Setting up a test for a population proportion

What the exam asks

On the AP exam, setting up a test for a single proportion requires you to:

• Identify the appropriate inference procedure by name: a one-sample $z$-test for a population proportion.
• Define the parameter $p$ in the context of the study (never as a sample statistic).
• State the null hypothesis ($H_0$) and alternative hypothesis ($H_a$) using proper notation and the hypothesized value $p_0$.
• Verify the three conditions for inference—randomization, the 10% condition, and large counts—using the specific numbers from the prompt.

Core ideas

Identifying the procedure and parameter

When a study collects one categorical variable from a single random sample and asks whether there is convincing statistical evidence to support a claim about the population proportion, the procedure is a one-sample $z$-test for a population proportion.

Before or right after writing the hypotheses, define the parameter $p$ in context:

Let $p$ = the true proportion of [population of interest] that [has the categorical trait of interest].

Describe the whole population (“the proportion of all customers who prefer...”) rather than the sample (“the proportion of the sampled customers who preferred...”), which describes the sample statistic $\hat{p}$.

Null and alternative hypotheses

Every significance test pits two competing claims about the population parameter $p$ against each other:

1. Null hypothesis ($H_0$): The statement of “no difference,” “no change,” or the baseline status quo. It is always an equality assumed to be true unless the sample provides convincing statistical evidence against it:

   $$H_0: p = p_0$$