Inference for Proportions: every key term you need

One-Proportion Confidence Interval

p̂ ± z*√(p̂(1-p̂)/n). Estimate population proportion p. Requires: np̂ ≥ 10 and n(1-p̂) ≥ 10 for normal approximation.

Sample Size for Proportions

n = (z*/ME)²·p(1-p). Use p=0.5 if unknown (most conservative). Larger ME = smaller n needed. Costly: doubling precision requires 4x sample.

Two-Proportion CI

(p̂₁ - p̂₂) ± z*√(p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂). Compares two populations. If CI doesn't contain 0, proportions differ significantly.

Hypothesis Test for Proportion

H₀: p = p₀. Test statistic: z = (p̂ - p₀)/√(p₀(1-p₀)/n). Use p₀ (hypothesized) in denominator, not p̂.

Two-Proportion Z-test

H₀: p₁ = p₂ vs Hₐ: p₁ ≠ p₂. Test statistic: z = (p̂₁ - p̂₂)/√(p̂(1-p̂)(1/n₁ + 1/n₂)) where p̂ = combined proportion.

Chi-Square Test

Tests association between two categorical variables. X² = Σ(observed - expected)²/expected. Higher values = stronger association.

Expected Frequencies

For independence: expected = (row total × column total)/total. Chi-square test valid if all expected ≥ 5.

Conditions for Chi-Square

Random sample, independence, expected frequencies ≥ 5 in each cell. If not met, may combine categories or use exact test.

Effect Size for Proportions

Sample size alone doesn't indicate importance; consider practical significance. Difference of 0.3% vs 30% in proportions very different.

Unit 7 Summary

Confidence intervals and tests for proportions use normal approximation. Chi-square test analyzes categorical association. Always check conditions.

More AP Statistics guides

• Designing Studies
• Exploring Bivariate Data
• Probability
• Random Variables & Distributions
• Sampling Distributions