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How do you state the hypotheses and decide between a chi-square test of homogeneity and one of independence?

Topic 8.5 Setting Up a Chi-Square Test for Homogeneity or Independence: distinguish a test of homogeneity from a test of independence based on the design, state the appropriate hypotheses, and check the conditions.

A focused answer to AP Statistics Topic 8.5, on distinguishing a chi-square test of homogeneity (several groups, same variable) from a test of independence (one sample, two variables), stating the right hypotheses, and checking the conditions.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
The design decides the test
Hypotheses for each test
Checking the conditions
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What this topic is asking

The College Board (Topic 8.5) wants you to distinguish a chi-square test of homogeneity from one of independence based on the design, state the appropriate hypotheses for each, and check the conditions.

The design decides the test

This is the central decision of the topic. The chi-square arithmetic is identical for the two tests (same expected counts, same statistic, same degrees of freedom), but the design, hypotheses, and conclusion wording differ. Homogeneity compares separate groups ("do these populations have the same distribution?"); independence examines one population ("are these two traits associated?"). Identify the sampling design before writing anything.

Hypotheses for each test

The wording must match the design. For homogeneity, name the variable and the groups ("the distribution of sport participation is the same across the three schools"). For independence, name the two variables ("education level and employment status are independent"). Hypotheses are stated in words about the categorical relationship, not as numeric parameter equalities, which is a notable contrast with the mean and proportion tests.

Checking the conditions

The condition check uses the expected counts (each $\ge 5$ ), exactly as in goodness of fit and as computed in Topic 8.4. The randomness requirement is read according to the design: homogeneity needs each group to be a random sample (or randomly assigned in an experiment); independence needs the single sample to be random. Verifying these is the gate to the test in Topic 8.6.

Setting up a test of homogeneity

A market researcher draws separate random samples of $120$ customers from each of three regions (North, South, East) and classifies each customer's preferred product (X, Y, Z). Set up the appropriate chi-square test.

step 1 Identify the design and test

Three separate random samples (one per region), each classified by one categorical variable (preferred product). This is a test of homogeneity (comparing the product-preference distribution across the three regions).

step 2 State the hypotheses

$H_0$ : the distribution of product preference (X, Y, Z) is the same across the three regions.
$H_a$ : the distribution of product preference is not the same for all three regions (at least one region differs).

step 3 Check the conditions

Random: each region's sample is a stated random sample. Large counts: compute expected counts via $E = \dfrac{\text{row total} \times \text{column total}}{\text{grand total}}$ and confirm each is at least $5$ (with $120$ per region and three products, expected counts will comfortably exceed $5$ unless a preference is very rare). $10\%$ : each sample of $120$ is plausibly under $10\%$ of its region's customers.

step 4 Readiness

Conditions met; a chi-square test of homogeneity is appropriate. The degrees of freedom (Topic 8.6) are $(\text{rows} - 1)(\text{columns} - 1)$ .

Try this

Q1. One sample of voters is classified by gender and party preference. Which test? [1 point]

Cue. Independence (one sample, two categorical variables); $H_0$ : gender and party preference are independent.

Q2. State the homogeneity null when comparing a response across four separate groups. [1 point]

Cue. $H_0$ : the distribution of the response variable is the same across all four groups.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2019 (style)1 marksSection I (multiple choice). A researcher takes one random sample and records two categorical variables on each person. The appropriate chi-square test is (A) goodness of fit (B) homogeneity (C) independence (D) a two-proportion z-test

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The correct answer is (C).

One sample with two categorical variables recorded on each unit calls for a chi-square test of independence (are the two variables associated?).

(A) is for one variable against a claimed distribution. (B) is for comparing one variable across several separate samples/groups. (D) compares only two proportions, not a full table.

AP 2021 (style)3 marksSection II (free response). Design 1: separate random samples of

100

students from each of three schools are classified by whether they participate in sport (yes/no). Design 2: one random sample of

300

adults is classified by both education level (three levels) and employment status (two levels). (a) Name the appropriate chi-square test for each design. (b) State the hypotheses for each. (c) Explain the design difference that determines the choice.

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A 3-point design-and-setup question.

(a) (1 point) Design 1: chi-square test of homogeneity (several separate samples, one variable). Design 2: chi-square test of independence (one sample, two variables).
(b) (1 point) Homogeneity: $H_0$ : the distribution of sport participation is the same across the three schools; $H_a$ : it is not the same for all schools. Independence: $H_0$ : education level and employment status are independent; $H_a$ : they are associated (not independent).
(c) (1 point) Homogeneity uses multiple separate samples with a fixed size from each group, comparing one variable's distribution; independence uses one sample and measures two variables on each unit.

Markers reward matching design to test, the distinct hypothesis wordings, and the multiple-samples versus one-sample distinction.

Related dot points

Sources & how we know this

AP Statistics Course and Exam Description — College Board (2020)