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 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 value of the F statistic to test this hypothesis?

A medical research team is interested in determining whether a new drug has an effect on creatine kinase (CK), which is often assayed in blood tests as an indicator of myocardial infarction. A random selection of 20 patients from a pool of possible subjects is selected, and each subject is given the medication. The subjects' CK levels are observed initially, after 3 weeks, and again after 6 weeks. The purpose is to study the CK levels over time. Here is a summary of the findings: In this example, what should we notice? Time (weeks) Mean CK Level (U/L) Standard Deviation (U/L) 0 121 20.37 3 106 16.09 6 100 10.21

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 null hypothesis for the ANOVA?

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 density (i.e., between-group) component?

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: 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 What is the alternative hypothesis for the appropriate ANOVA?

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: When checking the assumptions for this ANOVA test, what should we conclude? 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

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. What are the correct summary statistics for these data? (Use technology.)

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. 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 . $\sigma$ In the case of this ANOVA test, which of the following statements about this scenario is correct?

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. Based on this ANOVA test, and using a significance level of 0.05, what should you conclude?

Researchers wish to examine the effectiveness of a new weight-loss pill. A total of 200 obese adults are randomly assigned to one of four conditions: weight-loss pill alone, weight-loss pill with a low-fat diet, placebo pill alone, or placebo pill with a low-fat diet. The weight loss after six months of treatment is recorded in pounds for each subject. To analyze these data, which inference procedure would you use?

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 degrees of freedom in the numerator for this test?

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 alternative hypothesis for the ANOVA?

Heliconia is a genus of tropical plant with different varieties often fertilized by distinct species of hummingbirds. Researchers measured the flower length (in millimeters) of independent random samples of three varieties of Heliconia to see if the three varieties differ significantly in flower length. To analyze these data, which inference procedure would you use?

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. What is the mean square for groups to test this hypothesis? (Use technology.)

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 P-value for the ANOVA that tests for equality of the population means of the four densities?

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. What is the sum of squares for error to test this hypothesis? (Use technology.)

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. What is the P-value for this ANOVA test? (Use technology.)

Researchers examined a new treatment for advanced ovarian cancer in a mouse model. They created a nanoparticle-based delivery system for a suicide-gene therapy to be delivered directly to the tumor cells. The mice were randomly assigned to have their tumor injected with either the gene -- nanoparticle combination, the gene alone, or some buffer solution (placebo). The following table shows the tumor fold-increases after two weeks in a total of 29 mice. For reference, a fold-increase of 1 represents no change; a 2 represents a doubling in volume of the tumor. - Buffer Gene Gene+Nano 9.1 5.6 4.1 8.1 5.3 3.5 7.8 5.6 2.1 7.0 4.3 2.1 6.8 3.9 1.8 5.4 3.4 1.8 5.4 2.7 1.2 4.1 3.0 1.1 3.8 3.3 1.1 3.3 1.4 We want to test the null hypothesis that there is no difference in population mean tumor fold-increase for the three treatments. What is the value of the F statistic to test this hypothesis? (Use technology.)

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 null hypothesis for this test?

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: The numerical value of the test statistic is 5.795. What is the P-value for the ANOVA that tests for equality of the population mean weight gains under the four feed conditions? 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