On the Power Function of a Test of Significance for the Difference Between two Proportions

In medical science, especially while interpreting the results of expei_ments for the comparison of the therapeutic activities of two drugs, the statistical problem of testing if two samples come from the same Binomial population often arises. If, for instance, the number of individuals who recovered out of nx by the first treatment follows the Binomial distribution (qx+px)ni and the number who recovered out of n2 by the second treatment follows the distribution (q^^p^Y*, the problem can be defined mathematically as testing the statistical hypothesis H0(px = p2 = p), that the two parameters px and p2 have a common but unspecified value p. Alternative approaches to this problem depending on the type of experimental pro bability sot considered were examined by Pearson (1947) and Barnard (1947). The power function of the test based on the two-dimensional approach was studied by Patnaik (1948) who has given an approximate method of evaluating it. In this paper, exact values of power function of the test are given for a large range of the sample sizes. Also the usefulness of the concept of power in deciding the size of samples to be chosen in a planned experiment will be demonstrated.