AP Statistics / Inference for Categorical Data: Proportions
Lesson
What a sample proportion can do
Sampling distributions turn one p-hat into a model for many possible samples.
Learning goals
- Describe the center, shape, and spread of a sampling distribution for a proportion.
- Explain why conditions matter before using a normal model.
Explanation
Imagine drawing many random samples of the same size and recording each sample proportion. That collection of p-hat values is a sampling distribution. Its center sits at the true population proportion if the sample is random.
When the sample is large enough that both successes and failures are plentiful, the sampling distribution is approximately normal. The standard error shrinks as n grows. A bigger honest sample is quieter, not magically biased toward your favorite answer.
Key terms
- Sampling distribution. The distribution of a statistic across many random samples of the same size.
- Standard error. An estimate of the standard deviation of a statistic, used in intervals and tests.
Common mistakes
- Treating the sampling distribution as a histogram of the original data.
- Using a normal model when np or n(1 − p) is tiny.
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
Why does doubling the sample size not cut the standard error of p-hat in half?
Answer. Standard error uses a square root of n, so doubling n multiplies the error by 1/√2, about 0.71.
Variability of a mean or proportion shrinks with √n, not with n itself.
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