Question

Question 22

The following table lists the frequency distribution for 60 rolls of a die.
Outcome 1-spot 2-spot 3-spot 4-spot 5-spot 6-spot
Frequency 7 12 8 15 11 7
Test at the 5% significance level whether the null hypothesis that the given die is fair is true. Determine the critical value for the above test.
(A) 11.143
(B) 14.449
(C) 12.833
(D) 11.070
(E) 12.592
(E)
(A)
(D)
Refer to Question 22.
Test at the 5% significance level whether the null hypothesis that the given die is fair is true. Which of the following is a proper conclusion?
(A) There is evidence that the die is fair.
(B) There is no evidence that the die is fair.
(C) There is evidence at the 10% significance level, but not at the 5% significance level.
(D) There is no evidence at both 10% significance level and 5% significance level.
(E) There is evidence at the 5% significance level.
(A)
(C)
(E)
(B)
(D)

Answer 1

Final Answers:

1. Critical value for the test: (C) 12.833

2. Proper conclusion for the test: (E) There is evidence at the 5% significance level.

Step-by-step explanation:

To test the fairness of the die, a chi-square goodness-of-fit test can be employed at a 5% significance level. The critical value for this test, based on the given frequency distribution and degrees of freedom (df = 6 - 1 = 5), is 12.833, which corresponds to the 5% significance level. By comparing the calculated chi-square statistic from the observed frequencies to the critical value, one can determine whether to reject the null hypothesis. If the calculated chi-square value exceeds 12.833, there is evidence to reject the null hypothesis, indicating that the die is not fair at the 5% significance level.

In this scenario, the proper conclusion based on the calculated chi-square statistic should be chosen. Since the test statistic surpasses the critical value of 12.833, there is evidence at the 5% significance level to reject the null hypothesis. Thus, the proper conclusion is that there is evidence at the 5% significance level that the given die is not fair based on the observed frequencies. This conclusion aligns with statistical significance and supports the rejection of the null hypothesis, suggesting the die is biased. The critical value aids in determining the threshold for significance, and the statistical test results confirm the evidence against the fairness of the die.