To determine whether the distribution of a categorical outcome is the same across two populations, which test is appropriate?

To determine whether the distribution of a categorical outcome is the same across two populations, which test is appropriate?

• A. Chi-square test of independence
• B. 2-proportion z-test
• C. 1-proportion z-test
• D. Chi-square test of homogeneity

Answer: (D)

When you want to know if a categorical outcome occurs with the same probabilities in two different groups, you compare the category proportions across those groups. The chi-square test of homogeneity is designed for this: you lay out a contingency table with one axis for the populations (two groups) and the other axis for the outcome categories. Under the null, the distribution of categories is the same in both populations, so the expected counts come from applying the same category probabilities to both groups. The test looks at how far the observed counts deviate from these expectations using the sum of (O−E)²/E across all cells. A large deviation suggests the distributions differ between populations. The degrees of freedom are (number of populations − 1) times (number of categories − 1); with two populations and k categories, that’s k−1. Assumptions include random samples and sufficiently large expected counts in each cell (typically at least 5). The other tests either focus on a single binary proportion (2-proportion or 1-proportion z-test) or test independence between two variables in a pooled sample, rather than directly comparing the full distribution across populations. So, the chi-square test of homogeneity is the right choice for this question.

When you want to know if a categorical outcome occurs with the same probabilities in two different groups, you compare the category proportions across those groups. The chi-square test of homogeneity is designed for this: you lay out a contingency table with one axis for the populations (two groups) and the other axis for the outcome categories. Under the null, the distribution of categories is the same in both populations, so the expected counts come from applying the same category probabilities to both groups. The test looks at how far the observed counts deviate from these expectations using the sum of (O−E)²/E across all cells. A large deviation suggests the distributions differ between populations. The degrees of freedom are (number of populations − 1) times (number of categories − 1); with two populations and k categories, that’s k−1. Assumptions include random samples and sufficiently large expected counts in each cell (typically at least 5). The other tests either focus on a single binary proportion (2-proportion or 1-proportion z-test) or test independence between two variables in a pooled sample, rather than directly comparing the full distribution across populations. So, the chi-square test of homogeneity is the right choice for this question.