Deck 24: Nonparametric Tests

Question 1

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$ 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 91 cm?

A)8.0
B)8.5
C)9.0
D)9.5

Answer 1

C

Question 2

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$ 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?

A)4.0
B)5.0
C)5.5
D)6.0

Answer 2

C

Question 3

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$
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?

A)24.5
B)52.0
C)53.5
D)54.5

Answer 3

C

Question 4

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$ 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?

A)2.32
B)3.61
C)4.97
D)6.24

Question 5

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$ 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?

A)0.37
B)1.76
C)2.24
D)2.32

Question 6

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: $$\begin{array} { | l | l | } \hline \text { Herbicide } & \text { Height (cm) } \\\hline \text { A } & 97,93,91,91,89,87 \\\hline \text { B } & 91,87,84,78,74,69 \\\hline\end{array}$$ Let W, the Wilcoxon rank sum test statistic, be the sum of the ranks assigned to the heights of peach seedlings receiving Herbicide A.
The researcher is interested in testing the following two-sided test:
H₁: No difference in distribution of scores.
H_α: One of the herbicides produces seedlings that tend to have higher heights.
Using appropriate technology, what is the P-value for the test?

A)0.0102
B)0.0125
C)0.0204
D)0.0250

Question 7

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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?

A)2.0
B)2.5
C)3.0
D)4.0

Question 8

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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 16.7 g?

A)3.0
B)5.0
C)5.5
D)6.0

Question 9

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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.
What is the value of W?

A)6.5
B)11
C)22
D)35

Question 10

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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)2.5
B)5.5
C)30.0
D)35.0

Question 11

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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 mean of W?

A)2.5
B)5.5
C)30.0
D)35.0

Question 12

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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)0.18
B)0.91
C)2.14
D)2.47

Question 13

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: $$\begin{array} { | l | l l l l l l | } \hline \text { males } & 15.7 & 15.9 & 16.7 & 19.1 & 20.8 & \\\hline \text { females } & 14.7 & 15.4 & 15.5 & 16.8 & 18.6 & 19.3 \\\hline\end{array}$$
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.
The researcher is interested in testing the following two-sided test:
H₁: No difference in distribution of weights.
H_a: One of the sexes tends to have higher weights.
What is the P-value for the 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

Question 14

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:
$$\begin{array} {| l | l l l l l l| } \hline \text { Poisoned } & 8.456 & 12.207 & 16.869 & 20.589 & 22.429 & 25.050 \\\hline \text { Unpoisoned } & 6.642 & 8.182 & 9.351 & 9.686 & 11.074 & 12.064 \\\hline\end{array}$$
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.
When ranking all the observations, which rank should be assigned the relative spike height of 8.456?

Question 15

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:
$$\begin{array} {| l | l l l l l l| } \hline \text { Poisoned } & 8.456 & 12.207 & 16.869 & 20.589 & 22.429 & 25.050 \\\hline \text { Unpoisoned } & 6.642 & 8.182 & 9.351 & 9.686 & 11.074 & 12.064 \\\hline\end{array}$$
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?

Question 16

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:
$$\begin{array} {| l | l l l l l l| } \hline \text { Poisoned } & 8.456 & 12.207 & 16.869 & 20.589 & 22.429 & 25.050 \\\hline \text { Unpoisoned } & 6.642 & 8.182 & 9.351 & 9.686 & 11.074 & 12.064 \\\hline\end{array}$$
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.
If the two populations have the same continuous distribution, what is the mean of W?

Question 17

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 six rats poisoned with DDT and a control group of six unpoisoned rats. Here are the findings: $$\begin{array} {| l | l l l l l l |} \hline \text { Poisoned } & 8.456 & 12.207 & 16.869 & 20.589 & 22.429 & 25.050 \\\hline \text { Unpoisoned } & 6.642 & 8.182 & 9.351 & 9.686 & 11.074 & 12.064 \\\hline\end{array}$$
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 standardized value z of the Wilcoxon rank sum test, W?

A)0.16
B)2.24
C)6.24
D)9.10

Question 18

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 six rats poisoned with DDT and a control group of six unpoisoned rats. Here are the findings: $$\begin{array} {| l | l l l l l l |} \hline \text { Poisoned } & 8.456 & 12.207 & 16.869 & 20.589 & 22.429 & 25.050 \\\hline \text { Unpoisoned } & 6.642 & 8.182 & 9.351 & 9.686 & 11.074 & 12.064 \\\hline\end{array}$$
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.
The researcher is interested in testing the following two-sided test:
H₁: No difference in distribution of relative spike heights.
H_a: One of the conditions produces relative spike heights that tend to be larger.
What is the P-value for the 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

Question 19

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: $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}$$
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⁺?

A)6
B)7.5
C)13
D)27.5

Question 20

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: $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}$$
What is the value of the Wilcoxon signed rank test, W⁺?

A)7.5
B)13
C)21
D)34

Question 21

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: $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}$$
What is the standardized value z of the Wilcoxon signed rank test, W⁺?

A)0.40
B)0.95
C)1.48
D)3.51

Question 22

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: $$\begin{array} { | l | l | l | l | l | l | } \hline \text { Subject } & 1 & 2 & 3 & 4 & 5 \\\hline \text { Initial HDL } & 31 & 36 & 32 & 28 & 37 \\\hline \text { HDL after 6 months } & 39 & 32 & 44 & 34 & 39 \\\hline \text { Difference } & 8 & - 4 & 12 & 6 & 2 \\\hline\end{array}$$
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⁺?

A)0.0281
B)0.0329
C)0.0495
D)0.0694

Question 23

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: $$\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}$$
We choose to analyze these data using the Wilcoxon signed rank test.
If the bacterial counts have a continuous distribution that is not affected by the different treatments, then what is the mean of the Wilcoxon signed rank test statistic, W⁺?

A)4
B)4.5
C)5
D)7.5

Question 24

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: $$\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}$$
We choose to analyze these data using the Wilcoxon signed rank test.
If the bacterial counts have a continuous distribution that is not affected by the different treatments, what is the standard deviation of the Wilcoxon signed rank test statistic, W⁺?

A)3.71
B)4.50
C)7.50
D)13.75

Question 25

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: $$\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}$$
We choose to analyze these data using the Wilcoxon signed rank test.
What is the value of the Wilcoxon signed rank test, W⁺?

A)4
B)4.5
C)5
D)7.5

Question 26

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: $$\begin{array} { l l l l l l } & \text { Strain 1 } & \text { Strain 2 } & \text { Strain3 } & \text { Strain 4 } & \text { Strain 5 } \\\mathbf { 0 . 6 \% } \% \text { tryptone } & 66 & 71 & 93 & 102 & 62 \\\mathbf { 1 } \% \text { tryptone } & 147 & 63 & 89 & 106 & 98\end{array}$$
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⁺?

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

Question 27

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?
$$\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}$$
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⁺?

Question 28

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?
$$\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}$$
What is the value of the Wilcoxon signed rank test, W⁺?

Question 29

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? $$\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}$$
What is the standardized value z of the Wilcoxon signed rank test, W⁺?

A)0.21
B)2.20
C)2.31
D)4.77

Question 30

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? $$\begin{array} { | l | l | l | l | l | l | l | } \hline \text { Dog ID } & 1 & 2 & 3 & 4 & 5 & 6 \\\hline \text { Control } & 1.73 & 3.37 & 3.59 & 2.05 & 1.86 & 3.60 \\\hline \text { Phosphate } & 8.16 & 4.58 & 3.98 & 5.24 & 3.04 & 7.03 \\\hline\end{array}$$
The researchers are interested in testing the following one-sided test:
H₀: Bone formation with and without phosphate supplementation have the same distribution.
H_a: Bone formation is systematically higher with phosphate supplementation.
Using the Normal approximation, what is the P-value for the Wilcoxon signed rank test W⁺?

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

Question 31

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.
$$\begin{array} { | c | c | c | } \hline \text { Buffer } & \text { Gene } & \text { Gene+Nano } \\\hline 3.3 & 2.7 & 1.1 \\3.8 & 3.0 & 1.1 \\4.1 & 3.3 & 1.2 \\5.4 & 3.4 & 1.4 \\5.4 & 3.9 & 1.8 \\6.8 & 4.3 & 1.8 \\7.0 & 5.3 & 2.1 \\7.8 & 5.6 & 2.1 \\8.1 & 5.6 & 3.5 \\9.1 & & 4.1 \\\hline\end{array}$$ 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?

A)Not all three mean tumor fold-increases are equal.
B)The mean tumor fold-increase is larger for one treatment than for the other two treatments.
C)The mean tumor fold-increases are not Normally distributed in each group.
D)The tumor fold-increases are systematically larger in some treatments than in others.

Question 32

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.
$$\begin{array} { | c | c | c | } \hline \text { Buffer } & \text { Gene } & \text { Gene+Nano } \\\hline 3.3 & 2.7 & 1.1 \\3.8 & 3.0 & 1.1 \\4.1 & 3.3 & 1.2 \\5.4 & 3.4 & 1.4 \\5.4 & 3.9 & 1.8 \\6.8 & 4.3 & 1.8 \\7.0 & 5.3 & 2.1 \\7.8 & 5.6 & 2.1 \\8.1 & 5.6 & 3.5 \\9.1 & & 4.1 \\\hline\end{array}$$ What is the average rank for the gene nanoparticle combination?

A)6.8
B)15.0
C)22.2
D)66.5

Question 33

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.
$$\begin{array} { | c | c | c | } \hline \text { Buffer } & \text { Gene } & \text { Gene+Nano } \\\hline 3.3 & 2.7 & 1.1 \\3.8 & 3.0 & 1.1 \\4.1 & 3.3 & 1.2 \\5.4 & 3.4 & 1.4 \\5.4 & 3.9 & 1.8 \\6.8 & 4.3 & 1.8 \\7.0 & 5.3 & 2.1 \\7.8 & 5.6 & 2.1 \\8.1 & 5.6 & 3.5 \\9.1 & & 4.1 \\\hline\end{array}$$ Under the null hypothesis that the three populations have the same continuous distribution, which distribution does the test statistic H have?

A)Approximately a chi-square distribution with 2 degrees of freedom
B)Approximately an F(2, 26) distribution
C)A distribution that cannot be evaluated because the populations may not be Normal
D)Approximately a Normal distribution

Question 34

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.
$$\begin{array} { | c | c | c | } \hline \text { Buffer } & \text { Gene } & \text { Gene+Nano } \\\hline 3.3 & 2.7 & 1.1 \\3.8 & 3.0 & 1.1 \\4.1 & 3.3 & 1.2 \\5.4 & 3.4 & 1.4 \\5.4 & 3.9 & 1.8 \\6.8 & 4.3 & 1.8 \\7.0 & 5.3 & 2.1 \\7.8 & 5.6 & 2.1 \\8.1 & 5.6 & 3.5 \\9.1 & & 4.1 \\\hline\end{array}$$ Software gives H = 16.7 for these data. What is the P-value of the Kruskal-Wallis 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

Question 35

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:
$$\begin{array} { | l | l | } \hline \text { Substance type } & \text { Lifetime } \\\hline \text { Radioactive } & 26,29,33 \\\hline \text { Biological } & 27,29,31 \\\hline \text { Chemical } & 31,34,37 \\\hline\end{array}$$ The data are to be analyzed with the Kruskal-Wallis test. The null hypothesis is that the time until death has the same distribution in all groups. What is the alternative hypothesis?

A)Not all three mean lifetimes are equal.
B)The mean time until death is longer for one substance than for the other two substances.
C)The mean lifetimes are not Normally distributed in each group.
D)None of these choices are correct.

Question 36

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:
$$\begin{array} { | l | l | } \hline \text { Substance type } & \text { Lifetime } \\\hline \text { Radioactive } & 26,29,33 \\\hline \text { Biological } & 27,29,31 \\\hline \text { Chemical } & 31,34,37 \\\hline\end{array}$$ Using appropriate technology, what is the value of the Kruskal-Wallis statistic, H, for these data?

A)H = 2.77
B)H = 3.10
C)H = 3.76
D)H = 5.99

Question 37

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:
$$\begin{array} { | l | l | } \hline \text { Substance type } & \text { Lifetime } \\\hline \text { Radioactive } & 26,29,33 \\\hline \text { Biological } & 27,29,31 \\\hline \text { Chemical } & 31,34,37 \\\hline\end{array}$$ Under the null hypothesis that the three populations have the same continuous distribution, what is the best description of the distribution of H?

A)H has approximately a chi-square distribution with 3 degrees of freedom.
B)H has approximately an F(2, 6) distribution.
C)H has a distribution that cannot be evaluated because the populations may not be Normal.
D)None of the above

Question 38

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:
$$\begin{array} { | l | l | } \hline \text { Substance type } & \text { Lifetime } \\\hline \text { Radioactive } & 26,29,33 \\\hline \text { Biological } & 27,29,31 \\\hline \text { Chemical } & 31,34,37 \\\hline\end{array}$$ What is the P-value of the Kruskal-Wallis statistic, H, for these data?

A)Less than 0.025
B)Between 0.025 and 0.05
C)Between 0.05 and 0.10
D)Greater than 0.10

Question 39

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
$\begin{array} { l r r r r } \text { vaccine } & \mathrm { N } & \text { Median } & \text { Ave Rank } & \mathrm { Z } \\ 1 & 16 & 5.530 & 18.6 & - 0.18 \\ 2 & 13 & 5.030 & 11.2 & - 3.25 \\ \text { sham } & 8 & 6.860 & 32.5 & 3.98 \\ \text { Overall } & 37 & & 19.0 & \end{array}$
$$\begin{array} { l } \mathrm { H } = 19.29 \quad \mathrm { DF } = \quad \mathrm { P } = \\\mathrm { H } = 19.30 \quad \mathrm { DF } = \quad \mathrm { P } = \quad \text { (adjusted for ties) } \\\end{array}$$ The null hypothesis of the Kruskal-Wallis test is that viral load has the same distribution in all groups. What is the alternative hypothesis?

A)Not all three mean viral loads are equal.
B)The mean viral load is larger for the sham treatment than for the two vaccines.
C)The median viral load is larger for the sham treatment than for the two vaccines.
D)The viral loads are systematically larger in some treatments than in others.

Question 40

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:
​
$$\begin{array}{l}\text { Kruskal-Wallis Test on viral load }\\\begin{array} { l r r r r } \text { vaccine } & \mathrm { N } & \text { Median } & \text { Ave Rank } & Z \\1 & 16 & 5.530 & 18.6 & - 0.18 \\2 & 13 & 5.030 & 11.2 & - 3.25 \\\text { sham } & 8 & 6.860 & 32.5 & 3.98 \\\text { Overall } & 37 & & 19.0 & \\H = 19.29 & \mathrm { DF } = \quad \mathrm { P } = \\\mathrm { H } = 19.30 \quad \mathrm { DF } = \quad \mathrm { P } = \quad \text { (adjusted for ties) }\end{array}\end{array}$$ Under the null hypothesis that the three populations have the same continuous distribution, which type of distribution does the Kruskal-Wallis statistic, H, have?

A)Approximately a chi-square distribution with 2 degrees of freedom
B)Approximately an F(2, 34) distribution
C)Approximately the standard Normal distribution
D)A distribution that cannot be evaluated because the populations may not be Normal

Question 41

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:
​
$\begin{array}{l}\text { Kruskal-Wallis Test on viral load }\\\begin{array}{lrrrr}\text { vaccine } & \mathrm{N} & \text { Median } & \text { Ave Rank } & \mathrm{Z} \\1 & 16 & 5.530 & 18.6 & -0.18 \\2 & 13 & 5.030 & 11.2 & -3.25 \\\text { sham } & 8 & 6.860 & 32.5 & 3.98 \\\text { Overall } & 37 & & 19.0 &\end{array}\end{array}$

$\begin{array}{llll}\mathrm{H}=19.29 & \mathrm{DF}= & \mathrm{P}= & \\\mathrm{H}=19.30 & \mathrm{DF}= & \mathrm{P}= & \text { (adjusted for ties) }\end{array}$ What is the P-value of the Kruskal-Wallis test for these data?

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