Setting Up a Test for the Difference Between Two Population Proportions

A two-proportion test states $H_0: p_1 = p_2$, equivalently $p_1 - p_2 = 0$, against $H_a$ with $>$, $<$, or $\ne$, after both parameters are defined in context. Because the null says the two populations share one proportion, the test pools: $\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2} = \frac{228}{550} \approx 0.4145$, and $SE = \sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \approx 0.0422$. The conditions are random selection or assignment, independence between groups, the 10% condition for each sample, and all four counts at least 10.

The hypotheses go wrong in three ways: two separate values assigned to $p_1$ and $p_2$ from the samples, hypotheses written about $\hat{p}_1$ and $\hat{p}_2$ which are already known and unequal, or a null set to the observed difference of 0.10, which is the evidence rather than the claim. The standard error goes wrong by keeping the groups separate, which is the interval's formula and contradicts the null the test assumes. And the design goes unchecked, so paired before-and-after measurements get run through machinery whose derivation requires independence.