Question

Conduct the appropriate test of the specified probabilities using the given information. Use

α = 0.05.

The five categories are equally likely to occur, and the category counts are shown in the table

Category	1		2		3		4		5
Observed Count	48		62		75		50		65

Given:

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 1/5

H_a: At least one p_i is different from 1/5.

FInd:

Find the test statistic. (Round your answer to two decimal places.)

Χ² = ??

Find the rejection region. (Round your answer to two decimal places.)

Χ² > ??

Answer 1

Solution:

Given:

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = 1/5

H_a: At least one p_i is different from 1/5.

Part a) Find the test statistic.

Chi square test statistic for goodness of fit

Where

O_i = Observed Counts

E_i =Expected Counts = N / k = 300 / 5 =60

Thus we need to make following table

Number on Die	Oi: Observed frequency	Ei: Expected frequency	Oi^2/Ei
1	48	60	38.400
2	62	60	64.067
3	75	60	93.750
4	50	60	41.667
5	65	60	70.417
	N = 300

Thus

Part b) Find the rejection region.

Find Chi-square critical value:

df = k - 1 = 5 -1 = 4

Level of significance = 0.05

Chi-square critical value = 9.488 = 9.49

Thus rejection region is:

Reject H0 if   

Since < 9.49, we fail to reject H0.

Thus at 0.05 level of significance , we do not have sufficient evidence to reject the null hypothesis.

Thus we conclude that: the five categories are equally likely to occur