A student rolled a supposedly fair die 100 times, resulting in the distribution

Question

A student rolled a supposedly fair die 100 times, resulting in the distribution of dots shown. Research question: At the 5% significance level, can you reject the hypothesis that the die is fair?

At the 0.05 level of significance, determine if the die is fair. State your hypotheses and show all 7 steps clearly. Be sure to write a conclusion that clearly and completely addresses the research question.

Give and interpret the p-value.

Attach or include your Minitab output.

number of dots on the dice frequency 1 12 2 24 3 15 4 23 5 12 6 14 total 100

Explanation / Answer

Step I : H0 : The die is fair. p = 1/6 = 0.167

Ha : The dice is unfair. p 1/6

Step II : COnfidence level = 0.05

Step III : Here p is the probability of getting a roll of any number i.e. 1,2,3,4,5 or 6

Here the expected value of 100 die rolls.

Number of expected i's = 100/6 = 16.67 = E

Step IV: Here is the expected and chi - square table.

Where x2 = (O - E)2/E

x2 = (12 - 16.67)2 /16.67 + (24 - 16.67)2 /16.67 + (15 - 16.67)2 /16.67 + (23 - 16.67)2 /16.67 + (12 - 16.67)2 /16.67 + (14 - 16.67)2 /16.67 = 8.84

So here X2 = 8.84

Step V : Here Degree of freedom = 6 -1 = 5 and alpha = 0.05

so X2critical  = 11.0705

Step VI: p - value = 0.1156 > 0.05

so X2 <  X2critical  so we shalln't reject the null hypothesis .

Step VII : Conclusion : we can conclude that die is fair.

Roll Observed Expected Chi( O - E)^2/E 1 12 16.6667 1.3067 2 24 16.6667 3.2267 3 15 16.6667 0.1667 4 23 16.6667 2.4067 5 12 16.6667 1.3067 6 14 16.6667 0.4267 Sum 100 100 8.84

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