Question

Jeff and Mark are trying to decide whether a die is fair. They roll it 100 times, with the results
shown below:
19 ones, 13 twos, 15 threes, 21 fours, 18 fives, 14 sixes.
Average ≈ 3.43, SD ≈ 1.76.
Jeff wants to make a z-test, Mark wants to make a χ2 -test. Who is right? Explain briefly. Then, calculate the p-value of the respective test. Is the die fair?

Answer 1

Solution:

For the given scenario, Mark is right because we need to use the Chi-square test for the goodness of fit for checking the claim whether the die is fair or not.

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

Null hypothesis: H0: Data follows the uniform distribution. (Die is fair.)

Alternative hypothesis: Ha: Data do not follow the uniform distribution. (Die is not fair.)

We assume level of significance = α = 0.05

We are given

Number of categories = N = 6

Degrees of freedom = df = N - 1 = 5

α = 0.05

Critical value = 11.07049775

(by using Chi square table or excel)

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

Calculation tables for test statistic are given as below:

Result	O	E	(O - E)^2/E
1	19	16.66667	0.326666667
2	13	16.66667	0.806666667
3	15	16.66667	0.166666667
4	21	16.66667	1.126666667
5	18	16.66667	0.106666667
6	14	16.66667	0.426666667
Total	100	100	2.96

Test Statistic = Chi square = ∑[(O – E)^2/E] = 2.96

χ2 = 2.96

P-value = 0.706152775

(By using Chi square table or excel)

P-value > α = 0.05

So, we do not reject the null hypothesis.

There is sufficient evidence to conclude that the Die is fair.