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?

Question

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?

Quizplus · Accepted Answer

We can use the Wilcoxon rank sum test to compare the distributions of the relative spike heights in the poisoned and unpoisoned rats. The null hypothesis is that there is no difference in distribution between the two groups, and the alternative hypothesis is that the relative spike heights tend to be larger in one of the conditions.

We first rank all 12 observations from lowest to highest (ignoring group membership) and calculate the sum of ranks for the poisoned rats:
$$W = 1 + 2 + 3 + 4 + 5 + 6 = 21$$

We can use this value to calculate the test statistic U:
$$U = W - \frac{n_1(n_1+1)}{2}$$
where $n_1$ is the sample size of the first group (poisoned rats). Substituting in the values, we get:
$$U = 21 - \frac{6(6+1)}{2} = 0$$

To find the P-value for the test, we can use a normal approximation to the Wilcoxon test. Under the null hypothesis, the test statistic follows a normal distribution with mean $\mu_U = \frac{n_1n_2}{2}$ and variance $\sigma_U^2 = \frac{n_1n_2(n_1+n_2+1)}{12}$, where $n_2$ is the sample size of the second group (unpoisoned rats). Substituting in the values, we get:
$$\mu_U = \frac{6\times6}{2} = 18 \text{ and }\sigma_U^2 = \frac{6\times6\times(6+6+1)}{12} = 52.5$$
The z-score corresponding to U = 0 is:
$$z = \frac{0 - 18}{\sqrt{52.5}} \approx -2.08$$
Using a standard normal distribution table or calculator, we can find the two-sided P-value:
$$P = 2\times P(Z < -2.08) \approx 0.038$$
Since the P-value is between 0.01 and 0.05, we reject the null hypothesis at the 0.05 significance level and conclude that there is evidence of a difference in distribution between the poisoned and unpoisoned rats. The alternative hypothesis is that the relative spike heights tend to be larger in the poisoned rats.

Electrical Measurements of Nerve Activity Give a Clue to the Nature