Question

You are conducting a multinomial Goodness of Fit hypothesis test for the claim that the 4 categories occur with the following frequencies:

HoHo : pA=0.4pA=0.4;  pB=0.2pB=0.2;  pC=0.1pC=0.1;  pD=0.3

CATEGORY	OBSERVED FREQUENCY	EXPECTED FREQUENCY	RESIDUAL
A	41
B	9
C	25
D	47

What is the chi-square test-statistic for this data?
χ2

For significance level alpha 0.025, what is the chi-square Critical Value?

Answer 1

Solution:

Here, we have to use chi square test for goodness of fit.

Null hypothesis: H₀: The four categories follow the given distribution.

Alternative hypothesis: H_a: The four categories do not follow the given distribution.

We are given level of significance = α = 0.025

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

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

We are given

N = 4

Degrees of freedom = df = N – 1 = 4 – 1 = 3

α = 0.025

Critical value = 9.348404

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Category	Expected Proportion	Observed Frequency O	Expected Frequency E	Residual (O - E)^2/E
A	0.4	41	48.8	1.246721311
B	0.2	9	24.4	9.719672131
C	0.1	25	12.2	13.4295082
D	0.3	47	36.6	2.955191257
Total	1	122	122	27.3510929

Test statistic = Chi square = ∑[(O – E)^2/E] = 27.3510929

P-value = 0.000005

(By using Chi square table or excel)

P-value < α = 0.025

So, we reject the null hypothesis

There is insufficient evidence to conclude that the four categories follow the given distribution.