Inference for a Difference in Two Proportions — AP Statistics
1. Core Concepts and Sampling Distribution ★★☆☆☆ ⏱ 3 min
Inference for a difference in two proportions is a set of statistical methods used to compare the proportion of successes (a binary outcome) between two independent populations or two experimental treatment groups. The core goal is to use data from two independent samples to make claims about the true difference between the two population proportions.
The sampling distribution of $\hat{p}_1 - \hat{p}_2$ has the following key properties:
\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2
\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}
When conditions are met, the sampling distribution is approximately Normal, which allows us to use z-based inference for the difference.
2. Conditions for Valid Inference ★★☆☆☆ ⏱ 4 min
All valid inference requires four core conditions, with a variation for the normality check based on the type of inference:
- **Random**: Both groups come from independent random samples or a randomized controlled experiment, ensuring unbiased estimates.
- **10% Condition**: When sampling without replacement from a finite population, each sample size must be no more than 10% of its population to ensure within-group independence. Not required for randomized experiments.
- **Independent Groups**: The two groups are independent of each other (no matching or paired data).
- **Large Counts (Normality)**: For confidence intervals: all four observed counts $n_1\hat{p}_1, n_1(1-\hat{p}_1), n_2\hat{p}_2, n_2(1-\hat{p}_2) \geq 10$. For $H_0: p_1=p_2$ hypothesis tests: pooled counts $(n_1 + n_2)\hat{p}_{pooled}$ and $(n_1 + n_2)(1-\hat{p}_{pooled}) \geq 10$.
Exam tip: Always write out each condition explicitly when asked; AP graders award a point for each correctly checked condition, never skip any.
3. Confidence Intervals for $p_1 - p_2$ ★★★☆☆ ⏱ 4 min
A confidence interval for a difference in two proportions gives a range of plausible values for the true difference $p_1 - p_2$. We never pool sample proportions for confidence intervals, because we do not assume $p_1 = p_2$ when estimating the difference. We always use unpooled standard error.
(\hat{p}_1 - \hat{p}_2) \pm z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}
Where $z^*$ is the critical value for your desired confidence level (e.g., $z^* = 1.96$ for 95% confidence, $z^* = 1.645$ for 90% confidence). If 0 is inside the interval, 0 is a plausible value for the true difference, meaning there is no statistically significant evidence of a difference at significance level $\alpha = 1 - \text{confidence level}$.
Exam tip: If you swap the order of $p_1$ and $p_2$, the interval bounds flip sign but the conclusion about whether 0 is in the interval stays the same. Just ensure your interpretation matches your order of difference.
4. Hypothesis Tests for Difference in Proportions ★★★☆☆ ⏱ 5 min
We use hypothesis tests to test a claim about whether two population proportions differ. The most common null hypothesis is $H_0: p_1 - p_2 = 0$ (no difference between proportions). Because the null assumes $p_1 = p_2$, we pool the two samples to get a single estimate of the common population proportion, which gives a more accurate standard error for the test.
The pooled standard error and z-test statistic are:
SE_{pooled} = \sqrt{\hat{p}_{pooled}(1-\hat{p}_{pooled})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}
z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{SE_{pooled}}
Exam tip: Only pool when your null hypothesis is $p_1 - p_2 = 0$. Non-zero null differences (extremely rare on AP) require unpooled standard error.
Common Pitfalls
Why: Students confuse pooling rules for hypothesis tests and confidence intervals, incorrectly assuming pooling is always required
Why: Students forget that we use observed counts when population proportions are unknown
Why: Students automatically use two-sample methods for two proportions regardless of study design
Why: Students confuse the definition of confidence level with probability for a fixed interval
Why: Students memorize the 10% condition as required for all inference and forget it only applies to sampling without replacement from finite populations
Why: Students skip this step to save time, leading to unclear reasoning and lost points