Tests for a difference in population proportions

What the exam asks

• Section I (Multiple Choice): Recognize the correct test statistic formula using $\hat{p}_c$ instead of separate sample proportions, verify the normality condition using pooled expected counts, or interpret a given $p$-value in context.
• Section II (Free Response Question 3): Execute a complete hypothesis test across labeled parts, earning up to 10 analytic points (each earned independently unless the scoring guideline says otherwise) for tasks such as naming the procedure, defining parameters, stating hypotheses, checking conditions with explicit numbers, calculating $z$ and the $p$-value, and stating a conclusion in context.

Core ideas

Parameters and hypotheses

Define $p_1$ and $p_2$ in context as the true population proportions (for observational studies) or true treatment proportions (for experiments). Never refer to the sample in parameter definitions. State the order of subtraction clearly.

• Null hypothesis: $H_0: p_1 - p_2 = 0$ (or )

The combined (pooled) proportion and conditions

Because a hypothesis test operates under the assumption that $H_0$ is true ($p_1 = p_2$), both samples estimate the same common population proportion. We estimate this shared proportion by pooling the total successes and total sample sizes into the , :

Verify three conditions before carrying out the test: