Unit 6: Inference for Categorical Data: Proportions
AP Statistics: 51 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
Confidence intervals estimate a population proportion; hypothesis tests weigh evidence against a claimed value. Both rely on the normal approximation to p-hat.
Confidence intervals
point estimate ± margin of error, where margin = z* × √(p-hat(1-p-hat)/n). z* = 1.645 (90%), 1.96 (95%), 2.576 (99%). Bigger n narrows the interval; higher confidence widens it. Interpretation: the METHOD captures the parameter C% of the time — the parameter is fixed, not random.
Hypothesis tests
H0 states the assumed parameter value; Ha is the claim we seek evidence for. Test statistic z = (p-hat - p0) / √(p0(1-p0)/n) — note tests use p0, intervals use p-hat. The p-value is the probability of results this extreme ASSUMING H0 is true. p < α → reject H0; p > α → fail to reject. Never "accept" H0.
Errors and power
Type I: reject a true H0 (rate α). Type II: fail to reject a false H0 (rate β). Power = 1 - β: probability of rejecting a false H0. Larger n, larger α, or a bigger true effect raise power. A small p-value means strong evidence, not necessarily practical importance.
Two proportions
Compare p1 - p2 with z procedures. Tests POOL the proportions (H0 says they're equal); intervals do NOT pool. Conditions: random samples, independence, large counts in both groups.
Exam traps
Intervals use p-hat; tests use p0. Confidence describes the method, not one interval's probability of containing p. Failing to reject is not proving H0 true.