Question

Use software to test the null hypothesis of whether there is a relationship between the two classifications, A and B, of the 3×3 contingency table shown below.

	B1	B2	B3	Total
A1	44	68	42	154
A2	53	70	48	171
A3	72	56	75	203
Total	169	194	165	528

Test using ?=0.05.

(a) ?2=

(b) Find the degrees of freedom.

(c) Find the critical value. ?2=

Answer 1

Solution:

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

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: There is no relationship between the two classifications A and B.

Alternative hypothesis: H_a: There is a relationship between the two classifications A and B.

We are given

Level of significance = ? = 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

Excel output for this test is given as below:

Chi-Square Test
Observed Frequencies
	Column variable					Calculations
Row variable	B1	B2	B3	Total		(O - E)
A1	44	68	42	154		-5.29167	11.41667	-6.125
A2	53	70	48	171		-1.73295	7.170455	-5.4375
A3	72	56	75	203		7.024621	-18.5871	11.5625
Total	169	194	165	528
Expected Frequencies
	Column variable
Row variable	B1	B2	B3	Total		(O - E)^2/E
A1	49.29167	56.58333	48.125	154		0.568083	2.30351	0.779545
A2	54.73295	62.82955	53.4375	171		0.054869	0.818332	0.553289
A3	64.97538	74.58712	63.4375	203		0.759446	4.631913	2.107451
Total	169	194	165	528
Data
Level of Significance	0.05
Number of Rows	3
Number of Columns	3
Degrees of Freedom	4
Results
Critical Value	9.487729
Chi-Square Test Statistic	12.57644
p-Value	0.013542
Reject the null hypothesis
Expected frequency assumption
is met.

Answers:

Part a

Chi square = 12.57644

(From above output)

Part b

Number of columns = c = 3

Number of rows = r = 3

Degrees of freedom = (c – 1)*(r – 1)

Degrees of freedom = (3 – 1)*(3 – 1)

Degrees of freedom = 2*2 = 4

Degrees of freedom = 4

Part c

We are given

Level of significance = ? = 0.05

Degrees of freedom = 4

So,

Critical value = 9.487729

(By using Chi square table or excel)