Question

 

Two judges rate 10 contestant in a beauty contest.Test the hypothesis that the ratings are independent of one another versus the alternative that the judges tend to agree in th judgment. ( at 5% level of significance)

A B C D E F G H I J

Judge 1 1 2 3 4 5 6 7 8 9 10

Judge 2 2 3 1 4 6 5 9 10 8 7

Answer 1

A chi square test of independence is always a right tailed test.

The Observed and expected value tables are as below.

Each Expected Value = (Row Total * Column Total) / N, Where N = Total Observations

Observed
	A	B	C	D	E	F	G	H	I	J	Total
Judge 1	1	2	3	4	5	6	7	8	9	10	55
Judge 2	2	3	1	4	6	5	9	10	8	7	55
Total	3	5	4	8	11	11	16	18	17	17	110
Expected
	A	B	C	D	E	F	G	H	I	J	Total
Judge 1	1.5	2.5	2	4	5.5	5.5	8	9	8.5	8.5	55
Judge 2	1.5	2.5	2	4	5.5	5.5	8	9	8.5	8.5	55
Total	3	5	4	8	11	11	16	18	17	17	110

The Hypothesis:

H0: The ratings are independent of the Judges. (They do not agree)

Ha: The ratings are independent of the Judges. (They Tend to agree)

_______________________________________________________________

(b) The Test Statistic:

Observed	Expected	(O - E)2	(O-E)2/E
1	1.5	0.25	0.166667
2	1.5	0.25	0.166667
2	2.5	0.25	0.1
3	2.5	0.25	0.1
3	2	1	0.5
1	2	1	0.5
4	4	0	0
4	4	0	0
5	5.5	0.25	0.045455
6	5.5	0.25	0.045455
6	5.5	0.25	0.045455
5	5.5	0.25	0.045455
7	8	1	0.125
9	8	1	0.125
8	9	1	0.111111
10	9	1	0.111111
9	8.5	0.25	0.029412
8	8.5	0.25	0.029412
10	8.5	2.25	0.264706
7	8.5	2.25	0.264706
Total			2.775609

as found above = 2.78

The degrees of freedom, df = (r – 1) * (c -1) = (2 - 1) * (10 - 1) = 1 * 9 = 9

The Critical Value:   The critical value at = 0.05 , df = 9

critical = 16.92

(e) The p value: The p value at = 2.78, df = 9, is P value = 0.9724.

The Decision Rule: If test is > critical, then Reject H0.

If p value is < , Then Reject H0.

The Decision:    Since test (2.78) is < critical (16.92), We Fail to reject H0.

Since p value (0.9724) is > (0.05), We Fail To Reject H0.

The Conclusion: There is insufficient evidence at the 95% significance level to conclude that the ratings are independent of the Judges.