Review AP Statistics Unit 3 to build the full inference toolkit for categorical data, from sampling distributions and confidence intervals to hypothesis tests and chi-square procedures. This unit covers one-proportion and two-proportion inference plus chi-square tests for homogeneity and independence.
Use the topic guides, key terms, and FRQ practice available on Fiveable to work through every procedure from conditions to conclusions.
Unit 3 is the first full inference unit in AP Statistics. Every procedure here involves categorical data, meaning you are working with counts and proportions rather than means. The unit builds from the ground up: first you learn what makes a sample statistic a good estimator, then you describe sampling distributions for proportions, and then you use those distributions to construct intervals and run tests.
Unit 3 covers how to estimate and test claims about one or two population proportions using z-procedures, how to evaluate errors and power in hypothesis testing, and how to use chi-square tests to analyze two-way tables of categorical data.
Before any confidence interval or test, you need to know the shape, center, and spread of the sampling distribution of p-hat. The mean is p, the standard deviation is sqrt(p(1-p)/n), and the distribution is approximately normal when the Large Counts condition (np >= 10 and n(1-p) >= 10) is met. The same logic extends to the difference p-hat1 minus p-hat2 in Topics 3.9-3.13.
Every inference procedure in Unit 3 requires three conditions: randomization (random sample or randomized experiment), the 10% condition (n is at most 10% of N when sampling without replacement), and a normality condition (Large Counts). For the two-proportion z-test, the normality check uses the pooled proportion p-hat-c instead of separate sample proportions.
When you have a two-way table of categorical counts, z-procedures no longer apply. A chi-square test for homogeneity compares distributions of one variable across two or more populations. A chi-square test for independence checks whether two categorical variables are associated within one population. Both use the same statistic: chi-squared = sum of (Observed minus Expected) squared divided by Expected, with degrees of freedom equal to (rows minus 1)(columns minus 1).
Every confidence interval and hypothesis test in Unit 3 uses sample data to draw conclusions about a population parameter, either p, p1 minus p2, or a distribution of counts. A confidence interval gives a range of plausible values; a hypothesis test asks whether the data are consistent with a specific claim. Conclusions must always be stated in context, reference the population, and use non-definitive language because inference is probabilistic, not certain.
Introduces point estimators and the concept of bias. p-hat is an unbiased estimator of p because its sampling distribution is centered at p.
Describes the center, spread, and shape of the sampling distribution of p-hat, including the Large Counts and 10% conditions.
Covers the one-sample z-interval formula, conditions, standard error, margin of error, and sample size calculations.
Explains how to interpret a confidence interval in context and use it to support or refute a claim about a population proportion.
Covers writing H0 and Ha for a one-sample z-test, naming the correct procedure, and verifying conditions using p0.
Defines and interprets the p-value as the probability of a result as extreme or more extreme than observed, assuming H0 is true.
Calculates the z-test statistic using p0 in the denominator, finds the p-value, and states a conclusion in context.
Defines Type I and Type II errors, their probabilities in terms of alpha and power, and the factors that affect power.
Extends sampling distribution logic to p-hat1 minus p-hat2, including mean, standard deviation, and conditions for two independent samples.
Builds the two-sample z-interval using separate standard errors for each group and checks conditions for both samples.
Interprets the two-proportion interval in context and uses whether 0 is inside or outside the interval to justify a claim.
Writes H0: p1 = p2 and the appropriate Ha, names the two-sample z-test, and checks conditions using the pooled proportion.
Calculates the pooled proportion, the two-sample z-statistic, and the p-value, then states a conclusion in context.
Distinguishes homogeneity from independence, writes appropriate hypotheses, and verifies the expected counts condition.
Calculates expected counts, the chi-square statistic, degrees of freedom, and the p-value, then states a conclusion in context.
A point estimator uses a single sample statistic to estimate a population parameter. The sample proportion p-hat is the point estimator for the population proportion p. An estimator is unbiased if its sampling distribution is centered at the true parameter value, meaning it does not systematically overestimate or underestimate. Bias and variability are separate concerns: a large sample reduces variability but does not fix a biased estimator.
If a sampling method consistently produces p-hat values that are too high, the estimator is biased. Increasing sample size will not correct this.
| Property | Unbiased Estimator | Biased Estimator |
|---|---|---|
| Center of sampling distribution | Equal to true parameter | Shifted away from true parameter |
| Fixed by larger n? | Variability decreases; bias unchanged | No; bias persists regardless of n |
When you take a random sample of size n from a population with proportion p, the sample proportion p-hat varies from sample to sample. Its sampling distribution has mean mu-p-hat = p and standard deviation sigma-p-hat = sqrt(p(1-p)/n). The distribution is approximately normal when the Large Counts condition is satisfied. Two conditions support independence of observations: the randomization condition and the 10% condition.
Probabilities about p-hat are found using z = (p-hat minus p) divided by sqrt(p(1-p)/n), then using the standard normal distribution.
A one-sample z-interval for a population proportion estimates p using the formula p-hat plus or minus z-star times sqrt(p-hat(1-p-hat)/n). The standard error SE = sqrt(p-hat(1-p-hat)/n) replaces the true standard deviation because p is unknown. The margin of error equals z-star times SE. To find a required sample size, use n = (z-star)^2 times p-hat(1-p-hat) divided by (MOE)^2; if p-hat is unknown, substitute 0.5 for the most conservative estimate. Interpretation: we are C% confident the interval (a, b) contains the true population proportion. Increasing confidence level widens the interval; increasing sample size narrows it proportionally to 1/sqrt(n).
If a 95% confidence interval for p is (0.42, 0.58), you cannot say there is a 95% probability that p is in this specific interval. The interval either contains p or it does not; the 95% refers to the long-run capture rate of the procedure.
| Change | Effect on Margin of Error | Effect on Interval Width |
|---|---|---|
| Increase confidence level | Increases (larger z-star) | Wider |
| Increase sample size n | Decreases (larger denominator) | Narrower |
| Decrease sample size n | Increases | Wider |
A one-sample z-test for a population proportion tests H0: p = p0 against a one-sided or two-sided alternative. The test statistic is z = (p-hat minus p0) divided by sqrt(p0(1-p0)/n). Notice that the denominator uses p0, the null value, not p-hat, because you assume H0 is true when computing the statistic. The p-value is the probability of getting a z as extreme or more extreme than observed, assuming H0 is true. Compare the p-value to the significance level alpha: if p-value is less than or equal to alpha, reject H0 and conclude there is convincing evidence for Ha; otherwise, fail to reject H0. A test can never prove H0 true. The normality condition for the test uses np0 and n(1-p0), not p-hat.
A large p-value does not mean H0 is true. It means the data are consistent with H0, not that H0 has been proven.
Every hypothesis test risks two kinds of errors. A Type I error is rejecting H0 when it is actually true; its probability equals alpha. A Type II error is failing to reject H0 when Ha is actually true; its probability equals 1 minus power. Power is the probability of correctly rejecting a false H0. Power increases when sample size increases, when the true parameter is farther from the null value, when alpha increases, or when standard error decreases. Before a study, consider the real-world consequences of each error type to guide choices about alpha and sample size.
Decreasing alpha reduces the chance of a Type I error but increases the chance of a Type II error. The only way to reduce both simultaneously is to increase sample size.
| Decision | H0 True | H0 False |
|---|---|---|
| Reject H0 | Type I error (probability = alpha) | Correct decision (probability = power) |
| Fail to reject H0 | Correct decision | Type II error (probability = 1 minus power) |
When comparing two independent groups, the parameter of interest is p1 minus p2. The sampling distribution of p-hat1 minus p-hat2 has mean p1 minus p2 and standard deviation sqrt(p1(1-p1)/n1 + p2(1-p2)/n2). A two-sample z-interval estimates p1 minus p2 using (p-hat1 minus p-hat2) plus or minus z-star times sqrt(p-hat1(1-p-hat1)/n1 + p-hat2(1-p-hat2)/n2). Conditions mirror the one-proportion case but apply to both samples. Key interpretation: if the interval contains 0, there is insufficient evidence of a difference between the two population proportions; if the interval is entirely above or below 0, the data support a directional claim.
Keep the subtraction order consistent throughout: define which group is group 1 at the start and use that order in the parameter statement, the interval, and the conclusion.
A two-sample z-test for a difference of proportions tests H0: p1 = p2 (equivalently, p1 minus p2 = 0). Because H0 assumes the proportions are equal, the normality check uses the pooled proportion p-hat-c = (n1p-hat1 + n2p-hat2) / (n1 + n2) rather than separate sample proportions. The test statistic is z = (p-hat1 minus p-hat2) divided by sqrt(p-hat-c(1-p-hat-c)(1/n1 + 1/n2)). The p-value comes from the standard normal distribution. Conclusions follow the same reject or fail-to-reject logic as the one-proportion test, stated in context with non-definitive language.
Use p-hat-c only in the two-proportion z-test, not in the two-proportion confidence interval. The interval uses separate p-hat1 and p-hat2 in the standard error.
| Feature | Two-Proportion z-Interval | Two-Proportion z-Test |
|---|---|---|
| Goal | Estimate p1 minus p2 | Test whether p1 = p2 |
| Standard error uses | Separate p-hat1 and p-hat2 | Pooled proportion p-hat-c |
| Normality check uses | n1*p-hat1, n1*(1-p-hat1), n2*p-hat2, n2*(1-p-hat2) | n1*p-hat-c, n1*(1-p-hat-c), n2*p-hat-c, n2*(1-p-hat-c) |
Chi-square procedures handle two-way tables of categorical counts. A chi-square test for homogeneity compares the distribution of one categorical variable across two or more populations or treatment groups. A chi-square test for independence checks whether two categorical variables are associated within a single population. Both use the same mechanics: compute expected counts as (row total times column total) divided by table total, then calculate chi-squared = sum of (Observed minus Expected) squared divided by Expected. Degrees of freedom = (rows minus 1)(columns minus 1). The p-value comes from a chi-square distribution, which is right-skewed and has only positive values. The expected counts condition requires all expected counts to be greater than 5.
A large chi-square statistic means observed counts are far from expected counts, producing a small p-value. A small chi-square statistic means observed and expected counts are close, consistent with H0.
| Feature | Chi-Square Test for Homogeneity | Chi-Square Test for Independence |
|---|---|---|
| Research question | Do distributions of one variable differ across populations? | Are two categorical variables associated in one population? |
| Number of populations | Two or more | One |
| H0 | Distributions are the same across all populations | The two variables are independent (no association) |
| Ha | At least one distribution differs | The two variables are not independent (there is an association) |
| Term | Definition |
|---|---|
| randomness condition | Requires that data come from a random sample or randomized experiment before any inference procedure in Unit 3 is applied. |
| 10% Condition | When sampling without replacement, the sample size must be at most 10% of the population size to support approximate independence between observations. |
| success/failure condition | For proportion inference, the expected numbers of successes and failures must both be at least 10 (using p0 for tests, p-hat for intervals) for the normal approximation to apply. |
| Standard Error | Estimates the standard deviation of a sampling distribution using sample data; for p-hat, SE = sqrt(p-hat(1-p-hat)/n). |
| Critical Value | The z-star multiplier that determines the margin of error for a confidence interval; its value depends on the chosen confidence level. |
| Confidence Level | The long-run percentage of confidence intervals from repeated random samples that would capture the true population parameter. |
| One Proportion Z Test | Tests H0: p = p0 using z = (p-hat minus p0) divided by sqrt(p0(1-p0)/n), with the p-value found from the standard normal distribution. |
| Significance Level | The threshold alpha for rejecting H0; if the p-value is less than or equal to alpha, the result is statistically significant. |
| Statistically Significant | A result is statistically significant when its p-value is less than or equal to alpha, providing evidence against H0. |
| Power | The probability of correctly rejecting a false H0; power = 1 minus P(Type II error), and increases with larger sample size or larger effect size. |
| pooled proportion | p-hat-c = (n1*p-hat1 + n2*p-hat2) / (n1 + n2); used in the two-proportion z-test when H0 assumes p1 = p2. |
| Two Proportion Z Test | Tests H0: p1 = p2 using a pooled proportion in the standard error; z = (p-hat1 minus p-hat2) divided by sqrt(p-hat-c(1-p-hat-c)(1/n1 + 1/n2)). |
| Chi-Square Statistic | chi-squared = sum of (Observed minus Expected)^2 divided by Expected; measures how far observed counts in a two-way table are from what H0 predicts. |
| Independence Condition | Requires observations to be independent of one another; supported by the randomization condition and the 10% condition in proportion inference. |
| One-tailed test | A hypothesis test with a directional alternative hypothesis (Ha: p less than p0 or Ha: p greater than p0); the p-value is computed in one tail of the distribution. |
The test statistic is z = (p-hat minus p0) divided by sqrt(p0(1-p0)/n). You assume H0 is true when computing the statistic, so p0 goes in the denominator. Using p-hat there is a common error that produces the wrong z-value.
The two-proportion z-test assumes p1 = p2 under H0, so you must use the pooled proportion p-hat-c in both the normality check and the test statistic. The two-proportion confidence interval does not use p-hat-c; it uses separate p-hat1 and p-hat2.
The confidence level describes the long-run capture rate of the procedure across repeated samples, not the probability that this specific interval contains the parameter. Once an interval is computed, the parameter is either in it or not.
For a one-proportion confidence interval, use np-hat and n(1-p-hat). For a one-proportion z-test, use np0 and n(1-p0). For the two-proportion z-test, use the pooled proportion p-hat-c for both samples. Using the wrong value can lead to incorrectly claiming conditions are met.
The choice depends on the study design: multiple populations sampled separately means homogeneity; one population with two categorical variables recorded means independence. The hypotheses, parameter descriptions, and conclusions differ even though the calculation is identical.
AP Statistics free-response questions frequently ask you to carry out a complete inference procedure. For any confidence interval or hypothesis test in Unit 3, you are expected to name the procedure and parameter, verify all conditions with work shown, calculate the interval or test statistic and p-value, and write a conclusion in context using non-definitive language. Skipping or abbreviating any step typically costs points.
Multiple-choice and free-response items often describe a study and ask which inference procedure is appropriate. You need to distinguish one-proportion from two-proportion procedures, and both from chi-square tests. Key signals: one group with a yes/no variable means one-proportion; two independent groups with a yes/no variable means two-proportion; a two-way table of counts means chi-square, with the choice between homogeneity and independence depending on whether one or two populations were sampled.
A common task across Unit 3 is interpreting a p-value, a confidence interval, or a chi-square result and then using it to justify or refute a specific claim. For confidence intervals, this includes addressing whether a claimed value or zero falls inside the interval. For hypothesis tests, this includes comparing the p-value to alpha and stating what the result means for the populations involved, not just the samples.
For any confidence interval or test in Unit 3, check randomization, the 10% condition, and the Large Counts (normality) condition. For the two-proportion z-test, use the pooled proportion in the Large Counts check.
The one-proportion interval uses sqrt(p-hat(1-p-hat)/n). The two-proportion interval uses separate p-hat values. The two-proportion z-test uses the pooled proportion p-hat-c. Chi-square procedures use expected counts, not standard errors.
State whether you reject or fail to reject H0, reference the p-value and alpha, name the parameter and population, and use non-definitive language such as 'there is convincing evidence that...' Never say a test proves H0 true.
Say 'we are C% confident the interval (a, b) contains the true population proportion.' Do not say there is a C% probability the parameter is in this specific interval. For two-proportion intervals, address whether 0 is inside the interval.
P(Type I error) = alpha. P(Type II error) = 1 minus power. Increasing sample size increases power without changing alpha. Lowering alpha increases the risk of a Type II error.
Homogeneity: one categorical variable, two or more populations. Independence: two categorical variables, one population. Both use the same chi-square statistic and expected count formula.
Expected count = (row total times column total) divided by table total. Degrees of freedom = (rows minus 1)(columns minus 1). All expected counts must exceed 5.
Read the topic guides for 3.1 and 3.2 to solidify what p-hat estimates and how its sampling distribution behaves. Then preview 3.9 to see how the same logic extends to p-hat1 minus p-hat2. Practice computing the mean and standard deviation of each sampling distribution and checking conditions.
Use the topic guides for 3.3 through 3.7 in order. For each procedure, practice the full four-step structure: state the parameter and procedure, verify conditions, calculate the interval or test statistic and p-value, and write a conclusion in context. Pay close attention to which value goes in the standard error denominator.
Review the topic guide for 3.8 and practice identifying Type I and Type II errors in context. Given a scenario, be able to describe what each error would mean in plain language and explain which factors would increase the power of the test.
Work through the topic guides for 3.10 through 3.13. Focus on the key difference between the interval (separate standard errors) and the test (pooled proportion). Practice writing hypotheses, checking conditions with p-hat-c, computing the test statistic, and interpreting whether 0 is inside a confidence interval.
Use the topic guides for 3.14 and 3.15. Practice choosing between homogeneity and independence based on the study design, computing expected counts from a two-way table, calculating the chi-square statistic and degrees of freedom, and writing a conclusion. Use the AP Statistics score calculator to estimate how your overall performance maps to an AP score.
Open the individual guides for Unit 3 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Watch past review streams filtered to Unit 3 when you want a video walkthrough.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
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get printablesAP Stats Unit 3, Inference for Categorical Data: Proportions, contains 15 topics covering estimators; sampling distributions for one and two sample proportions; confidence intervals; one- and two-proportion hypothesis tests; p-values; Type I and Type II errors; and chi-square tests for homogeneity and independence. These are the Fall 2026 CED topics used for the May 2027 exam.
Unit 3 accounts for 15–25% of the multiple-choice section. That range is a section weight, not a promise that every practice form will use the same percentage, so review the full unit rather than trying to predict one exact question count.
By the end of Unit 3, you should be able to identify the parameter and procedure, state hypotheses, verify randomization, independence, and large-count conditions, calculate an interval or test result, and write a conclusion about the population in context. The exam rewards the method and the interpretation, so practice writing what a result means in the problem's context instead of stopping at a calculator output.
Common Unit 3 mistakes include using sample statistics in hypotheses, saying the p-value is the probability that the null hypothesis is true, skipping conditions, or using definitive language such as “proves” in a conclusion. Slow down long enough to identify the variables, population, and requested conclusion before calculating.
Start with the vocabulary and conditions, then mix short calculations with full-sentence interpretations. For each missed question, label whether the problem was choosing a method, doing the calculation, or interpreting the result. Finish with timed mixed practice so you have to recognize the method without a topic label.
The unit applies Unit 2 probability to categorical outcomes and establishes the inference workflow that Unit 4 repeats for quantitative means.