Carrying Out a Hypothesis Test for a Proportion — AP Statistics
1. What Is a One-Proportion Hypothesis Test? ★★☆☆☆ ⏱ 3 min
Carrying out a hypothesis test for a proportion (often called a one-proportion z-test, or 1-prop z-test for short) is the core inference procedure for testing a claim about an unknown population proportion when you have a single sample of binary categorical data. This topic makes up 6-8% of total AP exam points, appearing on both multiple-choice and free-response sections. The core goal is to quantify how unlikely your observed sample result is if the null claim about the population proportion is true, allowing you to make a statistically justified conclusion about the original claim.
2. Stating Hypotheses and Checking Inference Conditions ★★☆☆☆ ⏱ 4 min
The first step of any hypothesis test for a proportion is defining your parameter of interest, stating your hypotheses, then verifying that the conditions for z inference are satisfied. Hypotheses are always about the unknown population parameter $p$, not the observed sample $\hat{p}$.
- **Random**: The sample is drawn randomly from the population of interest, or comes from a randomized experiment.
- **Independence**: Individual observations are independent. When sampling without replacement, this requires the 10% condition: $n < 0.10N$, where $N$ is the population size.
- **Normal/Large Counts**: The sampling distribution of $\hat{p}$ under the null hypothesis is approximately normal. This requires $np_0 \geq 10$ and $n(1-p_0) \geq 10$ — we use $p_0$, not $\hat{p}$, here because we assume $H_0$ is true.
Exam tip: Never skip listing all three conditions on an FRQ—AP readers require you to explicitly check each to earn full credit for the test, even if one seems obvious.
3. Calculating the Test Statistic and P-Value ★★★☆☆ ⏱ 4 min
Once hypotheses are stated and conditions are confirmed, the next step is to calculate how far your observed sample result is from the hypothesized value, in units of standard deviation of the sampling distribution under $H_0$. This standardized value is called the z-test statistic.
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
Intuition: The numerator measures the gap between what we observed in our sample ($\hat{p}$) and what we expect to see if the null hypothesis is true ($p_0$). The denominator is the standard deviation of the sampling distribution of $\hat{p}$ when $H_0$ is true, so it tells us how much natural sampling variation we expect around $p_0$. We use $p_0$ (not $\hat{p}$) in the denominator because we calculate all probabilities under the assumption that $H_0$ is true.
After calculating $z$, we find the p-value: the probability of getting a sample result as extreme or more extreme than the one we observed, if $H_0$ is true. The p-value calculation depends on the alternative hypothesis:
- One-sided, $H_a: p < p_0$: p-value = $P(Z < z)$ (area left of $z$ in the standard normal distribution)
- One-sided, $H_a: p > p_0$: p-value = $P(Z > z)$ (area right of $z$)
- Two-sided, $H_a: p \neq p_0$: p-value = $2P(Z > |z|)$ (double the tail area, since extreme results can be in either direction)
Exam tip: On the AP exam, you do not need to show manual p-value calculation from a z-table if you correctly report the p-value from a calculator—just make sure you show the z calculation to earn the process point.
4. Making a Decision and Writing a Contextual Conclusion ★★★☆☆ ⏱ 3 min
The final step of a hypothesis test is comparing your p-value to the pre-specified significance level $\alpha$ (alpha), making a decision, and writing a conclusion that answers the original question in context. The significance level $\alpha$ is the threshold for how unlikely our result needs to be to reject $H_0$; if no $\alpha$ is given, the standard default is $\alpha = 0.05$.
- If p-value $< \alpha$: Reject $H_0$. We have convincing statistical evidence for the alternative hypothesis.
- If p-value $\geq \alpha$: Fail to reject $H_0$. We do not have convincing statistical evidence for the alternative hypothesis.
Exam tip: Always tie your conclusion back to the original problem's context—a conclusion that only says "reject H0" will not earn the conclusion point on an FRQ.
Common Pitfalls
Why: Students confuse hypothesis tests with confidence intervals, which use $\hat{p}$ to check normality.
Why: Students mix up the known sample statistic with the unknown population parameter we are actually testing.
Why: Students interpret failing to reject as confirmation of the null, when we can only rule out the null, not confirm it.
Why: Students again confuse hypothesis tests with confidence intervals, which use $\hat{p}$ for the standard error.
Why: Students often only remember the Large Counts condition and skip the independence check.
Why: Students forget that extreme results can occur in either tail for two-sided tests.