AP Statistics Unit 3 Cheat Sheet: Inference for Proportions

The basics

What it covers: Confidence intervals and significance tests for one and two population proportions, Type I/II errors and power, and chi-square tests for two-way tables.

Exam weight: About 15–25% of the AP Statistics exam.

The big question: Given a sample proportion, what can we conclude about the population proportion — and how sure can we be?

Statistical practices: Formulate Questions (P1), Collect Data (P2), Analyze Data (P3), Interpret Results (P4).

Key topics at a glance

Sampling Distribution of p̂

Mean = p, standard deviation = √(p(1 − p)/n), approximately normal when Large Counts holds. This is the engine behind all proportion inference.

Conditions

Random sample or randomized experiment; 10% (n < 10% of the population); Large Counts (np̂ ≥ 10 and n(1 − p̂) ≥ 10).

Confidence Interval

p̂ ± z*·√(p̂(1 − p̂)/n). Estimate ± margin of error. A larger n narrows it; a higher confidence level widens it.

Interpreting Intervals

Interval: 'We are C% confident the interval captures the true p (in context).' Level: 'C% of such intervals from repeated samples would capture p.'

Significance Test

H₀: p = p₀ vs. Hₐ. Test statistic z = (p̂ − p₀)/√(p₀(1 − p₀)/n); find the p-value. If p-value ≤ α, reject H₀.

Errors & Power

Type I: reject a true H₀ (probability α). Type II: fail to reject a false H₀ (probability β). Power = 1 − β rises with larger n, larger effect, or larger α.

Two Proportions

Interval uses separate SEs; the test pools: p̂_c = (x₁ + x₂)/(n₁ + n₂). If a difference interval contains 0, no significant difference.

Chi-Square Tests

Two-way tables. Expected = (row × column)/grand total. Homogeneity compares groups; independence tests association. Small p-value ⇒ association/difference.

The key terms you must know

Sampling distribution of p̂ — mean p, SD √(p(1 − p)/n), approximately normal under Large Counts.
Random / 10% / Large Counts — the three conditions for proportion inference.
Confidence interval — estimate ± margin of error; a plausible range for the parameter.
Margin of error — critical value × standard error; shrinks as n grows.
Critical value z* — the multiplier set by the confidence level (1.96 for 95%).
Null / alternative hypotheses — H₀: p = p₀ vs. Hₐ; always about the parameter.
Test statistic — z = (p̂ − p₀)/√(p₀(1 − p₀)/n) for one proportion.
p-value — probability, if H₀ is true, of a result at least as extreme as observed.
Significance level α — the pre-set threshold for rejecting H₀.
Type I / Type II error — reject a true H₀ (α) / fail to reject a false H₀ (β).
Power — 1 − β, the chance of detecting a real effect.
Pooled proportion — (x₁ + x₂)/(n₁ + n₂), used in the two-proportion test.
Chi-square test — compares observed and expected counts in a two-way table.

Key themes to remember

Inference rests on the sampling distribution. Everything follows from how p̂ varies from sample to sample.
Intervals estimate; tests decide. A confidence interval gives a range; a significance test weighs evidence against a claim.
Always check conditions and answer in context. Random, 10%, Large Counts — then interpret using the real variable and population.
A p-value is a conditional probability, not the probability H₀ is true. It assumes H₀ and measures how surprising the data are.
Intervals and tests agree. A value outside a C% interval is rejected by the matching two-sided test at α = 1 − C.

Common exam traps

The test's SE uses p₀; the interval's SE uses p̂. Don't swap them.
Never 'accept' H₀. You 'fail to reject' it — absence of evidence is not evidence of absence.
Confidence is about the method, not one interval. '95% confident' describes the long-run capture rate of the procedure.
A p-value is not P(H₀ is true). It is the probability of the data (or more extreme) assuming H₀.
If a two-proportion interval contains 0, there is no convincing evidence of a difference.
Pool only for the two-proportion test, not for the confidence interval.
Chi-square needs expected counts (usually ≥ 5), not observed, to satisfy its condition.