Question

A researcher wants to study the relationship between salary and gender. She randomly selects 364 individuals and determines their salary and gender. Can the researcher conclude that salary and gender are dependent?

Income Male Female Total

Below $25,000 54    46 100

$25,000-$50,000 41 30    71

$50,000-$75,000 56 41    97

Above $75,000    29 67 96

Total 180    184 364

Step 1: State the null and alternative hypothesis

Step 2: Find the expected value for the number of men with an income below $25,000. Round your answer to one decimal place.

Step 3: Find the expected value for the number of women with an income below $25,000. Round your answer to one decimal place.

Step 4: Find the value of the test statistic. Round your answer to three decimal places.

Step 5: Find the degrees of freedom associated with the test statistic for this problem

Step 6: Find the critical value of the test at the 0.01 level of significance. round your answer to three decimal places.

Step 7: Make the decision to reject or fail to reject the null hypothesis at the 0.01 level of significance

Step 8: State the conclusion of the hypothesis test at the 0.01 level of significance.

Answer 1

applying chi square test:

Expected	Ei=row total*column total/grand total	male	female	Total
	below 25000	49.5	50.5	100
	25000-5000	35.1	35.9	71
	50000-75000	48.0	49.0	97
	>75000	47.5	48.5	96
	total	180	184	364
chi square    χ2	=(Oi-Ei)2/Ei	male	female	Total
	below 25000	0.419	0.409	0.8280
	25000-5000	0.988	0.967	1.9548
	50000-75000	1.345	1.316	2.6613
	>75000	7.188	7.032	14.2198
	total	9.9400	9.7239	19.6639
test statistic X2 =		19.664

Step 1:

ho: salary and gender are independent

Ha: salary and gender are dependent

step 2:

expected value for the number of men with an income below $25,000 =49.5

step 3:

expected value for the number of women with an income below $25,000 =50.5

step 4:

test statistic X2 =

19.664

( please try 19.673 if this comes wrong due to rounding error)

step 5:

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

3

Step 6:

for 3 df and 0.01 level , critical value       χ2=

11.345

Step 7:

Decision rule : reject Ho if value of test statistic X2>11.345

step 8:

reject Ho since test statistic is in critical region,