Substances to be tested for cancer-causing potential are often painted on the skin of mice. The question arose Whether mice might get an additional dose of the substance by licking or biting their cagemates. To answer this question, the compound benzo(a)pyrene was applied to the backs of 10 mice: 5 were individually housed, and 5 were group-housed in a single cage. After 48 hours, the concentration of the compound in the stomach tissue of each mouse was determined. The results (nmol/gm) were as follows:

(a) What is the value of the Wilcoxon-Mann-Whitney test statistic for comparing the distributions?
(b) Let the alternative hypothesis be that benzo(a)pyrene concentrations tend to be high in group-housed mice than in singly housed mice. The P-value for the directional Wilcoxon-Mann-Whitney test is 0.004. If α = 0.01, what is your conclusion regarding H0?
(c) Why is a directional alternative valid in this case?

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Singly housed Group-housed 3.3 2.4 2.5 3.3 2.4 3.9 4.1 4.8 3.9 3.4