Interval estimation for the difference between proportions

Examples

$X \sim B i n (n, p_{1}), Y \sim B i n (m, p_{2}),$ assume $n, m \geq 25$ to apply Central Limit Theorem

\begin{aligned} {\hat{p}}_{1} = \frac{x}{n}, {\hat{p}}_{2} = \frac{y}{m}, {\hat{p}}_{1} - {\hat{p}}_{2} = \frac{x}{n} - \frac{y}{m} \\ E ({\hat{p}}_{1} - {\hat{p}}_{2}) = E (\frac{x}{n} - \frac{y}{m}) = \frac{n p_{1}}{n} - \frac{m p_{2}}{m} = p_{1} - p_{2} \\ V a r ({\hat{p}}_{1} - {\hat{p}}_{2}) = V a r ({\hat{p}}_{1}) + V a r ({\hat{p}}_{2}) = V a r (\frac{x}{n}) + V a r (\frac{y}{m}) \\ = \frac{n p_{1} (1 - p_{1})}{n^{2}} + \frac{m p_{2} (1 - p_{2})}{m^{2}} = \frac{p_{1} (1 - p_{1})}{n} + \frac{p_{2} (1 - p_{2})}{m} \\ Z & = \frac{{\hat{p}}_{1} - {\hat{p}}_{2} - E ({\hat{p}}_{1} - {\hat{p}}_{2})}{\sqrt{V \hat{a} r ({\hat{p}}_{1} - {\hat{p}}_{2})}} = \frac{{\hat{p}}_{1} - {\hat{p}}_{2} - (p_{1} - p_{2})}{\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{m}}} \sim N (0, 1) approx \\ D = p_{1} - p_{2}, P (D_{l} \leq D \leq D_{u}) = 1 - α \dots \\ CI for D : \\ ({\hat{p}}_{1} - {\hat{p}}_{2} - Z_{1 - \frac{α}{2}} \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{m}, {\hat{p}}_{1} - {\hat{p}}_{2} + Z_{1 - \frac{α}{2}} \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{m}) \end{aligned}