Statistical Inference: A Short Course

11.6 Hypothesis Tests for the Difference of Proportions when Sampling from Two Independent Binomial Populations

Let and be random samples taken from two independent binomial populations with p_X and p_Y representing the proportions of successes in the first and second binomial populations. In addition, let X and Y be independent random variables depicting the observed number of successes in the samples of sizes n_X and n_Y, respectively. To test hypotheses regarding the differences in population proportions , we must review the characteristics of the sampling distribution of the difference between two sample proportions. (Remember that these tests are only approximate and hold only for large samples.)

We know that the best estimators for p_X and p_Y are the sample proportions of successes , respectively, with

Hence the best estimator for p_X − p_Y is ; and the best estimators for and are, respectively, ...