Exam 24: Nonparametric Tests

In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol"), a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table: Subject 1 2 3 4 5 Initial HDL 31 36 32 28 37 HDL after 6 months 39 32 44 34 39 Difference 8 -4 12 6 2 What is the standardized value z of the Wilcoxon signed rank test, W⁺?

Electrical measurements of nerve activity give a clue to the nature of poisoning with the insecticide DDT. When a nerve is stimulated, its electrical response shows a sharp spike, which is followed by a much smaller second spike. In a randomized experiment, researchers measured the relative height of the second spike as a percent of the first spike, comparing 6 rats poisoned with DDT and a control group of 6 unpoisoned rats. Here are the findings: Poisoned 8.456 12.207 16.869 20.589 22.429 25.050 Unpoisoned 6.642 8.182 9.351 9.686 11.074 12.064 Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the relative spike height in rats poisoned with DDT. What is the value of W?

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 tumors 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. A fold-increase of 1 represents no change; a fold-increase of 2 represents a doubling in volume of the tumor. Buffer Gene Gene+Nano 3.3 2.7 1.1 3.8 3.0 1.1 4.1 3.3 1.2 5.4 3.4 1.4 5.4 3.9 1.8 6.8 4.3 1.8 7.0 5.3 2.1 7.8 5.6 2.1 8.1 5.6 3.5 9.1 4.1 Software gives H = 16.7 for these data. What is the P-value of the Kruskal-Wallis test?

Do male and female finches in the wild have similar weight distributions? Researchers measured the weight (in grams) of a random sample of finches caught in the wild. Here are the results: males 15.7 15.9 16.7 19.1 20.8 females 14.7 15.4 15.5 16.8 18.6 19.3 Given the small sample sizes, we prefer using a nonparametric procedure to determine whether males and females differ in weight. Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the males' weights. If the two populations have the same continuous distribution, what is the standard deviation of W?

A researcher is interested in the survival times of a particular organism after it is subjected to three substances. The first is a radioactive material, the second is a biological material, and the third is a chemical (nonradioactive) substance. Organisms are brought into contact with each of the three substances to see how long they will last (in hours) until they die. The times are recorded for each substance used in the study, and the results are given below: Substance type Lifetime Radioactive 26,29,33 Biological 27,29,31 Chemical 31,34,37 What is the P-value of the Kruskal-Wallis statistic, H, for these data?

In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol"), a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table: Subject 1 2 3 4 5 Initial HDL 31 36 32 28 37 HDL after 6 months 39 32 44 34 39 Difference 8 -4 12 6 2 The researcher is interested in testing the following one-sided test: H₁: Initial and final HDL measurements have the same distribution. H_a: Final HDL levels are systematically lower. What is the P-value for the Wilcoxon signed rank test, W⁺?

Some researchers have suggested that stem-pitting disease in peach tree seedlings can be controlled with soil treatment. An experiment was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In an experimental field containing 12 seedlings, 6 were randomly selected throughout the field to receive Herbicide A. The remaining 6 received Herbicide B. Soil and weeds for each seedling were treated with the appropriate herbicide for three months, at which point the height in centimeters was recorded for each seedling. The following results were obtained: Herbicide Height (cm) A 97,93,91,91,89,87 B 91,87,84,78,74,69 Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the heights of peach seedlings receiving Herbicide A. If the two populations have the same continuous distribution, what is the standard deviation of W?

Some researchers have suggested that stem-pitting disease in peach tree seedlings can be controlled with soil treatment. An experiment was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In an experimental field containing 12 seedlings, 6 were randomly selected throughout the field to receive Herbicide A. The remaining 6 received Herbicide B. Soil and weeds for each seedling were treated with the appropriate herbicide for three months, at which point the height in centimeters was recorded for each seedling. The following results were obtained: Herbicide Height (cm) A 97,93,91,91,89,87 B 91,87,84,78,74,69 Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the heights of peach seedlings receiving Herbicide A. Let z be the standardized value of W. What is the value of z?

Some researchers have suggested that stem-pitting disease in peach tree seedlings can be controlled with soil treatment. An experiment was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In an experimental field containing 12 seedlings, 6 were randomly selected throughout the field to receive Herbicide A. The remaining 6 received Herbicide B. Soil and weeds for each seedling were treated with the appropriate herbicide for three months, at which point the height in centimeters was recorded for each seedling. The following results were obtained: Herbicide Height (cm) A 97,93,91,91,89,87 B 91,87,84,78,74,69 Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the heights of peach seedlings receiving Herbicide A. What is the value of W?

Do male and female finches in the wild have similar weight distributions? Researchers measured the weight (in grams) of a random sample of finches caught in the wild. Here are the results: males 15.7 15.9 16.7 19.1 20.8 females 14.7 15.4 15.5 16.8 18.6 19.3 Given the small sample sizes, we prefer using a nonparametric procedure to determine whether males and females differ in weight. Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the males' weights. Let z be the standardized value of W. What is the value of z?

A research team investigated the protective effect of two variants of a vaccine against the simian immunodeficiency virus (SIV) in rhesus monkeys. They measured the viral load (in log scale) of monkeys randomly assigned to either a vaccine variant or a sham (fake) vaccine. Here is a partial software output for a Kruskal-Wallis test on the data: ​ Kruskal-Wallis Test on viral load vaccine Median Ave Rank 1 16 5.530 18.6 -0.18 2 13 5.030 11.2 -3.25 sham 8 6.860 32.5 3.98 Overall 37 19.0 =19.29 = = =19.30 = = (adjusted for ties) What is the P-value of the Kruskal-Wallis test for these data?

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 tumors 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. A fold-increase of 1 represents no change; a fold-increase of 2 represents a doubling in volume of the tumor. Buffer Gene Gene+Nano 3.3 2.7 1.1 3.8 3.0 1.1 4.1 3.3 1.2 5.4 3.4 1.4 5.4 3.9 1.8 6.8 4.3 1.8 7.0 5.3 2.1 7.8 5.6 2.1 8.1 5.6 3.5 9.1 4.1 We choose to analyze these data with the Kruskal-Wallis test. The null hypothesis is that tumor fold-increase has the same distribution in all groups. What is the alternative hypothesis?

Researchers examined the effect of phosphate supplementation on bone formation in six healthy adult dogs. For each dog, bone formation was measured twice: once after 12 weeks of phosphate supplementation and once after a 12-week control period. The results in percent bone growth per year are shown below. Do the data provide evidence that phosphate supplementation significantly stimulates bone formation? Dog ID 1 2 3 4 5 6 Control 1.73 3.37 3.59 2.05 1.86 3.60 Phosphate 8.16 4.58 3.98 5.24 3.04 7.03 The data are analyzed using the Wilcoxon signed rank test. If the responses have a continuous distribution that is not affected by the different treatments, what is the mean value of W⁺?

In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol"), a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table: Subject 1 2 3 4 5 Initial HDL 31 36 32 28 37 HDL after 6 months 39 32 44 34 39 Difference 8 -4 12 6 2 The data are to be analyzed using the Wilcoxon signed rank test. If the responses have a continuous distribution that is not affected by the different treatments, what is the mean of W⁺?

Five strains of the Staphylococcus aureus bacteria were grown at 35 degrees Celsius for 24 hours in a culture broth containing either 0.6% of the nutrient tryptone or 1% of tryptone. We want to know if these two tryptone concentrations affect the growth of S. aureus in culture. Here are the resulting bacterial counts for each condition and bacterial strain: Strain 1 Strain 2 Strain3 Strain 4 Strain 5 \% tryptone 66 71 93 102 62 \% tryptone 147 63 89 106 98 We choose to analyze these data using the Wilcoxon signed rank test. What is the value of the Wilcoxon signed rank test, W⁺?

In a study to investigate the effects of regular exercise on raising high-density lipoprotein (HDL) cholesterol levels ("good cholesterol"), a random sample of five male subjects known to have low HDL levels had their HDL measured at the beginning of the study and then again after six months on a regular exercise schedule. The data are given in the following table: Subject 1 2 3 4 5 Initial HDL 31 36 32 28 37 HDL after 6 months 39 32 44 34 39 Difference 8 -4 12 6 2 What is the value of the Wilcoxon signed rank test, W⁺?

Researchers examined the effect of phosphate supplementation on bone formation in six healthy adult dogs. For each dog, bone formation was measured twice: once after 12 weeks of phosphate supplementation and once after a 12-week control period. The results in percent bone growth per year are shown below. Do the data provide evidence that phosphate supplementation significantly stimulates bone formation? Dog ID 1 2 3 4 5 6 Control 1.73 3.37 3.59 2.05 1.86 3.60 Phosphate 8.16 4.58 3.98 5.24 3.04 7.03 What is the value of the Wilcoxon signed rank test, W⁺?

Do male and female finches in the wild have similar weight distributions? Researchers measured the weight (in grams) of a random sample of finches caught in the wild. Here are the results: males 15.7 15.9 16.7 19.1 20.8 females 14.7 15.4 15.5 16.8 18.6 19.3 Given the small sample sizes, we prefer using a nonparametric procedure to determine whether males and females differ in weight. Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the males' weights. When ranking all the observations, which rank should be assigned to the weight of 15.4 g?

Five strains of the Staphylococcus aureus bacteria were grown at 35 degrees Celsius for 24 hours in a culture broth containing either 0.6% of the nutrient tryptone or 1% of tryptone. We want to know if these two tryptone concentrations affect the growth of S. aureus in culture. Here are the resulting bacterial counts for each condition and bacterial strain: Strain 1 Strain 2 Strain3 Strain 4 Strain 5 \% tryptone 66 71 93 102 62 \% tryptone 147 63 89 106 98 We choose to analyze these data using the Wilcoxon signed rank test. The researcher is interested in testing the following one-sided test: H₀: Bacterial counts with 0.6% and 1% tryptone have the same distribution. H_a: Bacterial counts are systematically higher in one condition than the other. Using the Normal approximation, what is the P-value for the Wilcoxon signed rank test, W⁺?

Some researchers have suggested that stem-pitting disease in peach tree seedlings can be controlled with soil treatment. An experiment was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In an experimental field containing 12 seedlings, 6 were randomly selected throughout the field to receive Herbicide A. The remaining 6 received Herbicide B. Soil and weeds for each seedling were treated with the appropriate herbicide for three months, at which point the height in centimeters was recorded for each seedling. The following results were obtained: Herbicide Height (cm) A 97,93,91,91,89,87 B 91,87,84,78,74,69 Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the heights of peach seedlings receiving Herbicide A. When ranking all the observations, which rank should be assigned to the heights of 87 cm?