1. Section 13.2 Chi-Squared Test of Independence/Association

2. The Three Types of χ 2 tests Recall that there are three types of χ 2 tests: –Goodness of fit test Tests whether a sample distribution matches a hypothesized distribution (Does it fit GOOD?) –Homogeneity of proportions Tests whether p 1 = p 2 = p 3 = … –Association/Independence Tests whether two categorical variables are related

3. What are the assumptions? SRS from the population of interest 10% Rule for showing independent. None of the expected counts are less than 5.

4. χ 2 test of Association When a two-way table compares several populations, we use a χ 2 test for homogeneity of proportions. On the other hand, sometimes we want to classify observations FROM A SINGLE POPULATION by placing them in categories. Then we use a χ 2 test of association (also called the χ 2 test of independence).

5. Example SES SmokingHighMiddleLowTotal Current512243116 Former922128141 Never6892299 Total2115293356 My question: “Is there an association between SES level and smoking habits?”

6. Hypotheses The χ 2 test of association has hypotheses that are written in words: –H 0 : There is no relationship between smoking and SES level. –H a : There is some relationship between smoking and SES level.

7. In other words… –H 0 : Smoking and SES level are independent. –H a : Smoking and SES level are not independent. OR –H 0 : There is no association between smoking and SES level. –H a : There is some association between smoking and SES level.

8. Finishing up The rest of the test is carried out in the same manner as the χ 2 test for homogeneity of proportions. Just make sure your conclusion sentences match the H 0 and H a hypotheses.

9. χ 2 tests Comparison Chart Goodness of Fit Homogeneity of Proportions Association Hypotheses H 0 : The sample distribution matches the hypothesized distribution. H a : The sample distribution does not match the hypothesized distribution. H 0 : p1 = p2 = p3 = … H a : Not all of the proportions are equal. H 0 : There is no relationship between the two variables. H a : There is some relationship between the two variables. Assumptions SRS None of the expected counts are zero. No more than 20% of the expected counts are less than 5. Independent SRSs None of the expected counts are zero. No more than 20% of the expected counts are less than 5. SRS None of the expected counts are zero. No more than 20% of the expected counts are less than 5. Test Multiply the expected percents by your sample size. Calculate χ 2 statistic by hand. Look up p-value in Table E, df = # of categories minus 1. Enter observed counts in [A]. Use the calculator to find the p- value. *LOOK AT THE EXPECTED COUNTS IN [B] TO CHECK ASSUMPTIONS! So, the difference between the homogeneity of proportions test and the association test is ARE WE COMPARING SEVERAL POPULATIONS or DID THE DATA ARISE BY CLASSIFYING OBSERVATIONS INTO CATEGORIES.

10. HomeworkHomework Chapter 11 # 48, 49, 50, 51, 52