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Perform chi-square tests for goodness of fit, homogeneity, and independence.
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Chi-square () tests are used for categorical data. There are three types:
| Test | Purpose |
|---|---|
| Goodness of Fit | Does a distribution match a claimed distribution? |
| Homogeneity | Do different populations have the same distribution? |
| Independence | Are two categorical variables independent? |
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Always calculated the same way for all three tests.
Purpose: Test whether a categorical variable follows a specified distribution.
Hypotheses:
Degrees of freedom: (where = number of categories)
Expected counts: (sample size × hypothesized proportion)
Purpose: Test whether the distribution of a categorical variable is the same across different populations.
Hypotheses:
Degrees of freedom:
Expected counts:
Purpose: Test whether two categorical variables are independent within a single population.
Hypotheses:
Degrees of freedom:
Expected counts: Same formula as homogeneity
If you reject , identify which cells contribute most to by examining: for each cell. Large contributions indicate where the biggest discrepancies are.
AP Tip: Chi-square tests are ALWAYS right-tailed. There is no "left-tailed" or "two-tailed" chi-square test. Show expected counts and verify they are all ≥ 5.