3.14 Setting Up a Chi-Square Test for Homogeneity or Independence

A chi-square test for independence checks whether two categorical variables are associated within one population, while a chi-square test for homogeneity compares the distribution of one categorical variable across two or more populations or treatments. Setting one up means picking the right test, writing hypotheses in context, and verifying that your data came from the right kind of sampling and that all expected counts are large enough.

Multiple-choice questions test whether you can tell independence from homogeneity based on how the data were collected. The setup builds on the earlier two-way table work with expected counts, follows the same identify, hypothesize, check conditions, compute, conclude structure you used for proportion and mean tests, and leads into carrying out the test in the next topic.

Choosing Between Independence and Homogeneity Tests

• Use a test for independence when you have one sample from one population and you measure two categorical variables on each individual.
• Use a test for homogeneity when you compare the distribution of one categorical variable across two or more separate populations or treatment groups.
• Write hypotheses in context, naming the variables or populations from the question instead of using generic wording.
• The independence test needs a random sample from one population; the homogeneity test needs independent random samples from each population or a randomized experiment.
• Check the 10% condition when sampling without replacement, and confirm that all expected counts are at least 5.
• Conditions for chi-square tests use expected counts, not observed counts, for the large counts check.

Which Test to Run?

When you have categorical data in a two-way table, your first job is to decide between a test for independence and a test for homogeneity. The difference comes from how the data were collected.

• A chi-square test for independence fits when you take one sample from a single population and record two categorical variables on each individual. You are asking whether those two variables are associated.
• A chi-square test for homogeneity fits when you have two or more separate samples (or treatment groups) and you want to compare the distribution of one categorical variable across them.

A quick check: if every person in your data came from one group and you sorted them by two variables, think independence. If you started with separate groups and want to compare them, think homogeneity. Both tests use the chi-square statistic, which measures how far the observed counts fall from the expected counts, relative to the expected counts. Chi-square distributions take only positive values and are skewed right. They form a family of curves, and the skew lessens as the degrees of freedom increase.

Writing Hypotheses

Once you know which test is appropriate, write your hypotheses. For either test, include context by naming the actual variables or populations from the problem instead of using generic words.

Homogeneity Templates

• H0: There is no difference in distributions of a categorical variable across populations or treatments.
• Ha: There is a difference in distributions of a categorical variable across populations or treatments.

Independence Templates

• H0: There is no association between two categorical variables in a given population, or the two categorical variables are independent.
• Ha: Two categorical variables in a population are associated or dependent.

Example: Independence

Suppose you want to see whether a student's favorite sport is associated with their letter grade in an AP Statistics class. You take a random sample of 100 students from your school's AP Statistics class and record each student's favorite sport (football, basketball, or baseball) along with their letter grade.

Because you have one population and two variables measured on each student, this is a test for independence.

• H0: There is no association between sports preference and letter grade in AP Statistics for students at XYZ High School.
• Ha: There is an association between sports preference and letter grade in AP Statistics for students at XYZ High School.

Example: Homogeneity

Suppose you want to compare how sports preference is distributed among AP Statistics students versus AP Calculus students. You take a random sample of 100 Stats students and 100 Calculus students and record each student's preference among football, baseball, and basketball.

Because you have two separate populations and want to compare their distributions, this is a test for homogeneity.

• H0: There is no difference in sports preference between AP Statistics and AP Calculus students at XYZ High School.
• Ha: There is a difference in sports preference between AP Statistics and AP Calculus students at XYZ High School.

A test for homogeneity also fits a randomized experiment, since random assignment creates the separate groups you are comparing. For example, you might compare individuals receiving a new drug treatment against individuals receiving a placebo.

Conditions

Chi-square tests for two-way tables need two conditions you have seen before:

• Random data collection (the type depends on the test)
• Large counts

When sampling without replacement, also check the 10% condition (n ≤ 10%N).

For the large counts condition, verify that all expected counts are greater than or equal to 5. Use expected counts here, not the observed counts in the table.

Test for Independence

For a test for independence, the data should come from a random sample drawn from a single population. A simple random sample, where every individual had an equal chance of being selected, is the most common example. A representative random sample lets you generalize your conclusion back to that population.

Test for Homogeneity

For a test for homogeneity, the data should come from independent random samples from each population (for example, a stratified random sample) or from a randomized experiment.

A stratified random sample splits the population into non-overlapping groups (strata) based on some characteristic, then draws a simple random sample from each stratum. In an experiment, you instead verify that subjects were randomly assigned to treatment groups, which lets you attribute differences between groups to the treatments rather than to preexisting differences.

The 10% condition applies only when you sample without replacement from a population. If the data come from a randomized experiment, you do not need to check it, because no sample was drawn from a larger population. Subjects were randomly assigned to treatments instead.

Naming the Test and Justifying the Setup

Free Response:

• State which test you are using and why, based on how the data were collected. One sample with two variables points to independence; separate samples or treatment groups point to homogeneity.
• Write hypotheses in words with full context. Name the variables or populations from the prompt so a reader knows exactly what you are testing.
• List both conditions and tie them to the test. For independence, mention the random sample from one population; for homogeneity, mention the stratified random sample or random assignment.
• When you check large counts, say that all expected counts are at least 5. Showing this clearly is important for clean exam work.

Common Trap:

• Do not write hypotheses about a single proportion or mean here. Chi-square hypotheses for two-way tables are about association (independence) or differences in distributions (homogeneity).
• Do not use observed counts for the large counts check. The condition is about expected counts.

Common Misconceptions

• Misconception: Independence and homogeneity are the same.
  Actually: The test depends on data collection, not on the question wording alone. One sample with two variables means independence; two or more samples or treatment groups means homogeneity.
• Misconception: The large counts condition checks observed counts instead of expected counts.
  Actually: The large counts condition requires all expected counts to be at least 5, not the observed counts in your table.
• Misconception: Hypotheses do not need context.
  Actually: Generic hypotheses lose meaning. Always name the actual variables or populations from the problem.
• Misconception: At the setup stage, independence is a conclusion, not an assumption of the null hypothesis.
  Actually: That comes later when you interpret results. At the setup stage, independence is what the null hypothesis assumes, not something you have shown.
• Misconception: Any random sample works for homogeneity.
  Actually: Homogeneity needs separate samples from each population (often stratified) or a randomized experiment, not a single simple random sample.

Related AP Statistics Guides

• Unit 3 Review: Inference for Categorical Data: Proportions
• 3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence