Question

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

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

Complete the table. Report all answers accurate to three decimal places.

Category	Observed Frequency	Expected Frequency
A	30
B	4
C	22
D	19

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

What is the P-Value?
P-Value =

For significance level alpha 0.005,

What would be the conclusion of this hypothesis test?

• Reject the Null Hypothesis
• Fail to reject the Null Hypothesis

Answer 1

The null and alternative hypothesis is given as :

: Categories does not occurs with the claim frequency as given in null.

We have given the Observed frequencies for the different categories(A, B,C,D), and we have to calculate the Expected frequency.

The Expected frequency is calculated as-   

where, n is the total number of count and p_i is the probability associated with different categories.

Category	A	B	C	D	Total
Observed frequency	30	4	22	19	75
Expected frequency	30	7.5	22.5	15	75

Chi-square statistic:

Degrees of freedom: ; where k is the number of categories.

So, the chi-square statistic is calculated as

P-value: It gives the probability of observing the test-statistic one which we have calculated or one more extreme under the given condition that the null hypothesis is true.

For the calculated chi-square statistic of and degrees of freedom, df=4-1=3 , p-value is calculated as-

So, the p-value is calculated as

Decision:

Since,

Conclusion: Since we we fail to reject null hypothesis, so we conclude that at the sample data does not provide enough evidence to support the alternative hypothesis H₁. Hence, the four categories (A, B, C, D) occurs with the frequencies that is mention in the null hypothesis.