Chi-Square & Inference: every key term you need

Two-Sample t-Confidence Interval

(x̄₁ - x̄₂) ± t*·SE where SE = √(s₁²/n₁ + s₂²/n₂). Compares two population means. df approximated; often use smaller n-1.

Two-Sample t-test

H₀: μ₁ = μ₂ vs Hₐ: μ₁ ≠ μ₂. Test statistic: t = (x̄₁ - x̄₂)/√(s₁²/n₁ + s₂²/n₂). Compare to t-distribution.

Paired t-test

Compare dependent samples (before/after, matched pairs). Test: H₀: μ_d = 0. Use difference d = x₁ - x₂, treat as one-sample test.

Linear Regression Inference

Test H₀: slope β = 0 (no linear relationship). t = b/SE(b). If p < α, slope significantly ≠ 0.

Regression Slope CI

b ± t*·SE(b). Estimates true slope β. Interpretation: 95% CI for true change in y per unit x.

Regression Standard Error

SE = √(Σ(residuals)²/(n-2)). Measures typical distance points deviate from line. Smaller SE = better fit.

ANOVA (Analysis of Variance)

Tests if three or more means equal. H₀: μ₁ = μ₂ = μ₃. Test statistic F = MS_between / MS_within.

ANOVA Assumptions

Random samples from each group, independent observations, normal distributions within groups, equal variances. ANOVA robust to moderate violations.

Multiple Comparisons

If ANOVA rejects H₀, follow with pairwise comparisons (Tukey). Corrects for multiple testing (increases chance of false positive).

Effect Size (Cohen's d)

d = (x̄₁ - x̄₂)/s_pooled. Standardized difference. d<0.2 small, 0.2-0.8 medium, >0.8 large. Complements p-value.

Correlation & Causation

Strong correlation in observational data ≠ causation. Could be: reverse causation, confounding, lurking variable, coincidence.

Unit 8 Summary

Two-sample tests compare means. ANOVA extends to 3+ groups. Regression tests slope significance. Always consider practical significance beyond p-values.

More AP Statistics guides

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