11.87 Explaining Fisher’s exact test A pool of six candidates for three managerial positions includes three females and three males. Denote the three females by F1, F2, F3 and the three males by M1, M2, M3. The result of choosing three individuals for the managerial positions is (F2, M1, M3).

a. Identify the 20 possible samples that could have been selected. Explain why the contingency table relating gender to whether chosen for the observed sample is

b. For each of the 20 samples, find the corresponding contingency table as in part

a. Show that of these 20 tables, 1 has a cell count of 0, 9 have a cell count of 1, another 9 have a cell count of 2, and 1 has a cell count of 3 in the first cell. Hence, show that 10 of the 20 tables have a cell count in the first cell as large or larger than the observed table from part

a. (Note that, if the managers were randomly selected, each of the 20 tables is equally likely to be observed. Then, the probability is 10>20 = 0.5 of observing a table that shows equal or even stronger evidence that males are selected more often, and 0.5 is the P-value for Fisher’s exact test with Ha: p1 7 p2, where p1 is the proportion of males and p2 the proportion of females chosen for the position.)

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Chosen For Position Gender Yes No Male 2 1 Female 1 2