Stating Hypotheses for a Test of Homogeneity

From the Study Design to Hypotheses

In “Stating Hypotheses for a Test of Independence,” you learned to write hypotheses about whether two categorical variables are related in a population. This tutorial focuses on a different design: comparing separate populations or groups on one categorical response. For that question, a chi-square test of homogeneity asks whether the response has the same distribution across the populations.

A response distribution describes the proportions of individuals in each category of a categorical variable. For example, a survey might record one preferred option from three choices. The distribution consists of the population proportion choosing each option. Homogeneity means those category proportions are the same across the populations being compared.

The hypotheses concern population distributions, not the counts in the samples. Sample counts can differ simply because sample sizes differ. Even with equal sample sizes, sample proportions may vary by chance. The test uses the sample data to assess evidence about the population claim.

As explained in “Chi-Square Test for Homogeneity: Purpose and Setting,” this test fits separate random samples or an appropriate randomized experiment, with one categorical response recorded for each individual. The question is whether that response’s distribution is consistent across the groups.

Writing the Null and Alternative

Begin by naming the groups being compared and the response categories. Then say what the response distribution would look like under each hypothesis. The null describes matching distributions across all groups; the alternative says they are not all the same.

For a more formal statement, suppose there are \(k\) populations and the response has \(c\) categories. Let \(\pi_{ij}\) be the true proportion of individuals in population \(i\) who fall in response category \(j\). The null says that for each response category, its population proportion is equal across all populations:

The alternative says the distributions are not all identical: at least one population has a different proportion in at least one response category.

The category-by-category form makes “same distribution” precise. If a response has categories such as “walk,” “bike,” and “bus,” the null says the proportion choosing “walk” is the same in each population, the proportion choosing “bike” is the same in each population, and likewise for “bus.” Because the category proportions within each population add to 1, a difference in one category’s proportion means the distributions differ.

In an AP response, contextual wording is usually the clearest way to state the hypotheses. You do not need to list a separate equation for every category if the words clearly say the distribution is the same across all populations under \(H_0\), and differs for at least one population under \(H_a\).

What “Differs” Means

The alternative is non-directional. It does not specify which population has a larger proportion in any category, or which categories account for a difference. It says only that the population distributions are not all the same. A chi-square test of homogeneity can detect a difference in many patterns, so the alternative does not identify one particular pattern in advance.

Do not write the alternative as “the groups have different sample counts.” Counts depend on sample sizes and are observed data, not the population claim. Instead, refer to the population response distributions. Similarly, “the proportions are not equal” is incomplete unless it is clear which response proportions and populations are meant.

The test’s name can be a reminder: homogeneity asks whether the response distribution is homogeneous, or the same, across the populations. In “Independence Versus Homogeneity: Choosing the Right Test,” the design distinction was the key: separate groups compared on one response points to homogeneity, while one sample classified by two variables points to independence. The tables may look alike, but the hypotheses reflect the study design.

Worked Example: Preferred Library Study Space

Worked Example: Seasonal Water-Use Category

Worked Example: Randomized Reminder Methods

Common Mistakes and AP Exam Communication

A strong response makes the comparison explicit: which populations or groups, which categorical response, and whether its population distribution is the same or differs. Keep the null and alternative in context and make them match the homogeneity question.

• Writing “the populations are equal”: This does not identify what is being compared. Say that the distribution of the named response is the same across the populations.
• Writing “the sample distributions differ”: Sample distributions describe observed data. Hypotheses for an inference test make claims about the population distributions.
• Comparing counts instead of distributions: Equal distributions mean equal category proportions, not necessarily equal counts. Group sample sizes may differ.
• Leaving out “at least one” in the alternative: The alternative does not require every group to differ from every other group. Say the distributions differ for at least one population, or that they are not all the same.
• Making the alternative directional: A homogeneity alternative does not say that one group has more of a particular response. It allows any pattern of distributional difference.
• Confusing homogeneity with independence: Both procedures use categorical data, but the study designs and hypotheses differ. Use the design to identify the question, as in “Independence Versus Homogeneity: Choosing the Right Test.”
• Claiming cause and effect from the hypotheses: The hypotheses do not establish causation. The study design determines whether a causal conclusion may be supported.

Key Takeaway

A chi-square test of homogeneity evaluates whether one categorical response has the same distribution across separate populations or groups. The null says the category proportions match across all groups; the alternative says the distributions are not all identical.