deciding how to structure the testing process, an important issue is the possibility of interaction between test takers and test type. If there were no interaction, only one type of test would be needed. Three tests of each type are given to members of each of four groups of subject type. These were distinguished by ratings of poor, fair, good, and excellent in preliminary interviews. The scores obtained are listed in the following table:
a. Set up the analysis of variance table.
b. Test the null hypothesis of no interaction between subject type and test type.

James Kiss
James Kiss
Numerade Educator
01:32

Problem 47

Random samples of two freshmen, two sophomores, two juniors, and two seniors each from four dormitories were asked to rate, on a scale of 1 (poor) to 10 (excellent), the quality of the dormitory
environment for studying. The results are shown in the following table:
a. Set up the analysis of variance table.
b. Test the null hypothesis that the population mean ratings are the same for the four dormitories.
c. Test the null hypothesis that the population mean ratings are the same for the four student years.
d. Test the null hypothesis of no interaction between student year and dormitory rating.

Akhil Choudhary
Akhil Choudhary
Numerade Educator
03:37

Problem 48

In some experiments with several observations per cell the analyst is prepared to assume that there is no interaction between groups and blocks. Any apparen interaction found is then attributed to random error When such an assumption is made, the analysis is carried out in the usual way, except that what were previously the interaction and error sums of squares are now added together to form a new error sum of squares. Similarly, the corresponding degrees of freedom are added. If the assumption of no interaction is correct, this approach has the advantage of providing more error degrees of freedom and, hence, more powerful tests of the equality of group and block means? For the study of Exercise $15.47$, suppose that we now make the assumption of no interaction between dor mitory ratings and student years.
a. State, in your own words, what is implied by this assumption.
b. Given this assumption, set up the new analysis of variance table.
c. Test the null hypothesis that the population mean ratings are the same for all dormitories.
d. Test the null hypothesis that the population mean ratings are the same for all four student vears.

Jon Southam
Jon Southam
Numerade Educator
01:40

Problem 49

Refer to Exercise 15.31. Having carried out the experiment to compare mean yields per acre of four varieties of corn and three brands of fertilizer, an agricultural researcher suggested that there might be some interaction between variety and fertilizer. To check this
possibility, another set of trials was carried out, producing the yields in the table.
a. What would be implied by an interaction between variety and fertilizer?
b. Combine the data from the two sets of trials and set up an analysis of variance table.
c. Test the null hypothesis that the population mean yield is the same for all four varieties of corn.
d. Test the null hypothesis that the population mean yield is the same for all three brands of fertilizer.
e. Test the null hypothesis of no interaction between variety of corn and brand of fertilizer.

James Kiss
James Kiss
Numerade Educator
00:37

Problem 50

Refer to Exercise 15.33. Suppose that a second store for each region-can color combination is added to the study, yielding the results shown in the following table. Combining these results with those of Exercise 15.33, carry out the analysis of variance calculations and discuss your findings.

Victor Salazar
Victor Salazar
Numerade Educator
03:07

Problem 51

Having carried out the study of Exercise $15.34$, the instructor decided to replicate the study the following year. The results obtained are shown in the table. Combining these results with those of Exercise 15.34, carry out the analysis of variance calculations and discuss your findings.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
01:14

Problem 52

Carefully distinguish between the one-way analysis of variance framework and the two-way analysis of variance framework. Give examples different from those discussed in the text and exercises of business problems for which each might be appropriate.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:56

Problem 53

Carefully explain what is meant by the interaction effect in the two-way analysis of variance with more than one observation per cell. Give examples of this effect in business-related problems.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:27

Problem 54

Consider a study to assess the readability of financial report messages. The effectiveness of the written message is assessed using a standard procedure. Financial reports were given to independent random samples from three groups-certified public accountants, chartered financial analysts, and commercial bank loan officer trainees. The procedure was then administered, and the scores for the sample members were recorded. The null. hypothesis of interest is that the population mean scores for the three groups are identical. Test this hypothesis, given the information in the accompanying table.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:27

Problem 55

In an experiment designed to assess aids to the success of interviews of graduate students carried out by faculty mentors, interviewers were randomly assigned to one of three interview modes-feedback, feedback and goal setting, and control. For the feedback mode interviewers had the opportunity to examine and discuss their graduate students' reactions to previous interviews. In the feedback-and-goal-setting mode, faculty mentors were encouraged to set goals for the forthcoming interview. For the control group, interviews were carried out in the usual way, without feedback or goal setting. After the interviews were completed, the satisfaction levels of the graduate students with the interviews were assessed. For the 45 people in the feedback group, the mean satisfaction level was $13.98$. The 49 people in the feedback-and-goal-setting group had a mean satisfaction level of $15.12$, whereas the 41 control group members had a mean satisfaction level of $13.07$. The $F$ ratio computed from the data was $4.12$.
a. Prepare the complete analysis of variance table.
b. Test the null hypothesis that the population mean satisfaction levels are the same for all three types of interview.

Mohan Jain
Mohan Jain
Numerade Educator
02:36

Problem 56

A study classified each of 134 lawyers into one of four groups based on observation and an interview. The 62 lawyers in group A were categorized as having high levels of stimulation and support and average levels of public spirit. The 52 lawyers in group B had low stimulation, average support, and high public spirit. Group $C$ contained 7 lawyers with average
stimulation, low support, and low public spirit. The 13 lawyers in group D were assessed as low on all three criteria. Salary levels for these four groups were compared. The sample means were $7.87$ for group A, $7.47$ for group B, $5.14$ for group $C$, and $3.69$ for group D. The $F$ ratio calculated from these data was $25.60$.
a. Prepare the complete analysis of variance table.
b. Test the null hypothesis that the population mean salaries are the same for lawyers in these four groups.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:51

Problem 57

In a study to estimate the effects of smoking on routine health risk, employees were classified as continuous smokers, recent ex-smokers, long-term ex-smokers, and those who never smoked. Samples of $96,34,86$, and 206 members of these groups were taken. Sample mean numbers of mean health risk rates per month were found to be $2.15,2.21,1.47$, and $1.69$, respectively. The $F$ ratio calculated from these data was $2.56$.
a. Prepare the complete analysis of variance table.
b. Test the null hypothesis of equality of the four population mean health risk rates.

Nick Johnson
Nick Johnson
Numerade Educator
02:03

Problem 58

Michigan has had restrictions on price advertising for wine. However, for a period these restrictions were lifted. Data were collected on total wine sales over three periods of time-under restricted price advertising, with restrictions lifted, and after the re-imposition of restrictions. The accompanying table shows sums of squares and degrees of freedom. Assuming that the usual requirements for the analysis of variance are met-in particular, that sample observations are independent of one another-test the null hypothesis of equality of population mean sales in these three time periods.
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between groups } & 11,438.3028 & 2 \\
\text { Within groups } & 109,200.0000 & 15 \\
\text { Total } & 120,638.3028 & 17 \\
\hline
\end{array}
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
02:23

Problem 59

Independent random samples of the selling prices of houses in four districts were taken. The selling prices (in thousands of dollars) are shown in the accompanying table. Test the null hypothesis that population mean selling prices are the same in all four districts.
$$
\begin{array}{cccc}
\hline \text { District A } & \text { District B } & \text { District C } & \text { District D } \\
\hline 73 & 85 & 97 & 61 \\
63 & 59 & 86 & 67 \\
89 & 84 & 76 & 84 \\
75 & 70 & 78 & 67 \\
70 & 80 & 76 & 69 \\
\hline
\end{array}
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
02:29

Problem 60

For the data of Exercise 15.59, use the Kruskal-Wallis test to test the null hypothesis that the population mean selling prices of houses are the same in the four districts.

Adriano Chikande
Adriano Chikande
Numerade Educator
05:57

Problem 61

A study was aimed at assessing the class-schedule satisfaction levels, on a scale of 1 (very dissatisfied) to 7 (very satisfied), of nontenured faculty who were job-sharers, full time, or part-time. For a sample of 25 job-sharers, the mean satisfaction level was $6.60$; for a sample of 24 full-time faculty, the mean satisfaction level was $5.37 ;$ for a sample of 20 part-time faculty, the mean satisfaction level was $5.20$. The $F$ ratio calculated from these data was $6.62$.
a. Prepare the complete analysis of variance table.
b. Test the null hypothesis of equality of the three population mean satisfaction levels.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:00

Problem 62

Consider the one-way analysis of variance setup.
a. Show that the within-groups sum of squares can be written as follows:
$$
\mathrm{SSW}=\sum_{i=1}^{K} \sum_{j=1}^{n_{1}} x_{i j}^{2}-\sum_{i=1}^{K} n_{p} \bar{x}_{1}^{2}
$$
b. Show that the between-groups sum of squares can be written as follows:
$$
\mathrm{SSG}=\sum_{i=1}^{K} n_{i} \bar{x}_{l}^{2}-n \bar{x}^{2}
$$
c. Show that the total sum of squares can be written as follows:
$$
\mathrm{SST}=\sum_{i=1}^{K} \sum_{j=1}^{M_{1}} x_{i j}^{2}-n \bar{x}^{2}
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
02:00

Problem 63

Consider the two-way analysis of variance setup, with one observation per cell.
a. Show that the between-groups sum of squares can be written as follows:
$$
\mathrm{SSG}=H \sum_{i=1}^{K} \bar{x}_{i \cdot}^{2}-n \bar{x}^{2}
$$
b. Show that the between-blocks sum of squares can be written as follows:
$$
\mathrm{SSB}=K \sum_{=1}^{H} \bar{x}_{, j}^{2}-n \bar{x}^{2}
$$
c. Show that the total sum of squares can be written, as follows:
$$
\mathrm{SST}=\sum_{\mathrm{i}=1}^{\mathrm{K}} \sum_{j=1}^{H} x_{\hat{j}}^{2}-n \overline{\bar{x}}^{2}
$$
d. Show that the error sum of squares can be written as follows:
$$
\mathrm{SSE}=\sum_{i=1}^{K} \sum_{j=1}^{H} x_{i j}^{2}-H \sum_{i=1}^{K} \bar{x}_{i}^{2}-K \sum_{j=1}^{H} \bar{x}_{\nu}^{2}+n \bar{x}^{2}
$$

Adriano Chikande
Adriano Chikande
Numerade Educator
01:56

Problem 64

A survey indicates that soccer supporters can be divided into three spending categories when going to a game: high, medium, and low. These values were obtained from a sample of 235 people. The sums of squares for these levels of spending are given in the accompanying table. Complete the analysis of variance table, and test the null hypothesis that there is no difference in spending between supporter groups.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:08

Problem 65

Three real estate agents were each asked to assess the values of five houses in a neighborhood. The results, in thousands of dollars, are given in the table. Prepare the analysis of variance table, and test the null hypothesis that population mean valuations are the same for the three real estate agents.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:36

Problem 66

Students were classified according to three parental income groups and also according to three possible score ranges on the SAT examination. One student was chosen randomly from each of the nine cross-classifications, and the grade point averages of those sample members at the end of the sophomore year were recorded. The results are shown in the accompanying table.a. Prepare the analysis of variance table.
b. Test the null hypothesis that the population mean grade point averages are the same for all three income groups.
c. Test the null hypothesis that the population mean grade point averages are the same for all three SAT' score groups.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:58

Problem 67

For the two-way analysis of variance model with one observation per cell, write the observation from the ith group and $j$ th block as
$$
X_{i j}=\mu+G_{i}+B_{j}+\varepsilon_{i j}
$$
Refer to Exercise $15.65$ and consider the observation on agent $\mathrm{B}$ and house $1\left(x_{21}=218\right)$
a. Estimate $\mu$.
b. Estimate and interpret $G_{2}$.
c. Estimate and interpret $B_{1}$.
d. Estimate $\varepsilon_{21}$.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:06

Problem 68

Refer to Exercise $15.66$ and consider the observation on moderate-income group and high SAT score $\left(x_{22}=3.5\right)$.
a. Estimate $\mu$.
b. Estimate and interpret $\mathrm{G}_{2}$.
c. Estimate and interpret $B_{2}$.
d. Estimate $\varepsilon_{22}$.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:00

Problem 69

Consider the two-way analysis of variance setup, with n observations per cell.
a. Show that the between-groups sum of squares can be written as follows:
$$
\mathrm{SSG}=H m \sum_{i=1}^{K} \bar{x}_{i}^{2} .-H K m \bar{x}^{2}
$$
b. Show that the between-blocks sum of squares can be written as follows:
$$
\mathrm{SSB}=K m \sum_{j=1}^{H} \bar{x}_{\gamma}^{2}-H K m \bar{x}^{2}
$$
c. Show that the error sum of squares can be written as follows:
$$
\operatorname{SSE}=\sum_{i=1}^{K} \sum_{j=1}^{H} \sum_{i=1}^{m} x_{i j}^{2}-m \sum_{i=1}^{K} \sum_{j=1}^{H} \bar{x}_{i j}^{2}
$$
d. Show that the total sum of squares can be written as follows:
$$
\mathrm{SST}=\sum_{i=1}^{K} \sum_{j=1}^{H} \sum_{i=1}^{m} x_{i j}^{2}-H K m \bar{x}^{2}
$$
e. Show that the interaction sum of squares can be written as follows:

Adriano Chikande
Adriano Chikande
Numerade Educator
02:58

Problem 70

Purchasing agents were given information about a cellular phone system and asked to assess its quality. The information given was identical except for two factors - price and country of origin. For price there were three possibilities: $\$ 150, \$ 80$, and no price given. For country of origin there were also three possibilities: United States, Taiwan, and no country given. Part of the analysis of variance table for the quality assessments of the purchasing agents is shown here. Complete the analysis of variance table and provide a full analysis of these data.
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between prices } & 0.178 & 2 \\
\text { Between countries } & 4.365 & 2 \\
\text { Interaction } & 1.262 & 4 \\
\text { Error } & 93.330 & 99 \\
\hline
\end{array}
$$

Michaela Flitsch
Michaela Flitsch
Numerade Educator
01:15

Problem 71

In the study of Exercise 15.70, information on the cellular phone system was also shown to MBA students. Part of the analysis of variance table for their quality assessments is shown here. Complete the analysis of variance table and provide a full analysis of these data.
$$
\begin{array}{lcc}
\hline \begin{array}{l}
\text { Source of } \\
\text { Variation }
\end{array} & \begin{array}{c}
\text { Sum of } \\
\text { Squares }
\end{array} & \begin{array}{c}
\text { Degrees of } \\
\text { Freedom }
\end{array} \\
\hline \text { Between prices } & 0.042 & 2 \\
\text { Between countries } & 17.319 & 2 \\
\text { Interaction } & 2.235 & 4 \\
\text { Error } & 70.414 & 45 \\
\hline
\end{array}
$$

Nick Johnson
Nick Johnson
Numerade Educator
02:31

Problem 72

Having carried out the study of Exercise $15.66$, the investigator decided to take a second independent random sample of one student from each of the nine income-SAT score categories. The grade point averages found are given in the accompanying table.
a. Prepare the analysis of variance table.
b. Test the null hypothesis that the population mean grade point averages are the same for all three income groups.
c. Test the null hypothesis that the population mean grade point averages are the same for all three SAT ' score groups.
d. Test the null hypothesis of no interaction between income group and SAT score.

Karen Song
Karen Song
Numerade Educator
01:40

Problem 73

An experiment was carried out to test the effects on yields of five varieties of corn and five types of fertilizer. For each variety-fertilizer combination, six plots were used and the yields recorded, with the results. shown in the following table:
a. Test the null hypothesis that the population mean yields are the same for all five varieties of corn.
b. Test the null hypothesis that the population mean yields are the same for all five brands of fertilizer.
c. Test the null hypothesis of no interaction between variety and fertilizer.

James Kiss
James Kiss
Numerade Educator