Inference for Categorical Data: Proportions

What Unit 3 covers

Unit 3 is where AP Statistics turns from describing data and computing probabilities to statistical inference: using a sample to draw conclusions about a population. The categorical data here are counts of successes and failures, summarized as proportions. You estimate a population proportion $p$ (the true fraction of a population with some trait) using a sample proportion $\hat{p}$, read "p-hat."

Unit 3 is 15-25% of the multiple-choice section and about 30 class periods, which makes it the longest of the five units. It has 15 topics that build from inference for one proportion up to the difference between two proportions and then chi-square tests for two-way tables.

The official College Board framework frames the unit around questions like how to measure changing opinions over time and how to decide whether a difference between two groups is statistically significant.

Estimators and the sampling distribution of p-hat (topics 3.1 to 3.2)

3.1 Estimators. A sample statistic is a point estimator of the matching population parameter. An estimator is unbiased if, on average, it neither underestimates nor overestimates the parameter, and $\hat{p}$ is an unbiased point estimator of $p$.

3.2 Sampling distributions for sample proportions. When the sampled values are independent, the sampling distribution of $\hat{p}$ has mean $\mu_{\hat{p}} = p$ (the Greek letter mu, $\mu$, denotes a mean) and standard deviation $\sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size. That distribution is approximately normal when $np \ge 10$ and $n(1-p) \ge 10$. Sampling without replacement also needs a random sample and the 10% condition, meaning the population is at least 10 times the sample size.

Confidence intervals for one proportion (topics 3.3 to 3.4)

3.3 Constructing a confidence interval. The procedure is a one-sample $z$-interval for a population proportion, built as point estimate $\pm$ (critical value)(standard error), or $\hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. The standard error (SE) estimates the standard deviation of the sampling distribution, and $z^{*}$ is the critical value that encloses the middle $C\%$ of the standard normal curve. The three conditions are randomization, the 10% condition, and a normality condition that checks the observed successes and observed failures are each at least 10.

3.4 Justifying a claim. In repeated random sampling with the same $n$, about $C\%$ of intervals built this way capture the true $p$. You interpret a $C\%$ interval as "we are $C\%$ confident the interval from $a$ to $b$ captures the population proportion" in context. Raising the confidence level widens the interval, while raising the sample size narrows it, with the width roughly proportional to $\frac{1}{\sqrt{n}}$. The how to calculate a confidence interval guide walks through the arithmetic.

Significance tests for one proportion (topics 3.5 to 3.8)

3.5 Setting up the test. The procedure is a one-sample $z$-test for a population proportion. The null hypothesis $H_0: p = p_0$ states the status-quo value $p_0$, and the alternative $H_a$ is one-sided ($<$ or $>$) or two-sided ($\ne$). Here the normality condition uses the hypothesized value: $np_0 \ge 10$ and $n(1-p_0) \ge 10$.

3.6 p-values. The p-value is the probability, computed assuming $H_0$ is true, of getting a test statistic as extreme or more extreme than the one observed, in the direction of $H_a$. Small p-values mean the observed result would be unusual under $H_0$ and give evidence for $H_a$; a large p-value does not prove $H_0$ is true. See what does a p-value mean.

3.7 Carrying out the test. The test statistic is a standardized statistic of the form (statistic minus parameter) over standard error:

Compare the p-value to the significance level $\alpha$: if the p-value is below $\alpha$, reject $H_0$; otherwise fail to reject. Rejecting gives convincing evidence for $H_a$, but failing to reject never proves $H_0$.

3.8 Potential errors. A Type I error rejects a true $H_0$, and its probability equals $\alpha$. A Type II error fails to reject a false $H_0$, with probability $1 -$ power, where power is the chance of correctly rejecting a false $H_0$. Power increases when the sample size grows, the standard error shrinks, the true value sits farther from $H_0$, or $\alpha$ increases. See Type I vs Type II errors.

Inference for two proportions (topics 3.9 to 3.13)

3.9 Sampling distribution of the difference. For two independent samples, $\hat{p}_1 - \hat{p}_2$ has mean $p_1 - p_2$ and standard deviation $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$. It is approximately normal when the expected successes and expected failures in both samples are each at least 10.

3.10 and 3.11 Two-sample z-interval. The interval estimates $p_1 - p_2$ as $(\hat{p}_1 - \hat{p}_2) \pm z^{*}\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$. If the interval contains 0, there is insufficient evidence of a difference; if it excludes 0, there is evidence of a difference between the two population proportions.

3.12 and 3.13 Two-sample z-test. The null hypothesis is $H_0: p_1 = p_2$. Because $H_0$ assumes the proportions are equal, the test pools the two samples into a combined proportion $\hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$ and uses it in the standard error:

To choose between the one-proportion and two-proportion procedures, read one-proportion vs two-proportion z-test and practice with the proportion z-test calculator.

Chi-square tests for two-way tables (topics 3.14 to 3.15)

3.14 Setting up the test. The chi-square statistic (the Greek letter chi, written $\chi^2$) measures the distance between observed and expected counts relative to expected counts. Use a chi-square test for homogeneity to compare the distribution of one categorical variable across two or more populations or treatments, and a chi-square test for independence to check whether two categorical variables are associated within a single sampled population. The conditions are randomization, the 10% condition, and an expected-counts condition that every expected count is at least 5.

3.15 Carrying out the test. Each expected count is $\frac{\text{row total} \times \text{column total}}{\text{table total}}$, and the statistic sums over every cell of the table:

The degrees of freedom equal $(\text{rows} - 1)(\text{columns} - 1)$, and a larger $\chi^2$ gives a smaller p-value. Both tests share this arithmetic; the sampling design and the wording of the hypotheses are what differ. See chi-square tests explained and the chi-square calculator.

Conditions and how to study for the exam

Every procedure in Unit 3 begins by naming the method and verifying its conditions in context. The AP framework warns that students lose points for checking conditions superficially, such as writing "SRS" instead of stating that a simple random sample was taken. State each condition using the actual numbers from the problem.

Language matters as much as the arithmetic. Interpret a confidence interval as capturing a parameter, not a statistic, and never say a test "accepts" or "proves" $H_0$, because a test can only reject or fail to reject it. When a chi-square p-value is large, say the data do not give convincing evidence of an association rather than claiming there is none.

One change from the older 9-unit course: the chi-square goodness-of-fit test was removed from the AP exam, while the tests for homogeneity and independence stay in Unit 3. Goodness-of-fit is still standard in most college introductory statistics courses.