Two categorical variables from a single population, test whether they are related. Which test is appropriate?

Two categorical variables from a single population, test whether they are related. Which test is appropriate?

• A. Chi-square test of independence with 2 degrees of freedom
• B. Chi-square test of homogeneity
• C. Two-proportion z-test
• D. Chi-square test of independence

Answer: (D)

When two categorical variables come from the same population and you want to know if they are related, you use the chi-square test of independence. This test examines a contingency table of the two variables and checks whether the observed counts in each cell differ from what would be expected if the variables were independent. You compute expected counts under independence as (row total × column total) / grand total, then sum (observed − expected)² / expected across all cells. A small p-value indicates evidence of an association between the variables. This fits the scenario precisely, because you’re looking for a relationship between two categorical factors within a single population. The chi-square test of homogeneity would be used to compare distributions of a single categorical variable across different populations, which isn’t what's being asked here. The two-proportion z-test compares two proportions for a binary outcome between two groups and doesn’t assess the relationship between two categorical variables. Even though one option mentions a specific degrees-of-freedom value, the standard and most widely used label for this situation is the chi-square test of independence.

When two categorical variables come from the same population and you want to know if they are related, you use the chi-square test of independence. This test examines a contingency table of the two variables and checks whether the observed counts in each cell differ from what would be expected if the variables were independent. You compute expected counts under independence as (row total × column total) / grand total, then sum (observed − expected)² / expected across all cells. A small p-value indicates evidence of an association between the variables.

This fits the scenario precisely, because you’re looking for a relationship between two categorical factors within a single population. The chi-square test of homogeneity would be used to compare distributions of a single categorical variable across different populations, which isn’t what's being asked here. The two-proportion z-test compares two proportions for a binary outcome between two groups and doesn’t assess the relationship between two categorical variables. Even though one option mentions a specific degrees-of-freedom value, the standard and most widely used label for this situation is the chi-square test of independence.