Chapter 12 Lesson 12.2b Comparing Two Populations or Treatments 12.2: Test for Homogeneity and Independence in a Two-way Table.

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Presentation transcript:

Chapter 12 Lesson 12.2b Comparing Two Populations or Treatments 12.2: Test for Homogeneity and Independence in a Two-way Table

X 2 Test for Independence Null Hypothesis: H 0 : The two variables are independent Alternative Hypothesis: H a : The two variables are not independent Test Statistic:

X 2 Test for Independence Continued... Assumptions: 1)The data are in counts. 2)Data are from random samples or from subjects who were assigned at random to treatment groups. 3)All expected cell counts are at least 5. (Columns can be combined it this is not true)

X 2 Test for Independence Continued... Expected Counts: (assuming H 0 is true) P-value: df = (number of rows – 1)(number of columns – 1). The P-value associated with the computed test statistic value is the area to the right of  X  under the appropriate chi-square curve.

The paper “Contemporary College Students and Body Piercing” (Journal of Adolescent Health, 2004) described a survey of 450 undergraduate students at a state university in the southwestern region of the United States. Each student in the sample was classified according to class standing (freshman, sophomore, junior, senior) and body art category (body piercing only, tattoos only, both tattoos and body piercing, no body art). Is there evidence that there is an association between class standing and response to the body art question? Use  =.01. Body Piercing Only Tattoos Only Both Body Piercing and Tattoos No Body Art Freshman Sophomore Junior Senior State the hypotheses.

Body Art Continued... Body Piercing Only Tattoos Only Both Body Piercing and Tattoos No Body Art Freshman Sophomore Junior Senior H 0 : class standing and body art category are independent H a : class standing and body art category are not independent df = 9 Assuming H 0 is true, what are the expected counts? Body Piercing Only Tattoos Only Both Body Piercing and Tattoos No Body Art Freshman61 (49.7)7 (15.1)14 (18.5)86 (84.7) Sophomore43 (37.9)11 (11.5)10 (14.1)64 (64.5) Junior20 (23.4)9 (7.1)7 (8.7)43 (39.8) Senior21 (34.0)17 (10.3)23 (12.7)54 (58.0) How many degrees of freedom does this two-way table have?

Body Art Continued... Test Statistic: P-value <.001  =.01 Body Piercing Only Tattoos Only Both Body Piercing and Tattoos No Body Art Freshman61 (49.7)7 (15.1)14 (18.5)86 (84.7) Sophomore43 (37.9)11 (11.5)10 (14.1)64 (64.5) Junior20 (23.4)9 (7.1)7 (8.7)43 (39.8) Senior21 (34.0)17 (10.3)23 (12.7)54 (58.0)

Body Art Continued... Since the P-value < , we reject H 0. There is sufficient evidence to suggest that class standing and the body art category are not independent. Body Piercing Only Tattoos Only Both Body Piercing and Tattoos No Body Art Freshman61 (49.7)7 (15.1)14 (18.5)86 (84.7) Sophomore43 (37.9)11 (11.5)10 (14.1)64 (64.5) Junior20 (23.4)9 (7.1)7 (8.7)43 (39.8) Senior21 (34.0)17 (10.3)23 (12.7)54 (58.0) Which cell contributes the most to the X 2 test statistic? Seniors having both body piercing and tattoos contribute the most to the X 2 statistic.

Practice Handout A class collected data about eye color…

Homework Pg.722: #12.15, 19, 25