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2 test for independence Used with categorical, bivariate data from ONE sample Used to see if the two categorical variables are associated (dependent) or not associated (independent)

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Assumptions & formula remain the same!

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Hypotheses – written in words H 0 : two variables are independent H a : two variables are dependent Be sure to write in context!

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A beef distributor wishes to determine whether there is a relationship between geographic region and cut of meat preferred. If there is no relationship, we will say that beef preference is independent of geographic region. Suppose that, in a random sample of 500 customers, 300 are from the North and 200 from the South. Also, 150 prefer cut A, 275 prefer cut B, and 75 prefer cut C.

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If beef preference is independent of geographic region, how would we expect this table to be filled in? NorthSouthTotal Cut A150 Cut B275 Cut C75 Total300200500 9060 165110 4530

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Expected Counts Assuming H 0 is true,

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Degrees of freedom Or cover up one row & one column & count the number of cells remaining!

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Now suppose that in the actual sample of 500 consumers the observed numbers were as follows: (on your paper) Is there sufficient evidence to suggest that geographic regions and beef preference are not independent? (Is there a difference between the expected and observed counts?)

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Assumptions: Have a random sample of people All expected counts are greater than 5. H 0 : geographic region and beef preference are independent H a : geographic region and beef preference are dependent P-value =.0226df = 2 =.05 Since p-value < , I reject H 0. There is sufficient evidence to suggest that geographic region and beef preference are dependent. Expected Counts: N S A90 60 B165110 C45 30

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More Example Problems

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2 test for homogeneity single categorical two (or more) independent samplesUsed with a single categorical variable from two (or more) independent samples Used to see if the two populations are the same (homogeneous)

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Assumptions & formula remain the same! Expected counts & df are found the same way as test for independence. Only Only change is the hypotheses!

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Hypotheses – written in words H 0 : the two (or more) distributions are the same H a : the distributions are different Be sure to write in context!

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College Students’ Drinking Levels The data on drinking behavior for independently chosen random samples of male and female students was collected. Does there appear to be a gender difference with respect to drinking behavior?

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Homogeneity Test Gender DrinkingMenWomen None140186 Low478661 Moderate300173 High6316

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Assumptions: Have 2 random sample of students All expected counts are greater than 5. H 0 : drinking behavior is the same for female & male students H a : drinking behavior is not the same for female & male students P-value =.000df = 3 =.05 Since p-value < , I reject H 0. There is sufficient evidence to suggest that drinking behavior is not the same for female & male students. Expected Counts: M F 0158.6167.4 L554.0585.0 M230.1243.0 H38.440.6

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Titanic Moviemakers of Titanic imply that lower- class passengers were treated unfairly. Was that accurate?

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Likelihood of Survival on Titanic? H o : C = 109/1318, W = 402/1318, M = 807/1318 H a : at least one is different 2 = 225.16, df = 2, P( 2 > 225.16) = 0.000 Reject H o and conclude at least one proportion is different. ChildrenWomenMen Observed57296146 Expected41.269152.199305.533

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1 © 2008 Brooks/Cole, a division of Thomson Learning, Inc. 12.2 Tests for Homogeneity and Independence in a Two-Way Table Data resulting from observations.

1 © 2008 Brooks/Cole, a division of Thomson Learning, Inc. 12.2 Tests for Homogeneity and Independence in a Two-Way Table Data resulting from observations.

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