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

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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.

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

12. 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.

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

14. 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.

15. 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
(5.76) 14
(13.44)
(9.60) 5
(7.68)

16. The Test Statistic
The value of 2 is

17. 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.

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

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

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

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

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

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