Statistical Inference: A Short Course

11.3 Confidence Intervals for the Difference of Proportions When Sampling from Two Independent Binomial Populations

Let and be random samples drawn from two independent binomial populations, where p_X and p_Y are the proportions of successes in the first and second binomial populations, respectively. Additionally, let X and Y be independent random variables representing the observed number of successes in the samples of size n_X and n_Y, respectively.

To determine a 100(1 − α)% confidence interval for the difference in proportions , let us first discuss the characteristics of the sampling distribution of the difference between two sample proportions. We found in Section 9.1 that the best estimators for p_X and p_Y are the sample proportions or observed relative frequencies of successes and , respectively, with

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