Question

Explain how to perform a​ two-sample z-test for the difference between two population proportions.

Answer 1

Point estimate for the difference between two population proportions is the difference between the two sample proportions, written as:

$\hat{p}_1-\hat{p}_2$

and the standard deviation is given by

$\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

When np and n(1-p) is greater than 5, then the sampling distribution is approximately normal and we can use standard normal methods.

Thus, the propability of a test for the difference between two population proportions is given by

$P(X\ \textless \ x)=P\left(z\ \textless \ \frac{\hat{p}_1-\hat{p}_2}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}} \right)$