AP Statistics / Inference for Categorical Data: Proportions
Lesson
Intervals and tests for a proportion
Estimate a proportion with a confidence interval, or test a claimed value.
Learning goals
- Construct and interpret a one-proportion confidence interval.
- State hypotheses, compute a p-value idea, and write a decision in context.
Explanation
A confidence interval starts with p-hat and reaches out by a critical value times the standard error. The confidence level is about the method: in the long run, that many intervals would capture the true proportion. One interval either does or does not.
A test asks whether a claimed proportion is still believable. You measure how many standard errors the sample sits from that claim. A small p-value means data this extreme would be rare if the claim were true. That is not the probability the claim is true.
Key terms
- Confidence interval. A range of plausible values for a parameter, built from a sample and a stated confidence level.
- P-value. The probability, under the null model, of a statistic at least as extreme as the one observed.
Common mistakes
- Saying a 95% interval contains 95% of individuals.
- Treating a p-value as the probability the null hypothesis is true.
Practice
Original Marlow Works items. Check the answer explanation after you try.
A simple random sample of 400 city voters finds that 0.47 support a transit tax. A 95% confidence interval is 0.47 ± 0.049. Explain what this interval means, state one thing it does not mean, and say whether a similar sample of size 100 would tend to produce a wider or narrower interval.
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
A test of H0: p = 0.50 yields p-value 0.21. What should you conclude?
Answer. Fail to reject the null. The sample is not unusual enough to discard p = 0.50.
A large p-value means the data are compatible with the claimed proportion, not that you proved it.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Statistics course page
Official revised 2026–27 five-unit framework.
College Board · All rights reserved · accessed 2026-10-01
Open sourceCC BY — attribution required
Collaborative Statistics
Open statistics text for distributions, probability, and inference.
Open Textbook Library listing · CC BY · accessed 2026-10-01
Open source