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

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. Ha: 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 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 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 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

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

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

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

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

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. 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.