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. \[\begin{array}{l} \begin{array} { l | c c c c } & 6 \text { a.m. - } 9 \text { a.m. } &9 \text { a.m. } - 12 \text { p.m. } &12 \text { p.m. - 3 p.m. } &3 p.m. - 6 \text { p.m. } \\ \hline \text { Male } & 87 & 89 & 92 & 85 \\ \text { Female } & 72 & 84 & 94 & 89 \end{array}\\ \begin{array} { l c c l c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Gender } & 1 & 24.5 & 24.5 & 0.6652 & 0.4745 \\ \text { Time } & 3 & 183 & 61 & 1.6561 & 0.3444 \\ \text { Error } & 3 & 110.5 & 36.83 & & \\ \text { Total } & 7 & 318 & & & \end{array} \end{array}\]

@#LAT-DLM& : There is no gender effect.

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. \(\begin{array}{llll} & \text { Plot 1 } & \text { Plot 2 } & \text { Plot 3 } \\ \text { Variety 1 } & 156,167, & 162,160, & 145,151, \\ & 170,162 & 169,168 & 148,155 \\ \text { Variety 2 } & 172,176, & 179,186, & 161,162 \text {, } \\ & 166,179 & 160,176 & 165,170 \\ \text { Variety 3 } & 175,157, & 178,170, & 169,165, \\ & 179,178 & 172,174 & 170,169 \end{array}\)

H?: There is no interaction between vari

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.

@#LAT-DLM& : There is no employee effect

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) \(\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 52 & 25.2 & 0.48 & 374.40 \\ \hline \text { Error } & 24 & 84.0 & 3.5 & \\ \hline \text { Total } & 28 & 109.2 & & \\ \hline \end{array}\) 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) \(\begin{array}{|l|c|c|c|c|} \hline \text { Source } & \text { df } & \text { SS } & \text { MS=SS/df } & \text { F-statistic } \\ \hline \text { Treatment } & 52 & 25.2 & 0.48 & 374.40 \\ \hline \text { Error } & 24 & 84.0 & 3.5 & \\ \hline \text { Total } & 28 & 109.2 & & \\ \hline \end{array}\)

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. \[\begin{array}{l} \begin{array} { l | l l l } & \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \\ \hline \text { Car 1 } & 22.4 & 25.2 & 24.3 \\ \text { Car 2 } & 19 & 18.6 & 19.8 \\ \text { Car 3 } & 24.6 & 25 & 25.4 \\ \text { Car 4 } & 23.5 & 23.6 & 24.1 \end{array}\\\\ \begin{array} { l r r r r c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Car } & 3 & 61.249 & 20.416 & 39.033 & 0.000249 \\ \text { Gas } & 2 & 2.222 & 1.111 & 2.124 & 0.200726 \\ \text { Error } & 6 & 3.138 & 0.523 & & \\ \text { Total } & 11 & 66.609 & & & \end{array} \end{array}\]

H?: The brands of gas provide the same m

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. \[\begin{array}{l} \begin{array} { l | c c c c } & 6 \text { a.m. } - 9 \text { a.m. } &9 \text { a.m. } - 12 \text { p.m. } &12 \text { p.m. } - 3 \text { p.m.}& 3 p.m. - 6 \text { p.m. } \\ \hline \text { Male } & 87 & 89 & 92 & 85 \\ \text { Female } & 72 & 84 & 94 & 89 \end{array}\\ \begin{array} { l c l l c c } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { p } \\ \text { Gender } & 1 & 24.5 & 24.5 & 0.6652 & 0.4745 \\ \text { Time } & 3 & 183 & 61 & 1.6561 & 0.3444 \\ \text { Error } & 3 & 110.5 & 36.83 & & \\ \text { Total } & 7 & 318 & & & \end{array} \end{array}\]

@#LAT-DLM& : There is no effect due to the time of

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. 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.

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.

@#LAT-DLM& : There is no machine effect.

Exam 12: Analysis of Variance

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. 6 a.m. - 9 a.m. 9 a.m. -12 p.m. 12 p.m. - 3 p.m. 3p.m.-6 p.m. Male 87 89 92 85 Female 72 84 94 89 Source DF SS MS F p Gender 1 24.5 24.5 0.6652 0.4745 Time 3 183 61 1.6561 0.3444 Error 3 110.5 36.83 Total 7 318

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Question 21

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. Greenhouse \#1 Greenhouse \#2 Greenhouse \#3 73 71 67 72 69 63 73 72 62 66 72 61 68 65 60 71 73 62 72 71 59 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?

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Question 22

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. Plot 1 Plot 2 Plot 3 Variety 1 156,167, 162,160, 145,151, 170,162 169,168 148,155 Variety 2 172,176, 179,186, 161,162 , 166,179 160,176 165,170 Variety 3 175,157, 178,170, 169,165, 179,178 172,174 170,169

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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. Employee A B C I 16,18,19 15,17,20 14,18,16 Machine II 20,27,29 25,28,27 29,28,26 III 15,18,17 16,16,19 13,17,16 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 588.74 294.37 EMPLOYEE 2 2.07 1.04 INTERACTION 4 15.48 3.87 ERROR 18 98.67 5.48 TOTAL 26 704.96 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.

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Question 24

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. Source DF SS MS F p Teacher 2 56.93 28.47 1.018 0.404 Class Size 4 672.67 168.17 6.013 0.016 Error 8 223.73 27.97 Total 14 953.33

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Question 25

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. brandA brand B brand C brand D 17 18 21 22 20 18 24 25 21 23 25 27 22 25 26 29 21 26 29 35 29 36 37

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Question 26

Provide an appropriate response. -Fill in the missing entries in the following partially completed one-way ANOVA table. Source df SS MS=SS/df F-statistic Treatment 25.2 Error 24 3.5 Total 28

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Question 27

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. brand A brand B brand C 32 27 22 34 24 25 37 33 32 33 30 22 36 21 39

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Question 28

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. male female Machine 1 15,17 16,17 Machine 2 14,13 15,13 Machine 3 16,18 17,19 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.

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Question 29

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. Brand 1 Brand 2 Brand 3 Car 1 22.4 25.2 24.3 Car 2 19 18.6 19.8 Car 3 24.6 25 25.4 Car 4 23.5 23.6 24.1 Source DF SS MS F p Car 3 61.249 20.416 39.033 0.000249 Gas 2 2.222 1.111 2.124 0.200726 Error 6 3.138 0.523 Total 11 66.609

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Question 30

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. 6 a.m. -9 a.m. 9 a.m. -12 p.m. 12 p.m. -3 p.m. 3p.m.-6 p.m. Male 87 89 92 85 Female 72 84 94 89 Source DF SS MS F p Gender 1 24.5 24.5 0.6652 0.4745 Time 3 183 61 1.6561 0.3444 Error 3 110.5 36.83 Total 7 318

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Question 31

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. Female Male High school 23,27,24,26 25,26,22,24 College 28,36,31,33 35,32,39,28 Advanced degree 41,38,43,49 35,50,47,44

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Question 32

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. - Source DF SS MS F p Factor 3 30 10.00 1.6 0.264 Error 8 50 6.25 Total 11 80 What can you conclude about the equality of the population means?

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Question 33

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. Source DF SS MS F p Teacher 2 56.93 28.47 1.018 0.404 Class Size 4 672.67 168.17 6.013 0.016 Error 8 223.73 27.97 Total 14 953.33

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Question 34

Use the data given below to verify that the t test for independent samples and the ANOVA method are equivalent. 85 74 81 72 73 65 91 83 64 59 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.

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Question 35

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. - Source DF SS MS F p Factor 3 30 10.00 1.6 0.264 Error 8 50 6.25 Total 11 80 Identify the value of the test statistic.

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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. Employee A B C I 16,18,19 15,17,20 14,18,16 Machine II 20,27,29 25,28,27 29,28,26 III 15,18,17 16,16,19 13,17,16 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 588.74 294.37 EMPLOYEE 2 2.07 1.04 INTERACTION 4 15.48 3.87 ERROR 18 98.67 5.48 TOTAL 26 704.96 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.

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Question 37

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

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Question 38

Use the data given below to verify that the t test for independent samples and the ANOVA method are equivalent. 10 19 29 18 11 27 19 30 15 18 16 21 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.

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Question 39

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. Brand Brand Brand =10 =10 =10 =32.6 =31.7 =27.4 =4.51 =4.22 =4.69

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Question 40

Provide an appropriate response. -Fill in the missing entries in the following partially completed one-way ANOVA table. Source df SS MS=SS/df F-statistic Treatment 25.2 Error 24 3.5 Total 28

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

Introduction to Statistics

Summarizing and Graphing Data

Statistics for Describing, Exploring, and Comparing Data

Probability

Probability Distributions

Normal Probability Distributions

Estimates and Sample Sizes

Hypothesis Testing

Inferences From Two Samples

Correlation and Regression

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