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Exam 12: Analysis of Variance

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Question 21

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Select a property of the F distribution

A) It is symmetric.
B) The exact shape of the F distribution depends on two different degrees of freedom.
C) Values of the F distribution can be negative.
D) The total area under the curve is equal to 2.

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Question 22

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Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the value of the test statistic. $\begin{array} { l | r | c | c | c | c } \text { Source } & \boldsymbol { D F } & \boldsymbol { S S } & \boldsymbol { M S } & \boldsymbol { F } & \boldsymbol { p } \\\hline \text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\hline \text { Error } & 16 & 13.925 & 0.870 & & \\\hline \text { Total } & 19 & 27.425 & & &\end{array}$

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

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Question 23

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The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. $\begin{array} { l c c } & \text { Male } & \text { Female } \\\text { Machine 1 } & 15,17 & 16,17 \\\text { Machine 2 } & 14,13 & 15,13 \\\text { Machine 3 } & 16,18 & 17,19\end{array}$ 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.

Which of the following is NOT a requirement for one-way ANOVA?

Explain the procedure for two-way analysis of variance, and how it varies depending on whether there is an interaction between the two factors or not.

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

ANOVA requires usage of the _______ distribution

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. 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.

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. What can you conclude about the equality of the Population means? $\begin{array} { l | r | c | c | c | c } \text { Source } & \boldsymbol { D } \boldsymbol { F } & \boldsymbol { S S } & \boldsymbol { M S } & \boldsymbol { F } & \boldsymbol { p } \\\hline \text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\hline \text { Error } & 16 & 13.925 & 0.870 & & \\\hline \text { Total } & 19 & 27.425 & & &\end{array}$

Draw an example of an F distribution and list the characteristics of the F distribution

Select an appropriate null hypothesis for a one way analysis of variance test.

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 havebeen obtained. $\begin{array} { c c c c } \text { Brand A } & \text { Brand B } & \text { Brand C } & \text { Brand D } \\\hline 17 & 18 & 21 & 22 \\20 & 18 & 24 & 25 \\21 & 23 & 25 & 27 \\22 & 25 & 26 & 29 \\21 & 26 & 29 & 35 \\& & 29 & 36 \\& & & 37\end{array}$

Fill in the missing entries in the following partially completed one -way ANOVA table. $\begin{array} { l | c | c | c | c } \text { Source } & d \boldsymbol { f } & \boldsymbol { S S } & \boldsymbol { M S } = \boldsymbol { S S } / d f & \boldsymbol { F } \text {-statistic } \\\hline \text { Treatment } & 3 & & & 11.16 \\\hline \text { Error } & & 13.72 & 0.686 &\end{array}$

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. 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.

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. 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?

Fill in the missing entries in the following partially completed one-way ANOVA table $\begin{array} { l | l | l | l | l } \text { Source } & \boldsymbol { d f } & \boldsymbol { S S } & \boldsymbol { M S } & \boldsymbol { F } \\\hline \text { Treatment } & 3 & & & 13.89 \\\hline \text { Error } & & 13.58 & 0.617 & \\\hline \text { Total } & & & &\end{array}$

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the P-value. $\begin{array} { l | r | c | l | c | c } \text { Source } & \boldsymbol { D F } & \boldsymbol { S S } & \boldsymbol { M S } & \boldsymbol { F } & \boldsymbol { p } \\\hline \text { Factor } & 3 & 13.500 & 4.500 & 5.17 & 0.011 \\\hline \text { Error } & 16 & 13.925 & 0.870 & & \\\hline \text { Total } & 19 & 27.425 & & &\end{array}$

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. $\begin{array} { c c c } \text { Exercise A } & \text { Exercise B } & \text { Exercise } \mathbf { C } \\\hline 2.5 & 5.8 & 4.3 \\8.8 & 4.9 & 6.2 \\7.3 & 1.1 & 5.8 \\9.8 & 7.8 & 8.1 \\5.1 & 1.2 & 7.9\end{array}$ At the 1% significance level, does it appear that a difference exists in the true mean weight loss produced by the three exercise programs?

Identify the value of the test statistic. $\begin{array} { l | r | c | c | c | c } \text { Source } & \boldsymbol { D } \boldsymbol { F } & \boldsymbol { S S } & \boldsymbol { M S } & \boldsymbol { F } & \boldsymbol { p } \\\hline \text { Factor } & 3 & 30 & 10.00 & 1.6 & 0.264 \\\hline \text { Error } & 8 & 50 & 6.25 & & \\\hline \text { Total } & 11 & 80 & & &\end{array}$

The data below represent the weight losses for people on three diets $\begin{array} { l c c } \text { Diet A } & \text { Diet B } & \text { Diet C } \\\hline 1.75 & 3.8 & 4.8 \\8.8 & 4.9 & 6.2 \\7.3 & 1.1 & 5.8 \\9.8 & 7.8 & 8.1 \\5.1 & 1.2 & 6.9\end{array}$ If we want to test the claim that the three size categories have the same means, why don't we use three separate hypothesis tests for $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 2 } = \mu _ { 3 } , \text { and } \mu _ { 1 } = \mu _ { 3 } ?$