A Nonrandomized Unconditional Test for Comparing Two Proportions in 2×2 Contingency Tables

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Abstract
It is shown that the “usual” nonrandomized, conditional test for comparing proportions using independent binomial samples, is very conservative in the sense that the actual significance level attributable to an outcome is often one-fourth to one-half of the anticipated value. A nonrandomized unconditional test is proposed, and for sample sizes up to 15, tables are given in an appendix which specify one-sided critical regions of size less than or equal to the nominal values 0.05, and 0.01 (two-sided critical regions are also given). Numerical examples illustrating the use of the tables and a brief description of the algorithm used to generate the tables are included.