Topic 8.7 Notes – Skills Focus: Selecting an Appropriate Inference Procedure for Categorical Data

What categorical inference is trying to do

Categorical inference deals with counts in categories. We are making conclusions about:

• Proportions (percent who said yes, percent who prefer A)
• Distributions (how responses are spread across several categories)
• Associations (whether two categorical variables are related)

Before you even think about formulas, ask:

1. How many categorical variables are there?
2. How many categories does each variable have?
3. How many samples or groups are being compared?

Those three questions determine everything.

If a variable has only two categories (success/failure), that points toward a z-test for proportions.
If there are three or more categories, that pushes you toward a chi-square procedure.

The full set of categorical inference procedures

Here’s the entire menu for AP Stats:

1. One-proportion z-test
2. Two-proportion z-test
3. Chi-square goodness of fit test
4. Chi-square test for independence
5. Chi-square test for homogeneity

If you can quickly sort problems into one of these five, you’re in great shape.

One-proportion z-test

Structure:

• One sample
• One categorical variable
• Two categories
• Testing a claim about a single population proportion

Typical situation: A school claims 60% of students buy lunch. You sample students and test whether the true proportion equals 0.60.

If you see one sample + yes/no outcome → think one-proportion z-test.

Two-proportion z-test

Structure:

• Two independent samples or groups
• One categorical variable
• Two categories
• Comparing two population proportions

Example: Compare the proportion of voters who support a policy in City A vs City B.

If either variable has more than two categories, it is no longer a z-test.

Chi-square goodness of fit

Structure:

• One sample
• One categorical variable
• Three or more categories
• Comparing to a theoretical or claimed distribution

You’re checking whether observed counts “fit” expected proportions.

Example: A die is rolled 180 times. Do the results match a fair distribution?

This test compares:

• Observed counts
• Expected counts based on given proportions

There is no second sample here.

Independence vs Homogeneity

These two cause the most confusion because they look almost identical mathematically.

Both:

• Use a two-way table
• Use a chi-square statistic
• Have the same conditions and calculation

The difference is in how the data were collected.

Chi-square test for independence

Structure:

• One random sample
• Two categorical variables
• Asking whether the variables are associated

Example: A random sample of adults is asked about exercise level and coffee preference. You test whether exercise level and coffee preference are associated.

Hypotheses look like:

• $H_0$: The variables are independent (no association).
• $H_a$: The variables are associated.

You measured two traits on the same individuals.

Chi-square test for homogeneity

Structure:

• Two or more independent samples (or treatments)
• One categorical variable
• Comparing distributions across groups

Example: Three different schools are sampled, and students are asked their favorite subject. You test whether the distribution of favorite subjects is the same across schools.

Hypotheses look like:

• $H_0$: The population distributions are the same.
• $H_a$: At least one distribution differs.

You are comparing groups, not testing association within one sample.

How to choose the correct test

Keep these distinctions straight on test day:

Note: A two-proportion z-test is for one categorical variable with two categories, compared across two independent samples. A two-way table from one random sample with two variables calls for chi-square independence; multiple samples comparing one variable’s distribution across groups calls for homogeneity.

When stuck, slow down and label your situation before picking a test:

• Number of variables
• Number of categories
• Number of samples

Most multiple-choice questions on this topic are pure structure recognition.

Common mistakes that cost points

• Using a two-proportion z-test when a variable has 3+ categories.
• Calling a homogeneity test “independence” without checking whether there were multiple samples.
• Forgetting that goodness of fit compares to a theoretical distribution, not another group.
• Writing vague hypotheses like “there is a difference.” Always say what variables and what population in context.

On free-response questions, they often describe the sampling carefully. That wording is not filler. It tells you which chi-square test to use.