Test of Independence Lecture 43 Section 14.5 Mon, Apr 23, 2007

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 SmoothWrinkled Yellow93 Green31 That is, he observed the same ratios within categories that he observed for the totals.

Mendel’s Experiments SmoothWrinkled Yellow93 Green31 3 : 1 Ratio That is, he observed the same ratios within categories that he observed for the totals.

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. SmoothWrinkled Yellow93 Green31 3 : 1 Ratio

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. SmoothWrinkled Yellow93 Green31 3 : 1 Ratio

Mendel’s Experiments That is, he observed the same ratios within categories that he observed for the totals. SmoothWrinkled Yellow93 Green31 3 : 1 Ratio

Mendel’s Experiments Had the traits not been independent, he might have observed something different. SmoothWrinkled Yellow102 Green22

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? ABCDF – – Grade 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. H 0 : The variables are independent. H 1 : 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 Since the rows should all exhibit the same proportions, the method is the same as before.

Expected Counts ABCDF (8.64) 8 (17.28) 16 (20.16) 20 (14.40) 21 (11.52) 500 – (12.96) 28 (25.92) 32 (30.24) 22 (21.60) 13 (17.28) 600 – (8.64) 23 (17.28) 22 (20.16) 10 (14.40) 9 (11.52) (5.76) 13 (11.52) 14 (13.44) 8 (9.60) 5 (7.68)

The Test Statistic The value of  2 is

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  2 cdf( , E99, 12) = 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 Female50150 Male

Example Admissions figures for the Business School. Acceptance Status Accepted Not Accepted Race Female Male150200

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

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