Question

A researcher wants to study the relationship between salary and gender. She randomly selects 215 individuals and determines their salary and gender. Can the researcher conclude that salary and gender are dependent? Male Female Total 22 20 42 Income Below $25,000 $25,000-$50,000 $50,000-$75,000 Above $75,000 23 20 43 31 34 65 34 31 Total 110 TOS 215 < Copy Data Prev Next Step 6 of 8: Find the critical value of the test at the 0.05 level of significance. Round your answer to three decimal places

Answer 1

Answer:

Critical value = 7.815

Solution:

Here, we have to use chi square test for independence of two categorical variables.

Null hypothesis: H₀: Two categorical variables salary and gender are independent.

Alternative hypothesis: H_a: Two categorical variables salary and gender are dependent.

We are given level of significance = α = 0.05

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

E = row total * column total / Grand total

We are given

Number of rows = r = 4

Number of columns = c = 2

Degrees of freedom = df = (r – 1)*(c – 1) =3*1 = 3

α = 0.05

Critical value = 7.815

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Gender
Income	Male	Female	Total
Below $25k	22	20	42
$25k to $50k	23	20	43
$50k to $75k	31	34	65
above $75k	34	31	65
Total	110	105	215

Expected Frequencies
	Gender
Income	Male	Female	Total
Below $25k	21.48837	20.51163	42
$25k to $50k	22	21	43
$50k to $75k	33.25581	31.74419	65
above $75k	33.25581	31.74419	65
Total	110	105	215

Calculations
(O - E)
0.511628	-0.51163
1	-1
-2.25581	2.255814
0.744186	-0.74419

(O - E)^2/E
0.012182	0.012762
0.045455	0.047619
0.153017	0.160303
0.016653	0.017446

Test Statistic = Chi square = ∑[(O – E)^2/E] = 0.465436

χ² statistic = 0.465436

P-value = 0.926417

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that two categorical variables salary and gender are dependent.

The researcher cannot conclude that salary and gender are dependent.