Lecture 38 Section 14.5 Mon, Dec 4, 2006 Test of Independence Lecture 38 Section 14.5 Mon, Dec 4, 2006

Independence Only one sample is taken. For each subject in the sample, two observations are made (i.e., two variables are measured). We wish to determine whether there is a relationship between the two variables. The two variables are independent if there is no relationship between them.

Mendel’s Experiments In Mendel’s experiments, Mendel observed 75% yellow seeds, 25% green seeds. 75% smooth seeds, 25% wrinkled seeds. Because color and texture were independent, he also observed 9/16 yellow and smooth 3/16 yellow and wrinkled 3/16 green and smooth 1/16 green and wrinkled

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. Smooth Wrinkled Yellow 9 3 Green 1

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. Smooth Wrinkled Yellow 9 3 Green 1 3 : 1 Ratio

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. Smooth Wrinkled Yellow 9 3 Green 1 3 : 1 Ratio

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. Smooth Wrinkled Yellow 9 3 Green 1 3 : 1 Ratio

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. Smooth Wrinkled Yellow 9 3 Green 1 3 : 1 Ratio

Mendel’s Experiments Had the traits not been independent, he might have observed something different. Smooth Wrinkled Yellow 10 2 Green

Example Suppose a university researcher suspects that a student’s SAT-M score is related to his performance in Statistics. At the end of the semester, he compares each student’s grade to his SAT-M score for all Statistics classes at that university. He wants to know whether the student’s with the higher SAT-M scores got the higher grades.

Example Does there appear to be a difference between the rows? Or are the rows independent? Grade A B C D F 400 - 500 7 8 16 20 21 500 – 600 13 28 32 22 600 – 700 23 10 9 700 - 800 14 5 SAT-M

The Test of Independence The null hypothesis is that the variables are independent. The alternative hypothesis is that the variables are not independent. H0: The variables are independent. H1: The variables are not independent. Let  = 0.05.

The Test Statistic The test statistic is the chi-square statistic, computed as The question now is, how do we compute the expected counts?

Expected Counts Under the assumption of independence (H0), the rows should exhibit the same proportions. This is the same as when testing for homogeneity. Therefore, we may calculate the expected counts in the same way.

Expected Counts A B C D F 400 - 500 7 (8.64) 8 (17.28) 16 (20.16) 20 (14.40) 21 (11.52) 500 – 600 13 (12.96) 28 (25.92) 32 (30.24) 22 (21.60) 600 – 700 23 10 9 700 - 800 (5.76) 14 (13.44) (9.60) 5 (7.68)

The Test Statistic The value of 2 is 23.7603.

df = (no. of rows – 1)  (no. of cols – 1). Degrees of Freedom The degrees of freedom are the same as before df = (no. of rows – 1)  (no. of cols – 1). In our example, df = (4 – 1)  (5 – 1) = 12.

The p-value To find the p-value, calculate 2cdf(23.7603, E99, 12) = 0.0219. The results are significant at the 5% level.

TI-83 – Test of Independence The test for independence on the TI-83 is identical to the test for homogeneity.

Example Admissions figures for the School of Arts and Sciences. Acceptance Status Accepted Not Accepted Race Female 50 150 Male 500 1000

Example Admissions figures for the Business School. Acceptance Status Accepted Not Accepted Race Female 850 1500 Male 150 200

Example Admissions figures for the two schools combined. Acceptance Status Accepted Not Accepted Race Female 900 1650 Male 650 1200

Practice This is called Simpson’s paradox. It occurs whenever the aggregate population shows a different relationship than in the subpopulations.