Which statement distinguishes a chi-square test of independence from a chi-square goodness-of-fit test?

• A. Independence tests whether observed frequencies match a specified distribution for a single variable.
• B. Independence tests the relationship between two categorical variables; goodness-of-fit tests whether observed frequencies match the expected distribution.
• C. Independence tests the normality of each variable.
• D. Independence tests the variance across groups.

Answer: (B)

These chi-square tests address different questions about categorical data. A chi-square test of independence asks whether two categorical variables are related in the population. You build a contingency table of counts for the combinations of categories and test whether the observed pattern of counts would occur if the variables were independent. The null hypothesis is that the variables are independent; a significant result suggests a relationship between them.

A chi-square goodness-of-fit test, on the other hand, asks whether the distribution of a single categorical variable matches a specified (theoretical) distribution. You compare observed counts across the categories to the counts expected under that predefined distribution; the null is that the data follow that specified distribution.

The statement focusing on independence as simply matching a single variable to a distribution would be incorrect because that describes goodness-of-fit, not independence. Treating independence as a test of normality of each variable is off the mark, since independence concerns relationships between two variables, not the shape of their distributions. And testing variance across groups isn’t what independence concerns either; that’s about dispersion and is addressed by other tests, not by the chi-square test of independence.