Exact Distribution of Difference of Two Sample Proportions and Its Inferences

Abstract

Comparing two population proportions using confidence interval could be misleading in many cases, such as the sample size being small and the test being based on normal approximation. In this case, the only one option that we have is to collect a large sample. Unfortunately, the large sample might not be possible. One example is a person suffering from a rare disease. The main purpose of this journal is to derive a closed formula for the exact distribution of the difference between two independent sample proportions, and use it to perform related inferences such as a confidence interval, regardless of the sample sizes and compare with the existing Wald, Agresti-Caffo and Score. In this journal, we have derived a closed formula for the exact distribution of the difference between two independent sample proportions. This distribution doesn’t need any requirements, and can be used to perform inferences such as: a hypothesis test for two population proportions, regardless of the nature of the distribution and the sample sizes. We claim that exact distribution has the least confidence width among Wald, Agresti-Caffo and Score, so it is suitable for inferences of the difference between the population proportion regardless of sample size.