Question

A researcher wants to study the relationship between salary and gender. She randomly selects 368 individuals and determines their salary and gender. Can the researcher conclude that salary and gender are dependent?

Income	Male	Female	Total
Below $25,000	57	60	117
$⁢25,000-$50,000	48	44	92
$⁢50,000-$75,000	48	45	93
Above $75,000	3131	35	66
Total	184	184	368

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Step 1 of 8:

State the null and alternative hypothesis.

Step 2 of 8:

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

Step 3 of 8:

Find the expected value for the number of women with an income of $⁢25,000-$50,000. Round your answer to one decimal place.

Step 4 of 8:

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

Step 5 of 8:

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

Step 6 of 8:

Find the critical value of the test at the 0.005 level of significance. Round your answer to three decimal places.

Step 7 of 8:

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

Step 8 of 8:

State the conclusion of the hypothesis test at the 0.005 level of significance.

Answer 1

1) The null and alternative hypothesis is

H0: There is no relationship between salary and gender.

H1: There is a relationship between salary and gender.

2)

The formula for expected value.

E = ( Row total*Column total) / Grand total

E1 = (184*117)/368 = 58.5

3)

E4 = (184*92)/368 = 46

4)

Test statistic is

$\chi ^{2}=\sum \frac{(O-E)^{2}}{E}$

O: Observed frequency
E: Expected frequency.
E = ( Row total*Column total) / Grand total

O	E	(O-E)	(O-E)^2	(O-E)^2/E
57	58.5	-1.5	2.25	0.038462
60	58.5	1.5	2.25	0.038462
48	46	2	4	0.086957
44	46	-2	4	0.086957
48	46.5	1.5	2.25	0.048387
45	46.5	-1.5	2.25	0.048387
31	33	-2	4	0.121212
35	33	2	4	0.121212
			Total	0.590

(o- E)2

5)

Degrees of freedom = ( Number of rows - 1 ) * ( Number of column - 1) = ( 4 - 1) * (2 - 1) = 3 * 1 = 3

6)

Level of significance = 0.005

Critical value = 12.838

( From chi-square table)

7)

Test statistic < critical value we fail to reject null hypothesis.

8)

Conclusion:

There is no relationship between salary and gender.