Statistics · Inference for Categorical Data: Proportions · 14 min read · Updated 2026-05-11
Confidence Intervals for the Difference in Two Proportions — AP Statistics
AP Statistics · Inference for Categorical Data: Proportions · 14 min read
1. Core Concepts and Conditions for Inference★★☆☆☆⏱ 4 min
A confidence interval for $p_1 - p_2$ gives a range of plausible values for the true difference between the proportion of successes in two separate independent populations. This method is used to compare proportions from two distinct groups, such as pass rates between two prep courses or defect rates between two manufacturing lines. If 0 is not in the interval, we have evidence of a true difference at the corresponding significance level.
**Random**: Both samples must be independently drawn random samples from their populations, or from a randomized experiment.
**Independent Groups**: The two samples must be independent of each other, with no pairing or matching of observations.
**10% Condition**: When sampling without replacement, each sample size must be less than 10% of its population to ensure independence within samples.
**Large Counts**: Each sample must have at least 10 observed successes and 10 observed failures to ensure the sampling distribution is approximately normal: $n_1\hat{p}_1 \geq 10$, $n_1(1-\hat{p}_1) \geq 10$, $n_2\hat{p}_2 \geq 10$, $n_2(1-\hat{p}_2) \geq 10$.
Exam tip: On AP FRQs, you must explicitly name and verify every condition, not just say 'conditions are met'. You will lose an entire point if you do not show counts for the Large Counts condition.
2. Constructing the Confidence Interval★★★☆☆⏱ 4 min
Once conditions are verified, we always use the unpooled standard error for confidence intervals for the difference in two proportions (per AP CED requirements). Pooling is only used for hypothesis tests for two proportions, when we assume the null hypothesis $p_1 = p_2$ is true; for confidence intervals we make no such assumption, so we use individual sample proportions.
Key notation: $p_1$ = true proportion of successes for population 1, $p_2$ = true proportion for population 2, $\hat{p}_1 = x_1/n_1$, $\hat{p}_2 = x_2/n_2$, where $x_1, x_2$ are the number of observed successes and $n_1, n_2$ are sample sizes.
Where $\hat{p}_1 - \hat{p}_2$ is the point estimate of the true difference, $z^*$ is the critical z-value for your confidence level (common values: 90% = 1.645, 95% = 1.96, 99% = 2.576), and the term under the square root is the variance of the difference: for independent variables, the variance of a difference equals the sum of variances, so we add the two variance terms.
Exam tip: Always explicitly label which population is 1 and which is 2 at the start. This prevents sign errors that lead to wrong inference conclusions.
3. Interpretation and Inference Conclusions★★★☆☆⏱ 3 min
Interpretation is one of the most frequently tested skills on the AP exam for this topic. A correct interpretation requires context and correct phrasing: the true difference is a fixed value, so it is either in the interval or not; confidence refers to the long-run performance of the method, not the probability that the true value is in the interval.
For inference, use this simple rule: If 0 is not inside the confidence interval, we have convincing evidence at the $(100-C)$% significance level that the two population proportions differ. If 0 is inside the interval, we do not have convincing evidence of a difference. We can never conclude that the proportions are equal, because the interval contains many non-zero plausible values.
Exam tip: AP graders require full context for interpretation points. Generic interpretations without naming the populations and parameter will not earn full credit.
4. AP-Style Concept Check★★★☆☆⏱ 3 min
Common Pitfalls
Why: Confusion between confidence interval rules and hypothesis test rules, where pooling is sometimes used.
Why: Carrying over the pooled value from the standard error mistake to the condition check.
Why: Mixing up the probability of the method working with the probability of the fixed parameter being in the interval.
Why: Matching the subtraction in the point estimate to the variance calculation.
Why: Confusing 'no evidence of a difference' with 'evidence of no difference'.
Why: Automatically using two-sample method whenever two proportions are compared, even for dependent samples.