Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for AP statistics inference for proportions? Then this AP Statistics lesson on Inference for Proportions: Intervals and Tests explains each idea step by step, with worked examples.
Quick answer: When a categorical result is summarized by p̂, two tools answer questions about the true share p. An interval of plausible values is p̂ ± z*·SE. A significance test checks a claimed value p0 with a z statistic and a p-value, while Type I errors, Type II errors and power measure the risk of a wrong decision.
Standards: AP Statistics Unit 3 (2026-27) · Level: Grade 12 AP Statistics
What are the steps of inference for a proportion?
- Random: the data were produced by random sampling or a randomized experiment.
- 10%: the sample is less than 10% of the population when sampling without replacement.
- Large counts: for an interval, at least 10 successes and 10 failures in the sample; for a test, at least 10 expected successes and 10 expected failures under H0.
How do you build and read a confidence interval?
p̂ ± z* √(p̂(1 − p̂)n)
Critical values: z* = 1.645 (90%), 1.960 (95%), 2.576 (99%).
In a random sample of 640 teens, 211 own an e-scooter. Then p̂ = 0.3297, the standard error is 0.01858, and the margin of error is 1.96(0.01858) = 0.0364. The interval is (0.293, 0.366). We are 95% confident that the true proportion of teens who own an e-scooter is in this interval. The 95% describes the method: about 95% of intervals made this way capture the parameter.
The margin of error is z* × SE, so the sample size needed for a margin of error ME is n = (z*/ME)2 p̂(1 − p̂), using p̂ = 0.5 if no estimate is available. A difference of two proportions uses p̂1 − p̂2 ± z* √(p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2); if 0 is outside the interval, the data support a difference.
How does a significance test work?
- State H0: p = p0 and Ha (<, > or ≠), and a significance level α.
- Verify random sampling, the 10% rule and large counts, with p0 in the counts.
- Calculate z = (p̂ − p0)/√(p0(1 − p0)/n) and the p-value.
- If the p-value ≤ α, reject H0; otherwise fail to reject, and conclude in context.
An airline claims 80% of flights are on time. In 200 random flights, 148 (74%) were on time. For Ha: p < 0.80, z = (0.74 − 0.80)/0.0283 = −2.12 and the p-value is 0.0169. Since 0.0169 < 0.05, there is convincing evidence that fewer than 80% of flights are on time.
The p-value tells how likely it is, when H0 is true, to see a sample result as far from p0 as the one observed, or farther. It is not the probability that H0 is true.
What are Type I and Type II errors, and what is power?
| H0 is true | H0 is false | |
|---|---|---|
| Reject H0 | Type I error (probability α) | Correct decision (power = 1 − β) |
| Fail to reject H0 | Correct decision | Type II error (probability β) |
Power is the probability of rejecting H0 when a specific alternative is true. It increases with a larger sample, a larger α, or a true value farther from p0. For H0: p = 0.50, n = 400, α = 0.05 and true p = 0.56, the power is 0.777.
Never write that you accept H0. A large p-value only means the data do not provide convincing evidence against it.
- Check random, 10% and large counts before using z procedures.
- Interval: p̂ ± z*√(p̂(1 − p̂)/n); critical values 1.645, 1.960, 2.576.
- Sample size for margin of error ME: n = (z*/ME)² p̂(1 − p̂), rounded up.
- p-value: probability of a result at least as extreme when H0 is true.
- Type I error probability is α; power = 1 − β.
Inference for Proportions: Intervals and Tests: frequently asked questions
What does a 95% confidence level mean?
If we repeated the sampling many times and built an interval each time with the same method, about 95% of the intervals would capture the true parameter. It does not mean a 95% chance that one fixed interval contains the parameter.
How do I choose the sample size for a given margin of error?
Solve ME = z*√(p̂(1 − p̂)/n) for n, giving n = (z*/ME)² p̂(1 − p̂). Without a prior estimate use p̂ = 0.5, which gives the largest value, and always round n up to the next whole number.
What is the p-value?
It is the probability, computed assuming H0 is true, of getting a sample statistic at least as extreme as the one observed in the direction of Ha. A small p-value is evidence against H0, but it is not the probability that H0 is true.
How are Type I and Type II errors different?
A Type I error rejects a true H0, with probability α. A Type II error fails to reject a false H0, with probability β. Lowering α makes Type I errors rarer but makes Type II errors more likely unless the sample size increases.
Can a confidence interval be used for a two-sided test?
Yes. At the significance level α = 1 − confidence level, if the hypothesized value is outside the interval, reject H0 for a two-sided test. If it is inside, there is not convincing evidence against H0.