Question

A researcher wants to study the relationship between salary and gender. She randomly selects 384individuals and determines their salary and gender. Can the researcher conclude that salary and gender are dependent?

Income	Male	Female	Total
Below $25,000	31	84	115
$⁢25,000-$50,000	59	45	104
$⁢50,000-$75,000	30	20	50
Above $75,000	65	50	115
Total	185	199	384384

State the null and alternative hypothesis

-Find the expected value for the number of men with an income below $25,000?

Find the expected value for the number of women with an income above $⁢75,000. Round your answer to one decimal place.

-Find the value of the test statistic?

-Find the degrees of freedom associated with the test statistic for this problem?

-Find the critical value of the test at the 0.005 level of significance?

-Make the decision to reject or fail to reject the null hypothesis at the 0.005 level of significance?

-State the conclusion of the hypothesis test at the 0.005 level of significance? (Sufficient evidence or not sufficient evidence?)

Answer 1

Null hypotheisis:Ho:   salary and gender are independent.

alternate hypothesis:Ha: salary and gender are dependent.

  expected value for the number of men with an income below $25,000=(115*185/384)=55.4

expected value for the number of women with an income above $⁢75,000=(115*199/384)=59.6

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	male	female	Total
	<25000	55.40	59.60	115
	25000-50000	50.10	53.90	104
	50000-75000	24.09	25.91	50
	>75000	55.40	59.60	115
	total	185	199	384
chi square    χ2	=(Oi-Ei)2/Ei	male	female	Total
	<25000	10.7491	9.9929	20.7419
	25000-50000	1.5794	1.4683	3.0477
	50000-75000	1.4507	1.3486	2.7993
	>75000	1.6622	1.5452	3.2074
	total	15.4414	14.3550	29.7964
test statistic X2 =		29.796

degree of freedom(df) =(rows-1)*(columns-1)=

3

for 3 df and 0.005 level of signifcance critical value χ2=

12.838

reject the null hypothesis

Sufficient evidence to conclude that

salary and gender are dependent.