THE POWER FUNCTION OF THE TEST FOR THE DIFFERENCE BETWEEN TWO PROPORTIONS IN A 2×2 TABLE

Neyman & Pearson's (1933) conception of the power of a test of a statistical hypothesis, Ho, was developed, in the first instance as a means of guiding the choice between alternative tests. This, it was shown, could be done by comparing the effectiveness of the tests in discriminating between Ho and a set of admissible alternative hypotheses regarded as most relevant to the question under test. Where there is no doubt about the most appropriate test and no sequential scheme of sampling is possible, the power function may play a useful part in indicating, before the data are collected, how large the samples should be to avoid an inconclusive result. If this procedure is to be easily applied, a ready means must be available of calculating the power of the test for a given significance level and sample size. The tables of the power function of the t-test (Neyman & Tokarska, 1936) and Tang's Tables (1938) applicable in the Analysis of Variance, are examples of such aids. The present paper aims at providing in simple, if approximate, form a means of determining the power function of the test for the difference between two proportions. The test may be briefly outlined as follows. In two 'infinite' populations the proportions of individuals possessing a character A are pl(A) and p2(A) respectively. Random samples of m and n are drawn from the two populations and the result is represented in a 2 x 2 table, thus