Chi-Square Tests for Homogeneity of Categorical Distributions

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What does a chi-square test for homogeneity assess?

Whether two or more populations share the same distribution across categories of a categorical variable.

What are the hypotheses in a test for homogeneity?

$H_0$ states that the population distributions are the same; $H_a$ states that they are not the same.

How is an expected count calculated for a contingency-table cell?

$E=\frac{(\text{row total})(\text{column total})}{\text{grand total}}$.

What condition must expected counts meet for the procedure described here?

Every expected cell count must be at least 5.

How are the chi-square statistic and degrees of freedom calculated?

$\chi^2=\sum\frac{(O-E)^2}{E}$ across all cells; $df=(r-1)(c-1)$ for $r$ rows and $c$ columns.

Quiz questions

1. What does a chi-square test for homogeneity compare?Answer: Categorical distributions across two or more populations.
2. How is the expected count for a contingency-table cell calculated?Answer: Multiply its row and column totals, then divide by the grand total.
3. A contingency table has two population rows and four response-category columns. What is its degrees-of-freedom value?Answer: $df=(2-1)(4-1)=3$.
4. Which expression gives the chi-square statistic for a contingency table?Answer: $\sum\frac{(O-E)^2}{E}$, summed over all cells.
5. A test for homogeneity gives $\chi^2\approx22.50$ with 3 degrees of freedom. At significance level $0.05$, what conclusion is appropriate?Answer: Reject equal distributions; the data provide evidence that the population distributions differ.