Question

TEST THE APPROPRIATE HYPOTHESIS. Include the null and alternate hypotheses, degrees of freedom, test statistic, rejection region, and decision.

You roll a die 48 times. the results as followed

Number 1 2 3 4 5 6

Frequency 4 13 2 14 13 2

Use a significance level of 0.05 to test the claim that the die is fair

Answer 1

Solution:

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

Null hypothesis: H0: The die is fair.

Alternative hypothesis: Ha: The die is not fair.

We are 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

Number of outcomes = N = 6

Degrees of freedom = df = N – 1 = 6 – 1 = 5

α = 0.05

Critical value = 11.07049775

(by using Chi square table or excel)

Rejection region: Reject H0 if χ2 statistic > 11.07049775

Calculation tables for test statistic are given as below:

Outcome	O	E	(O - E)^2/E
1	4	8	2
2	13	8	3.125
3	2	8	4.5
4	14	8	4.5
5	13	8	3.125
6	2	8	4.5
Total	48	48	21.75

Test Statistic = Chi square = ∑[(O – E)^2/E] = 21.75

χ2 statistic = 21.75

P-value = 0.000584099

(By using Chi square table or excel)

P-value < α = 0.05

χ2 statistic > Critical value = 11.07049775

So, we reject the null hypothesis

Decision: Reject the null hypothesis

There is not sufficient evidence to conclude that the die is fair.