Deck 12: Analysis of Variance

Question 1

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-A consumer magazine wants to compare the lifetimes of ballpoint pens of three different types. The magazine takes a random sample of pens of each type in the following table.
$\begin{array}{lll}\text { brandA }&\text {brand B }&\text {brand C }\\\hline260 & 181 & 238 \\218 & 240 & 257 \\184 & 162 & 241 \\219 & 218 & 213\end{array}$
Do the data indicate that there is a difference in mean lifetime for the three brands of ballpoint pens? Use $\text { Use } \alpha = 0.0$

Answer 1

Test statistic: F = 1.620. Critical value: F = 8.02. p-value:0.251.
Fail to reject the claim of equal means. The data do not provide sufficient evidence to conclude that there is a difference in the mean lifetimes of the three brands of ballpoint pen.

Question 2

Provide an appropriate response.

-Describe the null and alternative hypotheses for one-way ANOVA. Give an example.

Answer 2

The null hypothesis for one-way ANOVA is that three or more means are equal. The alternative hypothesis is that the means are not all equal. Examples will vary.

Question 3

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-The data below represent the weight losses for people on three different exercise programs.
$\begin{array}{lll}\text { Exercise A}&\text { Exercise B}& \text {Exercise C }\\\hline2.5 & 5.8 & 4.3 \\8.8 & 4.9 & 6.2 \\7.3 & 1.1 & 5.8 \\9.8 & 7.8 & 8.1 \\5.1 & 1.2 & 7.9\end{array}$
At the 1% significance level, does it appear that a difference exists in the true mean weight loss produced by the three exercise programs?

Answer 3

Test statistic: F = 1.491. Critical value: F = 6.93. p-value:0.264.
Fail to reject the claim of equal means. The data do not provide sufficient evidence to conclude that there is a difference in the true mean weight loss produced by the three exercise programs.

Question 4

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results have been obtained. $\begin{array}{llll}\text { brandA }&\text {brand B }&\text {brand C }&\text {brand D }\\\hline15 & 20 & 21 & 15 \\25 & 17 & 22 & 15 \\21 & 22 & 20 & 14 \\23 & 23 & 19 & 23 \\22 & & 18 & 22 \\20 & & & 28\\&&&28\end{array}$

Question 5

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\\text { Error } & 8 & 50 & 6.25 & & \\\text { Total } & 11 & 80 & & &\end{array}$$ Identify the value of the test statistic.

A)30
B)10.00
C)0.264
D)1.6

Question 6

Provide an appropriate response.

Question 7

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\text { Error } & 16 & 13.925 & 0.870 & & \\\text { Total } & 19 & 27.425 & & &\end{array}$$ Identify the value of the test statistic.

A)13.500
B)0.011
C)5.17
D)4.500

Question 8

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-Given the sample data below, test the claim that the populations have the same mean. Use a significance level of 0.05. $$\begin{array} { l } \text { Brand A }& \text { Brand B } &\text { Brand C } & \text { Brand D } \\{ n = 16 } &{ n = 16 } & { n = 16 } &{ n = 16 } \\\bar { x } = 2.09& \bar { x } = 3.48 & \bar { x } = 1.86 & \bar { x } = 2.84 \\s = 0.37 & \mathrm {~s} = 0.61& \mathrm {~s} = 0.45 & \mathrm {~s} = 0.53 \\\end{array}$$

Question 9

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\text { Error } & 16 & 13.925 & 0.870 & & \\\text { Total } & 19 & 27.425 & & &\end{array}$$ What can you conclude about the equality of the population means?

A)Accept the null hypothesis since the p-value is less than the significance level.
B)Reject the null hypothesis since the p-value is greater than the significance level.
C)Reject the null hypothesis since the p-value is less than the significance level.
D)Accept the null hypothesis since the p-value is greater than the significance level.

Question 10

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-Given the sample data below, test the claim that the populations have the same mean. Use a significance level of 0.05. $\begin{array}{ccc}\text { Brand } \mathrm{A} &\text { Brand } \mathrm{B} &\text { Brand } \mathrm{C}\\\hline\mathrm{n}=10 & \mathrm{n}=10 & \mathrm{n}=10 \\\overline{\mathrm{x}}=32.6 & \overline{\mathrm{x}}=31.7 & \overline{\mathrm{x}}=27.4 \\\mathrm{~s}^{2}=4.51 & \mathrm{~s}^{2}=4.22 & \mathrm{~s}^{2}=4.69\end{array}$

Question 11

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c l c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\text { Error } & 16 & 13.925 & 0.870 & & \\\text { Total } & 19 & 27.425 & & &\end{array}$$ Identify the p-value.

A)0.011
B)4.500
C)5.17
D)0.870

Question 12

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\\text { Error } & 8 & 50 & 6.25 & & \\\text { Total } & 11 & 80 & & &\end{array}$$ Find the critical value.

A)1.6
B)8.85
C)4.07
D)7.59

Question 13

Provide an appropriate response.

- $$\text { The test statistic for one-way ANOVA is } F = \frac { \text { variance between samples } } { \text { variance within samples } } \text {. Describe variance within samples and }$$ variance between samples. What relationship between variance within samples and variance between samples would result in the conclusion that the value of F is significant?

Question 14

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-Random samples of four different models of cars were selected and the gas mileage of each car was measured. The results are shown below. $\begin{array}{llll}\text { Model A }&\text {Model B }&\text {Model C }&\text {Model D }\\\hline23 & 28 & 30 & 25 \\25 & 26 & 28 & 26 \\24 & 29 & 32 & 25 \\26 & 30 & 27 & 28\end{array}$ Test the claim that the four different models have the same population mean. Use a significance level of 0.05.

Question 15

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-At the 0.025 significance level, test the claim that the three brands have the same mean if the following sample results have been obtained. $\begin{array}{lll}\text { brand A }&\text {brand B }&\text {brand C }&\\\hline32 & 27 & 22 \\34 & 24 & 25 \\37 & 33 & 32 \\33 & 30 & 22 \\36 & & 21 \\39 & &\end{array}$

Question 16

Provide an appropriate response.

-When using statistical software packages, the critical value is typically not given. What method is used to determine whether you reject or fail to reject the null hypothesis?

Question 17

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\text { Error } & 16 & 13.925 & 0.870 & & \\\text { Total } & 19 & 27.425 & & &\end{array}$$ Find the critical value.

A)3.24
B)8.70
C)5.42
D)3.06

Question 18

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\\text { Error } & 8 & 50 & 6.25 & & \\\text { Total } & 11 & 80 & & &\end{array}$$ Identify the p-value.

A)0.264
B)10.00
C)6.25
D)1.6

Question 19

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean.

- $$\begin{array} { l r c c c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\\text { Error } & 8 & 50 & 6.25 & & \\\text { Total } & 11 & 80 & & &\end{array}$$ What can you conclude about the equality of the population means?

A)Reject the null hypothesis since the p-value is less than the significance level.
B)Reject the null hypothesis since the p-value is greater than the significance level.
C)Accept the null hypothesis since the p-value is greater than the significance level.
D)Accept the null hypothesis since the p-value is less than the significance level.

Question 20

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance.

-At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results have been obtained. $\begin{array}{llll}\text { brandA }&\text {brand B }&\text {brand C }&\text {brand D }\\\hline17 & 18 & 21 & 22 \\20 & 18 & 24 & 25 \\21 & 23 & 25 & 27 \\22 & 25 & 26 & 29 \\21 & 26 & 29 & 35 \\& & 29 & 36 \\& & & 37\end{array}$

Question 21

Provide an appropriate response.

-Fill in the missing entries in the following partially completed one-way ANOVA table. $\begin{array}{|l|r|r|l|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 3 & & & 11.16 \\\hline \text { Error } & & 13.72 & 0.686 & \\\hline \text { Total } & & & & \\\hline\end{array}$

A)
$\begin{array}{|l|r|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 3 & 0.184 & 0.061 & 11.16 \\\hline \text { Error } & 20 & 13.72 & 0.686 & \\\hline \text { Total } & 23 & 13.90 & & \\\hline\end{array}$
B)
$\begin{array}{|l|r|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 3 & 22.97 & 7.66 & 11.16 \\\hline \text { Error } & 20 & 13.72 & 0.686 & \\\hline \text { Total } & 23 & 36.69 & & \\\hline\end{array}$
C)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 3 & 48.80 & 16.27 & 11.16 \\\hline \text { Error } & 20 & 13.72 & 0.686 & \\\hline \text { Total } & 23 & 62.52 & & \\\hline\end{array}$
D)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 3 & 2.55 & 7.66 & 11.16 \\\hline \text { Error } & 20 & 13.72 & 0.686 & \\\hline \text { Total } & 23 & 16.27 & & \\\hline\end{array}$

Question 22

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week. $$\begin{array} { c | c | c } \text { Greenhouse \#1 } & \text { Greenhouse \#2 } & \text { Greenhouse \#3 } \\\hline 73 & 71 & 67 \\72 & 69 & 63 \\73 & 72 & 62 \\66 & 72 & 61 \\68 & 65 & 60 \\71 & 73 & 62 \\72 & 71 & 59\end{array}$$ i)Use a 0.05 significance level to test the claim that the average temperature is the same in each greenhouse. ii)How are the analysis of variance results affected if the same constant is added to every one of the original sample values?

Question 23

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r c c c } &&& { \text { Employee } } \\& & \text { A } & \text { B } & \text { C } \\& \text { I } & 31,34,32 & 29,23,22 & 21,20,24 \\\text { Machine } & \text { II } & 19,26,22 & 35,33,30 & 25,19,23 \\& \text { III } & 21,18,26 & 20,23,24 & 36,37,31\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & \text { MS } \\\text { MACHINE } & 2 & 1.19 & .59 \\\text { EMPLOYEE } & 2 & 5.85 & 2.93 \\\text { INTERACTION } & 4 & 710.81 & 177.70 \\\text { ERROR } & 18 & 160.00 & 8.89 \\\text { TOTAL } & 26 & 877.85 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced.

Question 24

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r c c c } & &&{ \text { Employee } } \\& & \text { A } & \text { B } & \text { C } \\& \text { I } & 16,18,19 & 15,17,20 & 14,18,16 \\\text { Machine } & \text { II } & 20,27,29 & 25,28,27 & 29,28,26 \\& \text { III } & 15,18,17 & 16,16,19 & 13,17,16\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } &{ \text { SS } } &{ \text { MS } } \\\text { MACHINE } & 2 & 588.74 & 294.37 \\\text { EMPLOYEE } & 2 & 2.07 & 1.04 \\\text { INTERACTION } & 4 & 15.48 & 3.87 \\\text { ERROR } & 18 & 98.67 & 5.48 \\\text { TOTAL } & 26 & 704.96 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced.

Question 25

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r c c c } && { \text { Employee } } \\&& { \text { A } } & \text { B } & \text { C } \\& \text { I } & 31,34,32 & 29,23,22 & 21,20,24 \\\text { Machine } & \text { II } & 19,26,22 & 35,33,30 & 25,19,23 \\& \text { III } & 21,18,26 & 20,23,24 & 36,37,31\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & \text { MS } \\\text { MACHINE } & 2 & 1.19 & .59 \\\text { EMPLOYEE } & 2 & 5.85 & 2.93 \\\text { INTERACTION } & 4 & 710.81 & 177.70 \\\text { ERROR } & 18 & 160.00 & 8.89 \\\text { TOTAL } & 26 & 877.85 &\end{array}\end{array}$$ Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced.

Question 26

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array}{rccc} &&\text { Employee }\\& \text { A } & \text { B } & \text { C } \\\text { I } & 23,27,29 & 30,27,25 & 18,20,22 \\\text { Machine II } & 25,26,24 & 24,29,26 & 19,16,14 \\\text { III } & 28,25,26 & 25,27,23 & 15,11,17\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & { \text { MS } } \\\text { MACHINE } & 2 & 34.67 & 17.33 \\\text { EMPLOYEE } & 2 & 504.67 & 252.33 \\\text { INTERACTION } & 4 & 26.67 & 6.67 \\\text { ERROR } & 18 & 98.00 & 5.44 \\\text { TOTAL } & 26 & 664.00 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced.

Question 27

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow.
$\begin{array}{rrccc} &&&{\text { Employee }} \\& & \text { A } & \text { B } & \text { C } \\\text { Machine } & \text { II } & 16,18,19 & 15,17,20 & 14,18,16 \\& \text { II } & 20,27,29 & 25,28,27 & 29,28,26 \\& \text { III } & 15,18,17 & 16,16,19 & 13,17,16\end{array}$

$\begin{array}{l}\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array}{lrrr}\text { SOURCE } & \text { DF } &{\text { SS }} & {\text { MS }} \\\text { MACHINE } & 2 & 588.74 & 294.37 \\\text { EMPLOYEE } & 2 & 2.07 & 1.04 \\\text { INTERACTION } & 4 & 15.48 & 3.87 \\\text { ERROR } & 18 & 98.67 & 5.48 \\\text { TOTAL } & 26 & 704.96 &\end{array}\end{array}$
Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced.

Question 28

Six independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 } = \mu _ { 5 } = \mu _ { 6 }$ .
If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 5 } = \mu _ { 6 }$ , how many ways can you pair off the means?
i) Assume that the tests are independent and that for each test of equality between two means, there is a $0.90$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probabi making no type I errors?
ii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 } = \mu _ { 5 } = \mu _ { 6 }$ at the $0.10$ level of significan what is the probability of not making a type I error?

Question 29

Four independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ .
If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 3 } = \mu _ { 4 }$ , how many ways can you pair off the means?
ii) Assume that the tests are independent and that for each test of equality between two means, there is a $0.99$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probabi making no type I errors?
iii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ at the $0.01$ level of significance, what is the probability of not making a type I error?

Question 30

The following table entries are the times in seconds for three different drivers racing on four different tracks. Assuming no effect from the interaction between driver and track, test the claim that the three drivers have the same mean time. Use a 0.05 significance level. $\begin{array}{c|cccc} & \text { Track 1 } & \text { Track 2 } & \text { Track 3 } & \text { Track 4 } \\\hline \text { Driver 1 } & 72 & 70 & 68 & 71 \\\text { Driver 2 } & 74 & 71 & 66 & 72 \\\text { Driver 3 } & 76 & 69 & 64 & 70\end{array}$
$$\begin{array} { l r c l r c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Driver } & 2 & 2 & 1 & 0.33 & 0.729 \\\text { Track } & 3 & 98.25 & 32.75 & 10.92 & 0.00763 \\\text { Error } & 6 & 18 & 3 & & \\\text { Total } & 11 & 118.25 & & &\end{array}$$

Question 31

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week. $$\begin{array} { c | c | c } \text { Greenhouse \#1 } & \text { Greenhouse \#2 } & \text { Greenhouse \# } \\\hline 73 & 71 & 67 \\72 & 69 & 63 \\73 & 72 & 62 \\66 & 72 & 61 \\68 & 65 & 60 \\71 & 73 & 62 \\72 & 71 & 59\end{array}$$ i)Use a 0.05 significance level to test the claim that the average temperature is the same in each greenhouse. ii)How are the analysis of variance results affected if 8° is added to each temperature listed for greenhouse #3?

Question 32

Provide an appropriate response.

-Fill in the missing entries in the following partially completed one-way ANOVA table. $\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & & 25.2 & & \\\hline \text { Error } & 24 & & 3.5 & \\\hline \text { Total } & 28 & & & \\\hline\end{array}$

A)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 4 & 25.2 & 6.30 & 1.80 \\\hline \text { Error } & 24 & 84.0 & 3.5 & \\\hline \text { Total } & 28 & 25.35 & & \\\hline\end{array}$
B)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 4 & 25.2 & 6.30 & 0.56 \\\hline \text { Error } & 24 & 84.0 & 3.5 & \\\hline \text { Total } & 28 & 109.2 & & \\\hline\end{array}$
C)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 4 & 25.2 & 6.30 & 1.80 \\\hline \text { Error } & 24 & 84.0 & 3.5 & \\\hline \text { Total } & 28 & 109.2 & & \\\hline\end{array}$
D)
$\begin{array}{|l|c|c|c|c|}\hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\\hline \text { Treatment } & 52 & 25.2 & 0.48 & 374.40 \\\hline \text { Error } & 24 & 84.0 & 3.5 & \\\hline \text { Total } & 28 & 109.2 & & \\\hline\end{array}$

Question 33

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r r c c } &&&{ \text { Employee } } \\& & \text { A } & \text { B } & \text { C } \\& \text { I } & 23,27,29 & 30,27,25 & 18,20,22 \\\text { Machine } & \text { II } & 25,26,24 & 24,29,26 & 19,16,14 \\& \text { III } & 28,25,26 & 25,27,23 & 15,11,17\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & { \text { MS } } \\\text { MACHINE } & 2 & 34.67 & 17.33 \\\text { EMPLOYEE } & 2 & 504.67 & 252.33 \\\text { INTERACTION } & 4 & 26.67 & 6.67 \\\text { ERROR } & 18 & 98.00 & 5.44 \\\text { TOTAL } & 26 & 664.00 &\end{array}\end{array}$$ Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced.

Question 34

The following table entries are test scores for males and females at different times of day. Assuming no effect from the interaction between gender and test time, test the claim that males and females perform the same on the test. Use a 0.05 significance level. $$\begin{array}{l}\begin{array} { l | c c c c } & 6 \text { a.m. - } 9 \text { a.m. } &9 \text { a.m. } - 12 \text { p.m. } &12 \text { p.m. - 3 p.m. } &3 p.m. - 6 \text { p.m. } \\\hline \text { Male } & 87 & 89 & 92 & 85 \\\text { Female } & 72 & 84 & 94 & 89\end{array}\\\begin{array} { l c c l c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Gender } & 1 & 24.5 & 24.5 & 0.6652 & 0.4745 \\\text { Time } & 3 & 183 & 61 & 1.6561 & 0.3444 \\\text { Error } & 3 & 110.5 & 36.83 & & \\\text { Total } & 7 & 318 & & &\end{array}\end{array}$$

Question 35

The following table entries are the times in seconds for three different drivers racing on four different tracks. Assuming no effect from the interaction between driver and track, test the claim that the track has no effect on the time. Use a 0.05 significance level. $$\begin{array}{l}\begin{array} { c | c c c c } & \text { Track 1 } & \text { Track 2 } & \text { Track 3 } & \text { Track 4 } \\\hline \text { Driver 1 } & 72 & 70 & 68 & 71 \\\text { Driver 2 } & 74 & 71 & 66 & 72 \\\text { Driver 3 } & 76 & 69 & 64 & 70\end{array}\\\\\begin{array} { l r c l r l } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Driver } & 2 & 2 & 1 & 0.33 & 0.729 \\\text { Track } & 3 & 98.25 & 32.75 & 10.92 & 0.00763 \\\text { Error } & 6 & 18 & 3 & & \\\text { Total } & 11 & 118.25 & & &\end{array}\end{array}$$

Question 36

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r c c c } &&&{ \text { Employee } } \\& & \text { A } & \text { B } & \text { C } \\& \text { I } & 31,34,32 & 29,23,22 & 21,20,24 \\\text { Machine } & \text { II } & 19,26,22 & 35,33,30 & 25,19,23 \\& \text { III } & 21,18,26 & 20,23,24 & 36,37,31\end{array}\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & \text { MS } \\\text { MACHINE } & 2 & 1.19 & .59 \\\text { EMPLOYEE } & 2 & 5.85 & 2.93 \\\text { INTERACTION } & 4 & 710.81 & 177.70 \\\text { ERROR } & 18 & 160.00 & 8.89 \\\text { TOTAL } & 26 & 877.85 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on the number of items produced.

Question 37

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { r r c c c } & && { \text { Employee } } \\& & \text { A } & \text { B } & \text { C } \\& \text { I } & 16,18,19 & 15,17,20 & 14,18,16 \\\text { Machine } & \text { II } & 20,27,29 & 25,28,27 & 29,28,26 \\& \text { III } & 15,18,17 & 16,16,19 & 13,17,16\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & { \text { MS } } \\\text { MACHINE } & 2 & 588.74 & 294.37 \\\text { EMPLOYEE } & 2 & 2.07 & 1.04 \\\text { INTERACTION } & 4 & 15.48 & 3.87 \\\text { ERROR } & 18 & 98.67 & 5.48 \\\text { TOTAL } & 26 & 704.96 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on the number of items produced.

Question 38

Use the data given below to verify that the t test for independent samples and the ANOVA method are equivalent. $$\begin{array} { c | c } \mathrm { A } & \mathrm { B } \\\hline 10 & 19 \\29 & 18 \\11 & 27 \\19 & 30 \\15 & 18 \\16 & 21\end{array}$$ i)Use a t test with a 0.05 significance level to test the claim that the two samples come from populations with the same means. ii)Use the ANOVA method with a 0.05 significance level to test the same claim. iii)Verify that the squares of the t test statistic and the critical value are equal to the F test statistic and critical value.

Question 39

Use the data given below to verify that the t test for independent samples and the ANOVA method are equivalent. $$\begin{array} { c | c } \mathrm { A } & \mathrm { B } \\\hline 85 & 74 \\81 & 72 \\73 & 65 \\91 & 83 \\64 & 59\end{array}$$ i)Use a t test with a 0.05 significance level to test the claim that the two samples come from populations with the same means. ii)Use the ANOVA method with a 0.05 significance level to test the same claim. iii)Verify that the squares of the t test statistic and the critical value are equal to the F test statistic and critical value.

Question 40

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. $$\begin{array}{l}\begin{array} { l r c c c } & &&{ \text { Employee } } \\&& { \text { A } } & \text { B } & \text { C } \\& \text { I } & 23,27,29 & 30,27,25 & 18,20,22 \\\text { Machine } & \text { II } & 25,26,24 & 24,29,26 & 19,16,14 \\& \text { III } & 28,25,26 & 25,27,23 & 15,11,17\end{array}\\\\\text { ANALYSIS OF VARIANCE ITEMS }\\\begin{array} { l r r r } \text { SOURCE } & \text { DF } & { \text { SS } } & \text { MS } \\\text { MACHINE } & 2 & 34.67 & 17.33 \\\text { EMPLOYEE } & 2 & 504.67 & 252.33 \\\text { INTERACTION } & 4 & 26.67 & 6.67 \\\text { ERROR } & 18 & 98.00 & 5.44 \\\text { TOTAL } & 26 & 664.00 &\end{array}\end{array}$$ Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on the number of items produced.

Question 41

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array} { l } & \text { male}& \text {female }\\ \text {Machine 1 }&15,17&16,17\\ \text {Machine 2}&14,13&15,13\\ \text { Machine 3}&16,18&17,19\\\end{array}$
Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if the first sample value in the first cell is changed to 30 minutes?

Question 42

The following data shows annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Assume that incomes are not affected by an interaction between gender and level of education, and test the null hypothesis that gender has no effect on income. Use a 0.05 significance level. $$\begin{array} { l c c } & \text { Female } & \text { Male } \\\text { High school } & 23,27,24,26 & 25,26,22,24 \\\text { College } & 28,36,31,33 & 35,32,39,28 \\\text { Advanced degree } & 41,38,43,49 & 35,50,47,44\end{array}$$

Question 43

The following data shows the yield, in bushels per acre, categorized according to three varieties of corn and three different soil conditions. Assume that yields are not affected by an interaction between variety and soil conditions, and test the null hypothesis that soil conditions have no effect on yield. Use a 0.05 significance level. $\begin{array}{cccc} & \text { Plot 1 } & \text { Plot 2 } & \text { Plot 3 } \\\text { Variety 1 } & 156,167, & 162,160, & 145,151 \text {, } \\& 170,162 & 169,168 & 148,155 \\\text { Variety 2 } & 172,176, & 179,186, & 161,162, \\& 166,179 & 160,176 & 165,170 \\\text { Variety 3 } & 175,157, & 178,170, & 169,165, \\& 179,178 & 172,174 & 170,169\end{array}$

Question 44

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array} { l } & \text { male}& \text {female }\\ \text {Machine 1 }&15,17&16,17\\ \text {Machine 2}&14,13&15,13\\ \text { Machine 3}&16,18&17,19\\\end{array}$
The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is no effect from the interaction between machine and gender, but there is an effect from the gender of the operator.

Question 45

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array}{lrl} & \text { Male } & \text { Female } \\\text { Machine 1 } & 15,17 & 16,17 \\\text { Machine 2 } & 14,13 & 15,13 \\\text { Machine 3 } & 16,18 & 17,19\end{array}$
Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if the times are all doubled?

Question 46

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array}{lrl} & \text { Male } & \text { Female } \\\text { Machine 1 } & 15,17 & 16,17 \\\text { Machine 2 } & 14,13 & 15,13 \\\text { Machine 3 } & 16,18 & 17,19\end{array}$
Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if 5 minutes is added to each completion time?

Question 47

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array}{lrl} & \text { Male } & \text { Female } \\\text { Machine 1 } & 15,17 & 16,17 \\\text { Machine 2 } & 14,13 & 15,13 \\\text { Machine 3 } & 16,18 & 17,19\end{array}$
Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if the times are converted to hours?

Question 48

The following data shows the yield, in bushels per acre, categorized according to three varieties of corn and three different soil conditions. Assume that yields are not affected by an interaction between variety and soil conditions, and test the null hypothesis that variety has no effect on yield. Use a 0.05 significance level.
$\begin{array}{clcc} & \text { Plot 1 } & \text { Plot 2 } & \text { Plot 3 } \\\text { Variety 1 } & 156,167, & 162,160, & 145,151 \\& 170,162 & 169,168 & 148,155 \\\text { Variety 2 } & 172,176, & 179,186, & 161,162 \\& 166,179 & 160,176 & 165,170 \\\text { Variety 3 } & 175,157, & 178,170, & 169,165 \\& 179,178 & 172,174 & 170,169\end{array}$

Question 49

The following data shows annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Assume that incomes are not affected by an interaction between gender and level of education, and test the null hypothesis that level of education has no effect on income. Use a 0.05 significance level. $$\begin{array} { l c c } & \text { Female } & \text { Male } \\\text { High school } & 23,27,24,26 & 25,26,22,24 \\\text { College } & 28,36,31,33 & 35,32,39,28 \\\text { Advanced degree } & 41,38,43,49 & 35,50,47,44\end{array}$$

Question 50

The following table entries are test scores for males and females at different times of day. Assuming no effect from the interaction between gender and test time, test the claim that time of day does not affect test scores. Use a 0.05 significance level. $$\begin{array}{l}\begin{array} { l | c c c c } & 6 \text { a.m. } - 9 \text { a.m. } &9 \text { a.m. } - 12 \text { p.m. } &12 \text { p.m. } - 3 \text { p.m.}& 3 p.m. - 6 \text { p.m. } \\\hline \text { Male } & 87 & 89 & 92 & 85 \\\text { Female } & 72 & 84 & 94 & 89\end{array}\\\begin{array} { l c l l c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Gender } & 1 & 24.5 & 24.5 & 0.6652 & 0.4745 \\\text { Time } & 3 & 183 & 61 & 1.6561 & 0.3444 \\\text { Error } & 3 & 110.5 & 36.83 & & \\\text { Total } & 7 & 318 & & &\end{array}\end{array}$$

Question 51

The following Minitab display results from a study in which three different teachers taught calculus classes of five different sizes. The class average was recorded for each class. Assuming no effect from the interaction between teacher and class size, test the claim that class size has no effect on the class average. Use a 0.05 significance level. $$\begin{array} { l r r r c c } \text { Source } & \text { DF } & \text { SS } & { \text { MS } } & \text { F } & \text { p } \\\text { Teacher } & 2 & 56.93 & 28.47 & 1.018 & 0.404 \\\text { Class Size } & 4 & 672.67 & 168.17 & 6.013 & 0.016 \\\text { Error } & 8 & 223.73 & 27.97 & & \\\text { Total } & 14 & 953.33 & & &\end{array}$$

Question 52

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array} { l } & \text { male}& \text {female }\\ \text {Machine 1 }&15,17&16,17\\ \text {Machine 2}&14,13&15,13\\ \text { Machine 3}&16,18&17,19\\\end{array}$
The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is an effect from the interaction between machine and gender.

Question 53

The following table shows the mileage for four different cars and three different brands of gas. Assuming no effect from the interaction between car and brand of gas, test the claim that the three brands of gas provide the same mean gas mileage. Use a 0.05 significance level. $$\begin{array}{l}\begin{array} { l | l l l } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \\\hline \text { Car 1 } & 22.4 & 25.2 & 24.3 \\\text { Car 2 } & 19 & 18.6 & 19.8 \\\text { Car 3 } & 24.6 & 25 & 25.4 \\\text { Car 4 } & 23.5 & 23.6 & 24.1\end{array}\\\\\begin{array} { l r r r r c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Car } & 3 & 61.249 & 20.416 & 39.033 & 0.000249 \\\text { Gas } & 2 & 2.222 & 1.111 & 2.124 & 0.200726 \\\text { Error } & 6 & 3.138 & 0.523 & & \\\text { Total } & 11 & 66.609 & & &\end{array}\end{array}$$

Question 54

The following data shows the yield, in bushels per acre, categorized according to three varieties of corn and three different soil conditions. Test the null hypothesis of no interaction between variety and soil conditions at a significance level of 0.05. $\begin{array}{llll} & \text { Plot 1 } & \text { Plot 2 } & \text { Plot 3 } \\\text { Variety 1 } & 156,167, & 162,160, & 145,151, \\& 170,162 & 169,168 & 148,155 \\\text { Variety 2 } & 172,176, & 179,186, & 161,162 \text {, } \\& 166,179 & 160,176 & 165,170 \\\text { Variety 3 } & 175,157, & 178,170, & 169,165, \\& 179,178 & 172,174 & 170,169\end{array}$

Question 55

The following data show annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Test the null hypothesis of no interaction between gender and level of education at a significance level of 0.05. $$\begin{array} { l c c } & \text { Female } & \text { Male } \\\text { High school } & 23,27,24,26 & 25,26,22,24 \\\text { College } & 28,36,31,33 & 35,32,39,28 \\\text { Advanced degree } & 41,38,43,49 & 35,50,47,44\end{array}$$

Question 56

The following table shows the mileage for four different cars and three different brands of gas. Assuming no effect from the interaction between car and brand of gas, test the claim that the four cars have the same mean mileage. Use a 0.05 significance level. $$\begin{array}{l}\begin{array} { l | l l l } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \\\hline \text { Car 1 } & 22.4 & 25.2 & 24.3 \\\text { Car 2 } & 19 & 18.6 & 19.8 \\\text { Car 3 } & 24.6 & 25 & 25.4 \\\text { Car 4 } & 23.5 & 23.6 & 24.1\end{array}\\\begin{array} { l r r r c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Car } & 3 & 61.249 & 20.416 & 39.033 & 0.000249 \\\text { Gas } & 2 & 2.222 & 1.111 & 2.124 & 0.200726 \\\text { Error } & 6 & 3.138 & 0.523 & & \\\text { Total } & 11 & 66.609 & & &\end{array}\end{array}$$

Question 57

The following Minitab display results from a study in which three different teachers taught calculus classes of five different sizes. The class average was recorded for each class. Assuming no effect from the interaction between teacher and class size, test the claim that the teacher has no effect on the class average. Use a 0.05 significance level. $$\begin{array} { l r r r c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\\text { Teacher } & 2 & 56.93 & 28.47 & 1.018 & 0.404 \\\text { Class Size } & 4 & 672.67 & 168.17 & 6.013 & 0.016 \\\text { Error } & 8 & 223.73 & 27.97 & & \\\text { Total } & 14 & 953.33 & & &\end{array}$$

Question 58

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.
$\begin{array} { l } & \text { male}& \text {female }\\ \text {Machine 1 }&15,17&16,17\\ \text {Machine 2}&14,13&15,13\\ \text { Machine 3}&16,18&17,19\\\end{array}$
The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is no effect from the interaction between machine and gender, there is no effect from the machine used, and there is no effect from the gender of the operator.

Question 59

The following results are from a statistics software package in which all of the F values and P-values are given. Is there a significant effect from the interaction? Should you test to see if there is a significant effect due to either A or B? If the answer is yes, is there a significant effect due to either A or B? $$\begin{array}{l}\text { ANOVA Table }\\\begin{array} { r | r | r | r | r | r } \text { Source } & \text { DF } & \text { Sum squares } & \text { Mean square } & \text { F test } & \text { P-value } \\\hline \mathrm { A } & 2 & 164.020 & 82.010 & 25.010 & < .0001 \\\mathrm {~B} & 4 & 230.786 & 57.697 & 18.002 & < .0001 \\\text { Interaction } & 8 & 80.879 & 10.110 & 3.154 & .0031 \\\text { Error } & 101 & 323.708 & 3.205 & & \\\text { Total } & 115 & 799.393 & & &\end{array}\end{array}$$

Question 60

The following results are from a statistics package in which all of the F values and P-values are given. Is there a significant effect from the interaction? Should you test to see if there is a significant effect due to either A or B? If the answer is yes, is there a significant effect due to either A or B? $$\begin{array}{l}\text { ANoVA 1able }\\\begin{array} { r | r | r | r | r | r } \text { Source } & \text { DF } & \text { Sum squares } & \text { Mean square } & \text { F test } & \text { P-value } \\\hline \mathrm { A } & 2 & 415.87305 & 207.93652 & 1.88259 & .1637 \\\mathrm {~B} & 3 & 2997.47186 & 999.15729 & 9.04603 & .0001 \\\text { Interaction } & 6 & 707.26626 & 117.87771 & 1.06723 & .3958 \\\text { Error } & 46 & 5080.81667 & 110.45254 & & \\\text { Total } & 57 & 9201.42784 & & &\end{array}\end{array}$$