Question

You are playing a dice game with your friend and he seems to be cheating (either that or you are really bad at this game). You deduce that the dice is not fair. This is, you expect each of the outcomes to be equally likely, but they do not seem to be coming up that way. In order to prove your point, you record the outcomes of 120 different die rolls and obtain the following frequencies. Run a hypothesis test to determine if the dice is truly fair using α = 0.01.

Outcome	1	2	3	4	5	6
Frequency	10	25	30	20	30	5

Answer 1

Chi-square test of goodness of fit

Solution:

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

Null hypothesis: H₀: The dice is truly fair.

Alternative hypothesis: H_a: The dice is not truly fair.

We are given level of significance = α = 0.01

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

N = 6

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

α = 0.01

Critical value = 15.08627

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Outcome	O	E	(O - E)^2	(O - E)^2/E
1	10	20	100	5
2	25	20	25	1.25
3	30	20	100	5
4	20	20	0	0
5	30	20	100	5
6	5	20	225	11.25
Total	120	120		27.5

Chi square = ∑[(O – E)^2/E] = 27.5

P-value = 0.0000456

(By using Chi square table or excel)

P-value < α = 0.01

So, we reject the null hypothesis

There is sufficient evidence to conclude that the dice is not truly fair.

There is insufficient evidence to conclude that the dice is truly fair.