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Testing Independence and Homogeneity of Proportions Using Chi-Squared Methods

This lesson covers the Chi-Squared Tests for Independence and Homogeneity of Proportions, explaining the purpose, vocabulary, and required conditions for these tests. Participants will learn how to determine if there is an association between row and column variables using a contingency table and if different populations show the same proportions for specific characteristics. The lesson includes an example of calculating the test statistic, expected frequencies, and interpreting the p-value. Homework exercises and answers reinforce the concepts discussed.

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Testing Independence and Homogeneity of Proportions Using Chi-Squared Methods

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  1. Lesson 12 - 3 Tests for Independence and the Homogeneity of Proportions

  2. Objectives • Perform a test for independence • Perform a test for homogeneity of proportions

  3. Vocabulary • Chi-Squared Test for Independence – used to determine if there is an association between a row variable and a column variable in a contingency table constructed from sample data • Expected Frequencies – row total * column total / table total • Chi-Squared Test for Homogeneity of Proportions – used to test if different populations have the same proportions of individuals with a particular characteristic

  4. Terms (row total)(column total) Expected Frequency = ---------------------------------- table total

  5. Requirements Goodness-of-fit test: • All expected counts are greater than or equal to 1 (all Ei ≥ 1) • No more than 20% of expected counts are less than 5

  6. Σ (Oi – Ei)2 Test Statistic: χ20 = ------------- Ei Goodness-of-Fit Test P-Value is thearea highlighted P-value = P(χ2 0) χ2α Critical Region where Oi is observed count for ith category and Ei is the expected countfor the ith category (Right-Tailed)

  7. Chi-Square Tests on TI • Access MATRX menu (2nd X-1) • Highlight EDIT menu and select 1: [A] • Enter the number of rows and columns of the matrix • Enter the cell entries for the observed data and press 2nd QUIT • Repeat steps for expected values in matrix B • Press STAT, highlight TESTS and select C: χ²-Test • With cursor after the Observed: enter matrix [A] by accessing the MATRX menu, highlighting NAMES, and selecting 1: [A] • With cursor after the Expected: enter matrix [B] • Highlight Calculate and press ENTER

  8. Example

  9. Summary and Homework • Summary • Often, in contingency tables, we wish to test specific relationships, or lack of, between the two variables • The test for independence analyzes whether the row and column variables are independent • The test for homogeneity analyzes whether the observed proportions are the same across the different categories • Homework • pg 662 - 667: 1, 4, 5, 11, 12, 16

  10. Even Homework Answers • 4: a) the chi-square test statistic is 13.049b) chi-square critical value (df = (3-1)*(2-1)=2, α=0.05) = 5.991 so we would rejectc) p-value = 0.001467 • 12: a) FTR H0, not enough evidence to support the claim that abortion opinion and gender are independent ; the chi-square test statistic is 0.036318; chi-square critical value (df = (2-1)*(2-1)=1, α=0.1) = 2.706 so we would reject b) p=value = 0.84886 Pro-Life M49.62% F48.98% M50.38% F51.02% Pro-Choice Male Female

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