6.11 Carrying Out a Test for the Difference of Two Population Proportions

A two-sample z test for a difference of proportions checks whether two population proportions are really different or whether the gap you see could just be random chance. You calculate a pooled proportion, find the z test statistic, get a p-value, then compare that p-value to your significance level to decide whether to reject the null hypothesis.

Why This Matters for the AP Statistics Exam

This is the payoff step of the two-proportion test. Earlier topics had you set up hypotheses and check conditions; here you actually run the numbers and state a conclusion. On multiple-choice questions, you might compute or interpret a test statistic, identify the pooled proportion, or match a p-value to a correct decision. On free-response, a full inference question often expects you to show the test statistic, report the p-value, compare it to alpha, and justify a claim about the two populations in context. Getting comfortable with this process also sets you up for two-sample mean tests and chi-square tests later, which follow the same logic.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculatorkey terms

Key Takeaways

• The test statistic uses a pooled (combined) proportion: $\hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$.
• The two-sample z-statistic is $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$.
• The p-value is calculated assuming the null hypothesis is true, meaning the two population proportions are equal.
• Decision rule: if p-value $\le \alpha$, reject $H_0$; if p-value $> \alpha$, fail to reject $H_0$.
• Always state your conclusion in context and connect it back to the research question about both populations.
• A calculator's "2-PropZTest" gives you both the z-statistic and the p-value, so you mostly need to interpret the output correctly.

Calculating the Test Statistic and p-value

You need two values to carry out this test: a z test statistic and a p-value. Both should appear in your written work on an inference free-response question.

The Pooled Proportion

Because the null hypothesis says the two population proportions are equal, you combine both samples into one estimate called the pooled proportion:

$\hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$

This is the same as adding up all the successes and dividing by the total of both sample sizes. You use $\hat{p}_c$ in the standard error because, under the null, both groups are assumed to share the same true proportion.

The z Test Statistic

The test statistic follows the general pattern of (statistic minus null value) divided by the standard deviation of the statistic:

$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

The formula looks heavy, but a graphing calculator's "2-PropZTest" reports the z-statistic for you. This formula is not printed on the AP formula sheet, but you can rebuild it from the general test statistic formula and the standard error pieces that are provided.

The p-value

The p-value uses the standard normal curve and your z-statistic to find the probability of getting a difference as extreme or more extreme than what you observed, assuming the null hypothesis is true. A calculator handles this quickly. Match the tail to your alternative hypothesis:

• For $H_a: p_1 > p_2$, use the area above your z-statistic.
• For $H_a: p_1 < p_2$, use the area below your z-statistic.
• For $H_a: p_1 \neq p_2$, use both tails.

Concluding the Test

Once you have the p-value, compare it to the significance level to decide on the null hypothesis.

Using the p-value

The significance level $\alpha$ is the probability of rejecting the null hypothesis when it is actually true, and it is often set at 0.05. Compare your p-value to $\alpha$:

• If the p-value $\le \alpha$, reject $H_0$. The difference is statistically significant and unlikely to be due to chance, so you have evidence supporting the alternative.
• If the p-value $> \alpha$, fail to reject $H_0$. The difference could reasonably be due to chance, so you do not have convincing evidence for the alternative.

A quick way to remember it: if the p is low, reject the $H_0$.

Remember that failing to reject is not the same as proving the null is true. It just means you do not have enough evidence to support the alternative.

Reading the z-statistic

If you only have the z-statistic, you can still judge how extreme the result is. On a normal curve, about 95% of values fall within two standard deviations of the mean and about 99.7% fall within three. A z-statistic far out in the tail (large in absolute value) points to a result that would be rare under the null, which supports rejecting it. The p-value is still the cleaner way to make and report the decision.

Worked Example: MJ vs. LeBron

Suppose Michael Jordan made 836 of 1623 shots and LeBron James made 622 of 1493 shots. You want to test whether the data give convincing statistical evidence that Jordan's true proportion of made shots is higher than LeBron's.

The hypotheses are $H_0: p_1 = p_2$ and $H_a: p_1 > p_2$, where group 1 is Jordan and group 2 is LeBron.

Running the Test

Rather than grinding through the formula by hand, use a graphing calculator's "2-PropZTest." Enter the successes and sample sizes for each player, then select the correct alternative ($p_1 > p_2$). The output gives you the z-statistic and the p-value.

The two values to record are the z-statistic and the p-value.

Stating the Conclusion

Use the p-value to write the conclusion:

In this example the z-statistic comes out around 5.5, which is far in the tail, and the p-value is essentially 0. Both point to the same decision: reject the null in favor of the alternative. This also matches what a confidence interval for the difference would tell you.

How to Use This on the AP Statistics Exam

Free Response

• Show the test statistic value and the p-value, not just a final yes/no answer.
• Explicitly compare the p-value to $\alpha$ before stating your decision.
• Write the conclusion in context, naming both populations and the direction of the alternative.
• Tie the result back to the original research question about the two groups.

MCQ

• Be ready to compute or identify the pooled proportion $\hat{p}_c$.
• Know that the p-value is found assuming the two population proportions are equal.
• Match a given p-value and $\alpha$ to the correct reject or fail-to-reject decision.
• Match the tail of the p-value to the alternative hypothesis.

Common Trap

Writing "we accept the null" or "we proved there is no difference" loses the point. Failing to reject only means there is not enough evidence for the alternative, never proof of the null.

Common Misconceptions

• Using separate sample proportions in the standard error instead of the pooled proportion. For the test, you assume the proportions are equal under the null, so you pool. The unpooled version belongs to the confidence interval, not the test.
• Thinking a small p-value proves a huge real-world difference. A small p-value means the result is statistically significant, but statistical significance is not the same as a large or practically important difference.
• Believing "fail to reject" proves the null hypothesis. It only means the evidence was not strong enough to support the alternative.
• Forgetting to state the conclusion in context. A bare "reject $H_0$" is incomplete; name the populations and what the difference means.
• Mixing up the tail for the p-value. A one-sided alternative uses one tail; a two-sided alternative ($p_1 \neq p_2$) uses both tails.
• Treating the significance level as fixed at 0.05. It is a common default, but the problem can set a different $\alpha$, and your decision depends on the one given.

Related AP Statistics Guides

• Unit 6 Overview: Inference for Categorical Data: Proportions
• 6.2 Constructing a Confidence Interval for a Population Proportion
• 6.1 Introducing Statistics: Why Be Normal?
• 6.4 Setting Up a Test for a Population Proportion
• 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion
• 6.5 Interpreting p-Values