15.35 Mann-Whitney statistic For the tanning experiment, Table 15.2 showed the sampling distribution of the difference between the sample mean ranks. Suppose you instead use as a test statistic the sample proportion of pairs of participants for which the tanning studio gave a better tan than the tanning lotion. This is the basis of the Mann-Whitney test, devised by two statisticians about the same time as Wilcoxon devised his test.

a. The table shows the proportion of pairs for which the tanning studio gave a better tan, for the possible sample results. Explain how to find these proportions.

b. Using this table, construct the sampling distribution that applies under the null hypothesis for this proportion.

c. For the one-sided alternative hypothesis of a better tan with the tanning studio, find the P-value for the observed sample proportion of 5/6. (Note: P-values based on this sample proportion are identical to P-values for the Wilcoxon test based on the mean ranks.)

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Sample Proportion of Pairs, One from Each Treatment, for which Tanning Studio Gave Better Tan Than Tanning Lotion Treatments Ranks Tanning lotion ranks (1,2,3) (1,2,4) (1,2,5) (1,3,4) (1,3,5) Tanning studio ranks (4,5) (3,5) (3.4) (2,5) (2,4) Proportion 0/6 1/6 2/6 2/6 3/6 Tanning lotion ranks (2,3,4) (1,4,5) (2,3,5) (2,4,5) (3,4,5) Tanning studio ranks (1,5) (2,3) Proportion (2.3) (1,4) (1,3) (1.2) 3/6 4/6 4/6 5/6 6/6