Question

Perform the indicated goodness-of-fit test.

You roll a die 48 times with the following results.

a. test statistic

b. critical value
Use a significance level of 0.05 to test the claim that the die is fair.

Answer 1

Solution:

Given: Level of significance = 0.05

Claim: The die is fair.

Hypothesis of the study are:

H₀: The die is fair.

Vs

H₁: The die is not fair.

Part a) Test statistic:

Chi square test statistic for goodness of fit

AN (=^

Where

O_i = Observed Counts

E_i =Expected Counts = N / k = 48 / 6 = 8

Thus we need to make following table

Number	Oi: Observed Frequency	Ei: Expected Frequencies	Oi^2/Ei
1	2	8	0.5
2	4	8	2
3	12	8	18
4	13	8	21.125
5	14	8	24.5
6	3	8	1.125
	N = 48		어 = 67.25

Thus

AN (=^

x = 67.250 - 48

X2 = 19.250

Part b) Critical value:

df = k - 1 = 6- 1 = 5

Level of significance = 0.05

X3995 X2990 X2975 x2050 0.000 0.010 0.072 0.207 0.000 0.020 0.115 0.297 0554 0.001 0.051 0.216 0.484 0.831 X2950 0.004 0.103 0.352 0.711 1.145 X2000 0.016 0.211 0.584 1.064 1.610 X2200 2.706 4.605 6.251 7.779 9.236 3. 41 5.901 7.815 9.488 11.070

Thus chi-square critical value = 11.070

Step 4) Decision Rule:

Reject null hypothesis H0, if Chi square test statistic > Chi-square critical value = 11.070 , otherwise we fail to reject H0.

Since Chi square test statistic = > Chi-square critical value = 11.070 , we reject null hypothesis H0.

X2 = 19.250

Conclusion: There is not sufficient evidence to support the claim that the die is fair.