Question

1. Chi-squared :

     X Y Row Marginals

A: 60, 30, 90

B: 18, 40, 58

Column Marginal: 78, 70, 140

Answer 1

Solution:

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

Null hypothesis: H₀: Column and row variables are independent.

Alternative hypothesis: H_a: Column and row variables are not independent.

We assume 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 = 2

Number of columns = c = 2

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

α = 0.05

Critical value = 3.841459

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Observed Frequencies
	Column variable
Row variable	X	Y	Total
A	60	30	90
B	18	40	58
Total	78	70	148

Expected Frequencies
	Column variable
Row variable	X	Y	Total
A	47.43243	42.56757	90
B	30.56757	27.43243	58
Total	78	70	148

Calculations
(O - E)
12.56757	-12.5676
-12.5676	12.56757

(O - E)^2/E
3.329868	3.710425
5.167037	5.757556

Chi square = ∑[(O – E)^2/E] = 17.96489

P-value = 0.0000225

(By using Chi square table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that Column and row variables are not independent.