Question

Suppose we have multinomial population with four categories: A,B,C, and D. The null hypothesis is that the proportion of items is the same in every category. The null hypothesis is H0: Pa =Pb=Pc=Pd =0.25 A sample of size 300 yielded the following results. A:80 B:85 C:60 D:70 Use alpha=0.05, determine whether H0 should be rejected?

Answer 1

Solution:

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

Null hypothesis: H0: Pa =Pb=Pc=Pd =0.25

Alternative hypothesis: Ha: At least one proportion is different.

We assume/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.

We are given

K = 4

Degrees of freedom = df = N - 1 = 3

α = 0.05

Critical value = 7.814727764

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

	O	E	(O - E)^2/E
A	80	73.75	0.529661017
B	85	73.75	1.716101695
C	60	73.75	2.563559322
D	70	73.75	0.190677966
Total	295	295	5

Test Statistic = Chi square = ∑[(O – E)^2/E] = 5

χ2 statistic = 5

P-value = 0.171797124

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis

H0 should not be rejected.

There is sufficient evidence to conclude that percentage of four categories is same.