4.2.2 Confidence Interval for Difference in Proportion

<< 4.2.1 Confidence Intervals for Difference in Means | 4.4 Order Statistics.md >>

Let

• $X\sim b(1,p_1)$ : Random variable (Bernoulli distribution)
• $Y\sim b(1,p_2)$ : Random variable
• $X,Y$ independent
• $X_1,\dots ,X_{n_1}$ : Random sample from $X$
• $Y_1,\dots ,Y_{n_2}$ : Random sample from $Y$
• $n=n_1+n_2$

Assume the random samples are independent of one another

$$\widehat{p_1}-\widehat{p_2}\pm z_{\alpha /2}\sqrt{\frac{\widehat{p_1}(1-\widehat{p_1})}{n_1}+\frac{\widehat{p_2}(1-\widehat{p_2})}{n_2}}$$