AP Statistics, AP exams

Unit 3: Inference for Categorical Data: Proportions

Unit 3 introduces statistical inference using proportions. Confidence intervals and significance tests are developed for one proportion and for the difference between two proportions, with the conditions that must be checked before either is valid. Chi-square procedures for categorical relationships appear here in the revised framework.

Position: 3 of 5 units
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Statistics. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Confidence intervals for a proportion: Constructing and interpreting an interval, and the meaning of the confidence level itself.
Significance tests for a proportion: Hypotheses, test statistic, p-value, and stating a conclusion in the context of the original question.
Conditions for inference: Randomness, independence, and the large-counts condition, and what happens when one fails.
Comparing two proportions: Intervals and tests for a difference in proportions, including the pooled approach for testing.
Errors and power: Type I and Type II errors in context, and the factors that increase the power of a test.
Chi-square procedures for categorical data: Testing for independence or homogeneity in a two-way table, including expected counts.

Study tips

How to practice Inference: Proportions.

Section-specific guidance that follows from how this material is usually tested.

Learn one inference template (state, plan, do, conclude) and apply it identically to every procedure.
Practice interpreting a confidence interval and a confidence level as two different statements, since both are asked and confusing them is common.
Write conclusions that reference the context and the significance level, not just the word significant.

Where this fits

Unit 3: Inference for Categorical Data: Proportions is 3 of 5 units in the AP Statistics guide. Use the full guide to balance it against everything else you have to review.

Back to the AP Statistics guide Official exam information

Sample Inference: Proportions questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
A 95% confidence interval for the proportion of voters who support a measure is 0.52 to 0.58. What is the correct interpretation of this interval?
1. AWe are 95% confident that the interval from 0.52 to 0.58 captures the population proportion
2. B95% of voters support the measure
3. CThere is a 95% probability that the sample proportion lies between 0.52 and 0.58
4. D95% of all samples produce a sample proportion between 0.52 and 0.58
Show answer
Answer: A. We are 95% confident that the interval from 0.52 to 0.58 captures the population proportion
A confidence interval is a statement about the unknown population parameter, expressed through the long-run success rate of the method. The sample proportion is already known to be 0.55, the interval's midpoint, so any interpretation about where the sample proportion lies misses the point of inference.
Key insight: Confidence intervals estimate the parameter; the confidence level describes the method, not one interval.
Question 2Foundational
Which condition must be checked before using a normal model for a one-sample z-interval for a population proportion?
1. ABoth n times p-hat and n times (1 - p-hat) are at least 10
2. BThe population itself is normally distributed
3. CThe sample size is at least 100
4. DThe population standard deviation is known
Show answer
Answer: A. Both n times p-hat and n times (1 - p-hat) are at least 10
The large-counts condition requires at least about 10 expected successes and 10 expected failures so the sampling distribution of p-hat is approximately normal. The shape of the population itself does not matter for a proportion, and no fixed sample size such as 100 is required.
Key insight: For proportions, check random selection, independence, and large counts before any normal calculation.
Question 3Intermediate
A researcher rejects the null hypothesis in a significance test, but the null hypothesis is in fact true. What kind of error has occurred?
1. AA Type I error
2. BA Type II error
3. CA sampling error caused by a small sample
4. DNo error, because the data supported the decision
Show answer
Answer: A. A Type I error
Rejecting a true null hypothesis is a Type I error, and its long-run probability equals the significance level α. A Type II error is the opposite mistake, failing to reject a null hypothesis that is actually false.
Key insight: Type I rejects a true null hypothesis; Type II fails to reject a false one.
Question 4Intermediate
All else being equal, how does increasing the sample size from 400 to 1600 change the margin of error of a confidence interval for a proportion?
1. AIt is cut in half
2. BIt is cut to one fourth
3. CIt doubles
4. DIt does not change
Show answer
Answer: A. It is cut in half
The margin of error is proportional to 1/√n, and √1600/√400 = 40/20 = 2, so quadrupling the sample size halves the margin of error. Expecting one fourth treats the margin of error as proportional to 1/n instead of 1/√n.
Key insight: To halve a margin of error you must quadruple the sample size.
Question 5Advanced
In a chi-square test on a two-way table, one cell lies in a row with total 40 and a column with total 30, and the table's grand total is 120. What is the expected count for that cell?
1. A10
2. B12
3. C35
4. D70
Show answer
Answer: A. 10
The expected count is (row total times column total)/grand total = (40 times 30)/120 = 10. Averaging the two totals gives 35, and adding them gives 70; neither uses the independence model the chi-square statistic is built on.
Key insight: Expected counts come from assuming independence: row total times column total, divided by the grand total.

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Unit 3: Inference for Categorical Data: Proportions