Unit 8: Inference for Categorical Data: Chi-Square
AP Statistics: 51 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
χ-square tests compare counts: goodness of fit checks one distribution; independence and homogeneity test relationships in two-way tables.
The three tests
Goodness of fit: one categorical variable vs a claimed distribution. Independence: two variables from ONE sample — are they associated? Homogeneity: one variable across several populations — are the distributions the same? All three share the same statistic, computed from counts.
The statistic
χ-square = sum of (observed - expected)² / expected, over all cells. Expected counts come from H0: row total × column total / grand total (two-way), or claimed proportion × total (goodness of fit). Large values mean observed counts are far from expectation → evidence against H0.
Conditions and df
Random sampling and all expected counts ≥ 5. Degrees of freedom: k - 1 for goodness of fit; (r - 1)(c - 1) for two-way tables. The χ-square distribution is right-skewed and nonnegative; larger df makes it more symmetric.
Conclusions
Reject H0 → significant evidence of poor fit (goodness of fit), association (independence), or different distributions (homogeneity). Fail to reject → insufficient evidence — never proof of independence. Tests use counts, not proportions, because counts carry sample size.
Exam traps
Independence vs homogeneity differ only in how data was collected. Expected counts, not observed, must be ≥ 5. df = (r-1)(c-1), not r×c.