Two-Sample Proportion Test

Used to compare the proportion of successes between two populations. Suppose the $p_1$, $p_2$ are the true proportions of two populations.

Suppose the sample proportions from independent random samples of sizes $n_1$ and $n_2$ are:

$$\hat{p}_1 = \frac{x_1}{n_1}, \quad \hat{p}_2 = \frac{x_2}{n_2}$$

Here $x_1$, $x_2$ are successes on each sample.

Hypotheses Setup

The general null and alternative hypotheses are:

$H_0 : p_1 - p_2 = d_0$

$H_1 : p_1 - p_2 < d_0\ \text{or}\ p_1 - p_2 > d_0\ \text{or}\ p_1 - p_2 \neq d_0$

Test Statistic

$$T_\text{cal} = \frac{ \hat{p_1} - \hat{p_2} - (p_1 - p_2) }{ \sqrt{ \frac{p_1^2 (1 - p_1)^2}{n_1} + \frac{p_2^2 (1 - p_2)^2}{n_2} } } \sim N(0,1)$$