# You are playing a dice game wi

You are playing a dice game with your friend and he seems to becheating (either that or you are really bad at this game). Youdeduce that the dice is not fair. This is, you expect each of theoutcomes to be equally likely, but they do not seem to be coming upthat way. In order to prove your point, you record the outcomes of120 different die rolls and obtain the following frequencies. Run ahypothesis 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

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: Ha: The dice is not trulyfair.

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 expectedfrequencies.

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 nottruly fair.

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

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