AP Statistics (AP Stats): Chi?Square Test for Homogeneity (Two?Way Table, Multiple Populations)

AP Statistics – Chi?Square Test for Homogeneity (Two?Way Table, Multiple Populations)

AP Statistics: Chi-Square Test for Homogeneity – Exam-Ready Study Guide

What This Is

The Chi-Square Test for Homogeneity determines whether the distribution of a categorical variable is the same across multiple populations (e.g., "Do high school students, college students, and adults have the same preferences for social media platforms?"). Unlike the Chi-Square Test for Independence (which analyzes one sample with two categorical variables), homogeneity compares two or more independent samples on one categorical variable. This test is essential for the AP exam because it appears in FRQs (often with two-way tables) and requires careful interpretation of hypotheses, conditions, and conclusions.

Real-world example: A researcher wants to know if voter support for a new policy (Support/Oppose/Neutral) differs across three regions (North, South, West). The homogeneity test answers: "Is the distribution of support the same in all three regions?"

Key Terms & Formulas

• Chi-Square Test for Homogeneity: Tests whether the distribution of a categorical variable is the same across multiple populations.
• H?: The distribution of the categorical variable is the same for all populations.

• H?: The distribution of the categorical variable is not the same for at least one population.

• Expected Count Formula: [ E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}} ]

• E = expected count for a cell in the two-way table.

• Chi-Square Test Statistic (?²): [ \chi^2 = \sum \frac{(O - E)^2}{E} ]

• O = observed count, E = expected count.

• Degrees of Freedom (df) for Homogeneity: [ df = (\text{number of rows} - 1) \times (\text{number of columns} - 1) ]

• Rows = categories of the response variable (e.g., Support/Oppose/Neutral).

• Columns = populations being compared (e.g., North/South/West).

• Conditions for Chi-Square Tests (RICE):

• Random: Data comes from random samples or randomized experiments.
• Independent: Samples are independent (no overlap between populations).
• Categorical: Data is categorical (counts or proportions).

• Expected Counts-5: All expected counts must be at least 5 (check with ?²-Test output).

• Calculator Command (TI-84):

• STAT-TESTS-?²-Test (for homogeneity or independence).
• Input observed counts in a matrix (2nd-x?¹-EDIT).

• Output: ?² test statistic, p-value, and expected counts (store in a matrix).

• P-value Interpretation:

• If p-value-? (e.g., 0.05), reject H?-distributions are not the same.

• If p-value > ?, fail to reject H?-no evidence that distributions differ.

• Follow-Up Analysis (if H? is rejected):

• Compare residuals (O – E) or standardized residuals to identify which cells contribute most to the difference.

Step-by-Step / Process Flow

How to Solve an AP FRQ on Homogeneity

1. State Hypotheses in Context
2. H?: The distribution of [categorical variable] is the same for [list populations].
3. H?: The distribution of [categorical variable] is not the same for at least one population.

4. Example: H?: The distribution of voter support (Support/Oppose/Neutral) is the same in the North, South, and West.

5. Check Conditions (RICE)

6. Random: "The samples were randomly selected from each region."
7. Independent: "The samples are independent (no overlap between regions)."
8. Categorical: "Voter support is a categorical variable."

9. Expected Counts-5: Use calculator output to verify (or show expected counts table).

10. Compute Test Statistic & P-value

11. Enter observed counts into a matrix (TI-84: 2nd-x?¹-EDIT).
12. Run ?²-Test (STAT-TESTS-?²-Test).

13. Record ?² statistic and p-value.

14. Make a Conclusion in Context

15. Compare p-value to ? = 0.05 (or given significance level).

16. Example: "Since the p-value (0.02) < 0.05, we reject H?. There is convincing evidence that the distribution of voter support differs across the three regions."

17. Follow-Up (If Required)

18. If H? is rejected, identify which cells contribute most to the difference (e.g., "The South has a higher-than-expected proportion of 'Oppose' responses").

Common Mistakes

• Mistake: Confusing homogeneity with independence.
• Correction: Homogeneity compares multiple populations on one variable; independence tests one population on two variables.

• Why? The hypotheses and interpretations differ (e.g., "same distribution" vs. "no association").

• Mistake: Forgetting to check expected counts-5.

• Correction: Always verify this condition using the calculator’s expected counts matrix.

• Why? The test is invalid if expected counts are too small.

• Mistake: Miscalculating degrees of freedom.

• Correction: Use (rows – 1) × (columns – 1), where rows = response categories and columns = populations.

• Why? Incorrect df leads to wrong p-values and conclusions.

• Mistake: Writing a generic conclusion (e.g., "Reject H?").

• Correction: Always state the conclusion in context (e.g., "There is evidence that the distribution of support differs across regions").

• Why? AP graders deduct points for lack of context.

• Mistake: Using proportions instead of counts in the matrix.

• Correction: The ?² test requires raw counts, not percentages or proportions.
• Why? The formula relies on counts for expected values.

AP Exam Insights

• FRQ Setup: Expect a two-way table with 2–4 populations and 2–4 response categories. You’ll need to:
• State hypotheses.
• Check conditions (especially expected counts).
• Run the test and interpret the p-value.

• Follow up (e.g., "Which group differs most?").

• Tricky Distinction: Homogeneity vs. Independence

• Homogeneity: "Are the distributions the same across populations?" (e.g., "Do men and women prefer the same brands?")

• Independence: "Is there an association between two variables in one population?" (e.g., "Is there an association between gender and brand preference in adults?")

• Calculator Pitfall: Forgetting to store expected counts in a matrix.

• Always check 2nd-x?¹-[B] (or another matrix) to verify expected counts-5.

• Common Follow-Up: If H? is rejected, AP may ask:

• "Which cell contributes most to the ?² statistic?" (Look for largest (O – E)²/E.)
• "Describe the difference in distributions." (Compare observed vs. expected counts.)

Quick Check Questions

Question 1 (Multiple Choice)

A study compares the distribution of favorite ice cream flavors (Vanilla, Chocolate, Strawberry) across three age groups (Kids, Teens, Adults). The ?² test for homogeneity yields a p-value of 0.03. Which conclusion is correct? (A) There is no association between age group and ice cream preference. (B) The distribution of ice cream preferences is the same for all age groups. (C) There is convincing evidence that the distribution of ice cream preferences differs across age groups. (D) The sample size was too small to draw a conclusion.

Answer: (C) Explanation: A p-value of 0.03 < 0.05 means we reject H?, concluding the distributions differ.

Question 2 (FRQ Part)

A researcher surveys 500 people from three cities (A, B, C) about their primary mode of transportation (Car, Public Transit, Bike). The observed counts are:

Part (a): State the hypotheses for a ?² test for homogeneity. Part (b): Are the conditions for inference met? Justify your answer.

Answer (a): - H?: The distribution of transportation modes is the same for all three cities. - H?: The distribution of transportation modes is not the same for at least one city.

Answer (b): - Random: Assume the samples are random (if not stated, note the assumption). - Independent: The cities are independent populations. - Categorical: Transportation mode is categorical. - Expected Counts-5: All expected counts (e.g., for City A, Car: (250×310)/750-103.3) are-5.

Last-Minute Cram Sheet

1. Homogeneity vs. Independence: Homogeneity = multiple populations, one variable; Independence = one population, two variables.
2. Hypotheses: H?: Distributions are the same; H?: At least one differs.
3. Expected Count Formula: (row total × column total) / grand total.
4. Degrees of Freedom: (rows – 1) × (columns – 1).
5. Conditions (RICE): Random, Independent, Categorical, Expected counts-5.
6. Calculator: STAT-TESTS-?²-Test (use matrix for observed counts).
7. P-value-?: Reject H?-distributions differ.
8. Follow-Up: Check residuals or standardized residuals for largest differences.
9. Always state conclusions in context (AP graders deduct for generic answers).
10. Never use proportions in the matrix—only raw counts!