AP Stats 3.5: Setting Up a Proportion Test

What topic 3.5 covers

Topic 3.5 is the setup stage of a significance test for one proportion. A hypothesis test is a procedure for deciding about the value of a population parameter. The correct method here is the one-sample z-test for a population proportion.

You identify the parameter, state the hypotheses, and check conditions, but you do not compute the test statistic until topic 3.7. Doing the setup carefully is worth real points on the free-response section, where a missing parameter definition or an unchecked condition costs credit.

Null and alternative hypotheses

The null hypothesis $H_0$ is the status quo, assumed true unless the evidence is convincing otherwise. The alternative hypothesis $H_a$ is the researcher's claim, the statement you collect evidence for.

For a proportion, $H_0\colon p = p_0$, where $p_0$ is the hypothesized value. The alternative is one of $H_a\colon p < p_0$, $H_a\colon p > p_0$ (one-sided), or $H_a\colon p \ne p_0$ (two-sided). Hypotheses are always about the parameter $p$, never about the sample statistic $\hat{p}$.

Conditions to justify the test

The one-sample z-test requires three conditions:

1. Randomization: the data come from a random sample.
2. 10% condition: when sampling without replacement, $n \le 0.10N$.
3. Normality: the expected successes $np_0$ and expected failures $n(1-p_0)$ are both at least 10.

The test uses the hypothesized value $p_0$ in the normality check, unlike the confidence interval in topic 3.3, which uses $\hat{p}$. That difference is small on paper but is a frequent source of lost points, so decide which procedure you are running before you plug in numbers.

Choosing one-sided or two-sided

Let the research question, not the data, decide the direction of the alternative. Use a one-sided alternative when the question asks whether the proportion is specifically greater than or specifically less than $p_0$, and a two-sided alternative when it asks whether the proportion simply differs from $p_0$. Even when the alternative is one-sided, the null is tested at the boundary of equality, so you still calculate as if $p = p_0$.

Writing the direction before seeing $\hat{p}$ keeps the test honest and prevents you from cherry-picking the tail that happens to look significant. If the question gives no directional language and only asks whether the proportion differs from the claimed value, default to a two-sided alternative.