Question

(1 point) The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below. Dept. Math Physics Chemistry Linguistics English Men 87 41 24 43 Women 10 7 7 14 20 Test the claim that the gender of a professor is independent of the department. Use the significance level a = 0.025. (a) The test statistic is x2 =

Answer 1

(a)

H0: Null Hypothesis: The gender of a professor is independent of the department (Claim)

HA: Alternative Hypothesis: The gender of a professor is not independent of the department

Observed Frequencies:

Dept.	Math	Physics	Chemistry	Linguistics	English	Total
Men	69	87	41	24	43	264
Women	10	7	7	14	20	58
Total	79	94	48	38	63	322

Expected Frequencies:

Dept.	Math	Physics	Chemistry	Linguistics	English	Total
Men	79X264/322=64.77	94X264/322=77.068	48X264/322=39.354	38X264/322=31.155	63X264/322=51.652	264
Women	79X58/322=14.23	94X58/322=16.952	48X58/322=8.646	38X58/322=6.845	63X58/322=11.348	58
Total	79	94	48	38	63	322

Test Statistic ($\chi ^{2}$) is calculated as follows:

Observed (O)	Expected (E)	(O - E)2/E
69	64.77	0.276
87	77.068	1.28
41	39.354	0.069
24	31.155	1.643
43	51.652	1.449
10	14.23	1.257
7	16.932	5.826
7	8.646	0.313
14	6.845	7.48
20	11.348	6.597
Total = Test Statistic ($\chi ^{2}$)   =		26.191

Test Statistic ($\chi ^{2}$)   = 26.191

(b)

$\alpha$ = 0.025

df = (2 - 1) X (5 - 1) = 4

By Technology,

critical value of $\chi ^{2}$ = 11.143

Since calculated value of $\chi ^{2}$ = 26.191 is greater than critical value of $\chi ^{2}$ = 11.143, the difference is sigificant. Reject null hypothesis.

Conclusion:
The data do not support the claim that the gender of a professor is independent of the department.

Answer to question asked:

Test Statistic ($\chi ^{2}$)   = 26.191