Topic 8.7 Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data: choose among the one-proportion, two-proportion, and chi-square (goodness of fit, homogeneity, independence) procedures based on the scenario.

What this topic is asking

The College Board (Topic 8.7) is a skills focus: given a categorical-data scenario, choose the correct procedure among the one-proportion z, two-proportion z, and the three chi-square tests (goodness of fit, homogeneity, independence), and justify the choice. It synthesizes the categorical inference of Units 6 and 8.

The categorical decision tree

These questions partition every categorical scenario. The trickiest boundaries are: (a) two-proportion z versus homogeneity, both compare groups, but two-proportion z is for a two-category variable across two groups, while homogeneity handles two or more categories and two or more groups (a $2 \times 2$ homogeneity test and a two-proportion two-sided z-test actually agree); and (b) homogeneity versus independence, the same table arithmetic, distinguished by several samples (homogeneity) versus one sample, two variables (independence).

Why goodness of fit is different

Goodness of fit stands apart: it tests one categorical variable's distribution against a claimed distribution (equal, a ratio, or stated percentages), using a one-way list of counts, not a two-way table. The cue is a sentence like "the company claims colors appear in a $2:3:5$ ratio", a single variable measured against an external claim. There are no groups to compare and no second variable; that is what separates it from homogeneity and independence.

Implement and communicate

Implementation details follow the choice: a proportion test uses $z$ and the large-counts condition with counts of $10$; a chi-square test uses $\sum (O-E)^2/E$, expected counts, the $\ge 5$ condition, and degrees of freedom ($k - 1$ for goodness of fit, $(r-1)(c-1)$ for a table). The conclusion wording must match: "the proportion differs," "the distributions differ across groups" (homogeneity), or "the variables are associated" (independence). As always, association is not causation, and statistical significance is not practical importance.

Try this

Q1. Separate random samples from four hospitals are classified by infection outcome (yes/no). Which procedure? [1 point]

• Cue. Chi-square test of homogeneity (several separate samples, one variable across groups).

Q2. What single cue separates goodness of fit from the two-way chi-square tests? [1 point]

• Cue. Goodness of fit tests one variable against a claimed distribution (a one-way list of counts), with no second variable and no separate groups.