In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. \[\begin{array}{l} \begin{array} { l c c c } { \text { Treatment } } & \text { Mean } & \text { Standard Deviation } & \text { Sample Size } n \\ \text { Raw garlic } & 142 & 22 & 49 \\ \text { Garlic powder } & 137 & 25 & 47 \\ \text { Garlic extract } & 137 & 22 & 48 \\ \text { Placebo } & 133 & 21 & 48 \end{array}\\ \begin{array} { l c c c c } \text { Source } & \text { df } & \text { Sums of Squares } & \text { Mean Square } & \text { F-Ratio } \\ \text { Treatment } &&& 959.46 & \\ \text { Error } && 95,457.00 & \end{array} \end{array}\] The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? Based on this ANOVA test, and using a significance level of 0.05, what should you conclude?

A) Reject the null hypothesis and conclude that there is significant evidence that the four treatments do not all lead to the same population mean LDL level after six months. B) Reject the null hypothesis and conclude that there is significant evidence that raw garlic for six months leads to a higher mean LDL level in this population. C) Fail to reject the null hypothesis and conclude that there is not enough evidence to say that the treatments influence the population mean LDL level after six months. D) Do not reach a conclusion because the conditions for ANOVA are not all satisfied in this case. A) Reject the null hypothesis and conclude that there is significant evidence that the four treatments do not all lead to the same population mean LDL level after six months. B) Reject the null hypothesis and conclude that there is significant evidence that raw garlic for six months leads to a higher mean LDL level in this population. C) Fail to reject the null hypothesis and conclude that there is not enough evidence to say that the treatments influence the population mean LDL level after six months. D) Do not reach a conclusion because the conditions for ANOVA are not all satisfied in this case. C

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. \[\begin{array} { | c |ccc| } \hline \text { Plants (per acre) } & \text { Yield (bushels per acre) } \\ \hline 12,000 & 150.1 & 113.0 & 118.4 \\ \hline 16,000 & 166.9 & 120.7 & 135.2 \\ \hline 20,000 & 165.3 & 130.1 & 139.6 \\ \hline 24,000 & 134.7 & 138.4 & 156.1 \\ \hline \end{array}\] Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. \[\begin{array}{l} \text { One-Way ANOVA: Yield Versus Density }\\ \\ \begin{array} { l r r r r r } \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Density } & & 589 & & & \\ \text { Error } & & & 356 & & \\ \text { Total } & & & & & \end{array}\\ \\ \begin{array} { l c r r } \text { Density } & \mathrm { N } & \text { Mean } & \text { StDev } \\ 12,000 & 3 & 127.17 & 20.04 \\ 16,000 & 3 & 140.93 & 23.63 \\ 20,000 & 3 & 145.00 & 18.21 \\ 24,000 & 3 & 143.07 & 11.44 \end{array} \end{array}\] What is the degrees of freedom for the error component?

A) 2 B) 3 C) 8 D) 11 A) 2 B) 3 C) 8 D) 11 C

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] The ANOVA procedure relies on three important assumptions to be valid: (1) The data are independent random samples. (2) The populations are Normally distributed or the sample sizes are large enough. (3) The populations have the same standard deviation σ . In the case of this ANOVA test, which of the following statements is correct?

A) Assumption (1) is the only assumption not satisfied. B) Assumption (2) is the only assumption not satisfied. C) Assumption (3) is the only assumption not satisfied. D) All three test assumptions are satisfied. A) Assumption (1) is the only assumption not satisfied. B) Assumption (2) is the only assumption not satisfied. C) Assumption (3) is the only assumption not satisfied. D) All three test assumptions are satisfied. D

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. \[\begin{array} { | c | ccc | } \hline \text { Plants (per acre) } & \text { Yield (bushels per acre) } \\ \hline 12,000 & 150.1 & 113.0 & 118.4 \\ \hline 16,000 & 166.9 & 120.7 & 135.2 \\ \hline 20,000 & 165.3 & 130.1 & 139.6 \\ \hline 24,000 & 134.7 & 138.4 & 156.1 \\ \hline \end{array}\] Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the sum of squares for the error component?

A) 196 B) 356 C) 589 D) 2848 A) 196 B) 356 C) 589 D) 2848

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. \[\begin{array} { | c | ccc | } \hline \text { Plants (per acre) } & \text { Yield (bushels per acre) } \\ \hline 12,000 & 150.1 & 113.0 & 118.4 \\ \hline 16,000 & 166.9 & 120.7 & 135.2 \\ \hline 20,000 & 165.3 & 130.1 & 139.6 \\ \hline 24,000 & 134.7 & 138.4 & 156.1 \\ \hline \end{array}\] Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the value of the F statistic for this test?

A) 0.55 B) 4.73 C) 1.82 D) 4.83 A) 0.55 B) 4.73 C) 1.82 D) 4.83

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] The denominator degrees of freedom for the ANOVA is ________.

The answer of A study randomly assigned adult subjects to...

In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. $\begin{array} { l c c c } \text { Treatment } & \text { Mean } & \text { Standard Deviation } & \text { Sample Size } n \\ \text { Raw garlic } & 142 & 22 & 49 \\ \text { Garlic powder } & 137 & 25 & 47 \\ \text { Garlic extract } & 137 & 22 & 48 \\ \text { Placebo } & 133 & 21 & 48 \end{array}$ \[\begin{array}{l} \begin{array} { l l r } \text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square }&\text {F-Ratio }\\ \text { Treatment}&&&659.46\\ \text {Error }&&95,457.00 \end{array} \end{array}\] The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? What is the mean square for error to test this hypothesis?

A) MSE = 144.75 B) MSE = 507.75 C) MSE = 659.46 D) MSE = 31,819.00 A) MSE = 144.75 B) MSE = 507.75 C) MSE = 659.46 D) MSE = 31,819.00

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. \[\begin{array} { | c | ccc | } \hline \text { Plants (per acre) } & \text { Yield (bushels per acre) } \\ \hline 12,000 & 150.1 & 113.0 & 118.4 \\ \hline 16,000 & 166.9 & 120.7 & 135.2 \\ \hline 20,000 & 165.3 & 130.1 & 139.6 \\ \hline 24,000 & 134.7 & 138.4 & 156.1 \\ \hline \end{array}\] Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ For this example, which of the following statements is correct?

A) These data were collected as part of an observational study. B) The data show no evidence of a violation of the assumption that the four populations have the same standard deviation. C) ANOVA can be used on these data because ANOVA requires that the sample sizes be equal. D) None of the above statements is correct. A) These data were collected as part of an observational study. B) The data show no evidence of a violation of the assumption that the four populations have the same standard deviation. C) ANOVA can be used on these data because ANOVA requires that the sample sizes be equal. D) None of the above statements is correct.

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] The numerator degrees of freedom for the ANOVA is _______.

The answer of A study randomly assigned adult subjects to...

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] What is the P-value for the ANOVA that tests for equality of the population means of the three treatments?

A) Less than 0.01 B) Between 0.01 and 0.05 C) Between 0.05 and 0.10 D) Greater than 0.10 A) Less than 0.01 B) Between 0.01 and 0.05 C) Between 0.05 and 0.10 D) Greater than 0.10

In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. \[\begin{array}{l} \begin{array} { l c c c } { \text { Treatment } } & \text { Mean } & \text { Standard Deviation } & \text { Sample Size } n \\ \text { Raw garlic } & 142 & 22 & 49 \\ \text { Garlic powder } & 137 & 25 & 47 \\ \text { Garlic extract } & 137 & 22 & 48 \\ \text { Placebo } & 133 & 21 & 48 \end{array}\\ \begin{array} { l c c c c } \text { Source } & \text { df } & \text { Sums of Squares } & \text { Mean Square } & \text { F-Ratio } \\ \text { Treatment } &&& 959.46 & \\ \text { Error } && 95,457.00 & \end{array} \end{array}\] The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? What is the P-value for this ANOVA test?

A) Less than 0.01 B) Between 0.01 and 0.05 C) Between 0.05 and 0.10 D) Greater than 0.10 A) Less than 0.01 B) Between 0.01 and 0.05 C) Between 0.05 and 0.10 D) Greater than 0.10

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. \[\begin{array} { | c | ccc | } \hline \text { Plants (per acre) } & \text { Yield (bushels per acre) } \\ \hline 12,000 & 150.1 & 113.0 & 118.4 \\ \hline 16,000 & 166.9 & 120.7 & 135.2 \\ \hline 20,000 & 165.3 & 130.1 & 139.6 \\ \hline 24,000 & 134.7 & 138.4 & 156.1 \\ \hline \end{array}\] Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the best estimate of the pooled standard deviation for yield?

A) 11.44 B) 18.87 C) 23.63 D) 22.48 A) 11.44 B) 18.87 C) 23.63 D) 22.48

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] The mean square for groups, MSG, is _______________.

The answer of A study randomly assigned adult subjects to...

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: \[\begin{array} { | l | c | c | c | } \hline \text { Treatment } & \text { Mean } & \text { Std. Dev. } & \boldsymbol { n } \\ \hline \text { Long periods } & 10.2 & 4.2 & 37 \\ \hline \text { Multiple short periods } & 9.3 & 4.5 & 36 \\ \hline \text { Multiple short periods with treadmill } & 10.2 & 5.2 & 42 \\ \hline \end{array}\] Here is a partial ANOVA table based on these data: \[\begin{array}{l} \begin{array} { l | l r } &\text {Source }&\text {df }&\text {Sums of Squares }&\text {Mean Square}&\text { F-Ratio}&\\ \hline\text {group }&&&20.032\\ \text { Error} && & &21.8967\\ \hline\text {Total }& \end{array} \end{array}\] What is the value of the F statistic?

A) 0.46 B) 1.09 C) 1.84 D) 2.19 A) 0.46 B) 1.09 C) 1.84 D) 2.19

Exam 22: One-Way Analysis of Variance: Comparing Several Means

In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. Treatment Mean Standard Deviation Sample Size n Raw garlic 142 22 49 Garlic powder 137 25 47 Garlic extract 137 22 48 Placebo 133 21 48 Source df Sums of Squares Mean Square F-Ratio Treatment 959.46 Error 95,457.00 The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? Based on this ANOVA test, and using a significance level of 0.05, what should you conclude?

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Question 1

Correct Answer:

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C

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. Plants (per acre) Yield (bushels per acre) 12,000 150.1 113.0 118.4 16,000 166.9 120.7 135.2 20,000 165.3 130.1 139.6 24,000 134.7 138.4 156.1 Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error 356 Total Density Mean StDev 12,000 3 127.17 20.04 16,000 3 140.93 23.63 20,000 3 145.00 18.21 24,000 3 143.07 11.44 What is the degrees of freedom for the error component?

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Question 2

Correct Answer:

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C

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total The ANOVA procedure relies on three important assumptions to be valid: (1) The data are independent random samples. (2) The populations are Normally distributed or the sample sizes are large enough. (3) The populations have the same standard deviation σ . In the case of this ANOVA test, which of the following statements is correct?

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Question 3

Correct Answer:

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D

The F distributions are a family of distributions that take on only positive values and are skewed right.

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Question 4

Does fibrolytic enzyme level in feedlot cattle affect cattle weight gain, on average? Researchers randomly assigned 20 feedlot cattle to four groups, with each group being fed a diet containing a different enzyme level. Cattle weight gain (in kilograms) was recorded over the time of the study. There was no obvious deviation from Normality for these data. The findings are summarized below: What are the degrees of freedom for the appropriate ANOVA test? Enzyme level n mean std. dev. None 5 51.8 4.9 Low 5 55.0 7.8 Medium 5 67.4 9.6 High 5 48.8 7.3

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Question 5

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. Plants (per acre) Yield (bushels per acre) 12,000 150.1 113.0 118.4 16,000 166.9 120.7 135.2 20,000 165.3 130.1 139.6 24,000 134.7 138.4 156.1 Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the sum of squares for the error component?

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Question 6

A study of the effects of sedating and nonsedating antihistamines on driving impairment was done in a driving simulator. Volunteers were randomly assigned to take either a sedating antihistamine, a nonsedating antihistamine, or a placebo. Their steering instability in the simulator was recorded on a quantitative scale. To analyze these data, which inference procedure would you use?

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Question 7

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. Plants (per acre) Yield (bushels per acre) 12,000 150.1 113.0 118.4 16,000 166.9 120.7 135.2 20,000 165.3 130.1 139.6 24,000 134.7 138.4 156.1 Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the value of the F statistic for this test?

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Question 8

Interchanging the degrees of freedom for the F distribution changes the distribution, so order of parameters is very important.

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Question 9

An experiment examined the psychophysiological effect of THC, the active ingredient in marijuana. The study recruited a sample of 18 young adults who were habitual marijuana smokers. Subjects came to the lab three times, each time completing the same questionnaire, but each time smoking a different marijuana cigarette: one with 3.9% THC, one with 1.8% THC, and one with no THC (a placebo). The order of the conditions was randomized in a double-blind design. Why can we not use a one-way ANOVA procedure here to test whether the mean "feeling of high" is the same for all three THC amounts?

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Question 10

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total The denominator degrees of freedom for the ANOVA is ________.

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Question 11

In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. Treatment Mean Standard Deviation Sample Size n Raw garlic 142 22 49 Garlic powder 137 25 47 Garlic extract 137 22 48 Placebo 133 21 48 Source df Sums of Squares Mean Square F-Ratio Treatment 659.46 Error 95,457.00 The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? What is the mean square for error to test this hypothesis?

(Multiple Choice)

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Question 12

ANOVA is not too sensitive to violations of Normality if all samples have similar sizes and no sample is very small.

(True/False)

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Question 13

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. Plants (per acre) Yield (bushels per acre) 12,000 150.1 113.0 118.4 16,000 166.9 120.7 135.2 20,000 165.3 130.1 139.6 24,000 134.7 138.4 156.1 Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ For this example, which of the following statements is correct?

(Multiple Choice)

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Question 14

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total The numerator degrees of freedom for the ANOVA is _______.

(Short Answer)

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Question 15

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total What is the P-value for the ANOVA that tests for equality of the population means of the three treatments?

(Multiple Choice)

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Question 16

In an experiment on the effect of garlic on blood lipid concentrations, adult volunteers with slightly elevated cholesterol levels were randomly assigned to one of four treatments taken daily for six months: raw garlic, garlic powder, garlic extract, or a placebo. The participants' LDL levels (low-density lipoprotein, or "bad" cholesterol, in mg/dL) were assessed at the end of the six-month study period. Summary statistics and a partial ANOVA table for this study are shown here. Treatment Mean Standard Deviation Sample Size n Raw garlic 142 22 49 Garlic powder 137 25 47 Garlic extract 137 22 48 Placebo 133 21 48 Source df Sums of Squares Mean Square F-Ratio Treatment 959.46 Error 95,457.00 The research question is: Do the data provide evidence that the treatments affect the mean LDL level in this population? What is the P-value for this ANOVA test?

(Multiple Choice)

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Question 17

How much corn should be planted per acre for a farmer to get the highest yield? Too few plants will give a low yield, while too many plants will result in plants competing for moisture and nutrients, resulting in a lower yield. Four levels of planting density are to be studied: 12,000, 16,000, 20,000, and 24,000 plants per acre. The experimenters had 12 acres available for the study; 3 acres were assigned at random to each of the planting densities. The data follow. Plants (per acre) Yield (bushels per acre) 12,000 150.1 113.0 118.4 16,000 166.9 120.7 135.2 20,000 165.3 130.1 139.6 24,000 134.7 138.4 156.1 Assume the data can be considered four independent SRSs, one from each of the four populations of planting densities, and that the distribution of the yields is Normal. A partial ANOVA table produced by Minitab follows, along with the means and standard deviations of the yields for the four groups. One-Way ANOVA: Yield Versus Density Source DF SS MS F P Density 589 Error $\quad 356$ Total Density N Mean StDev $12,000 \quad 3127.1720 .04$ $16,000 \quad 3140.9323 .63$ $20,000 \quad 3145.0018 .21$ $24,000 \quad 3143.0711 .44$ What is the best estimate of the pooled standard deviation for yield?

(Multiple Choice)

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Question 18

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total The mean square for groups, MSG, is _______________.

(Short Answer)

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Question 19

A study randomly assigned adult subjects to one of three exercise treatments: (1) a single long exercise period five days per week; (2) several ten-minute exercise periods five days per week; and (3) several ten-minute periods five days per week using a home treadmill. The study report contains the following summary statistics about weight loss (in kilograms) after six months of treatment: Treatment Mean Std. Dev. Long periods 10.2 4.2 37 Multiple short periods 9.3 4.5 36 Multiple short periods with treadmill 10.2 5.2 42 Here is a partial ANOVA table based on these data: Source df Sums of Squares Mean Square F-Ratio group 20.032 Error 21.8967 Total What is the value of the F statistic?

(Multiple Choice)

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Question 20

The F distributions are a family of distributions that take on only positive values and are skewed right.

Interchanging the degrees of freedom for the F distribution changes the distribution, so order of parameters is very important.

ANOVA is not too sensitive to violations of Normality if all samples have similar sizes and no sample is very small.

Picturing Distributions With Graphs

Describing Quantitative Distributions With Numbers

Scatterplots and Correlation

Regression

Two-Way Tables

Samples and Observational Studies

Designing Experiments

Essential Probability Rules

Independence and Conditional Probabilities

The Normal Distributions

Discrete Probability Distributions

Sampling Distributions

Introduction to Inference

Exercises

Inference About a Population Mean

Comparing Two Means

Inference About a Population Proportion

Comparing Two Proportions

The Chi-Square Test for Goodness of Fit

The Chi-Square Test for Two-Way Tables

Inference for Regression

More About Analysis of Variance: Follow-Up Tests and Two-Way Anova

Nonparametric Tests

Multiple and Logistic Regression

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