Two dice are thrown together 360 times and the total score is recorded for each

Question

Two dice are thrown together 360 times and the total score is recorded for each throw. The possible totals are 2, 3, 4...12 and their number of occurrences are shown below.   Calculate the probabilities of each total, and hence the number of expected occurrences for each, assuming the die are fair. At the 5% level of significance, can you reject the hypothesis that the die are fair? At the 1% level?

Identify the type of problem and statistics to be used and justify/explain your answer.

totals occurrences 2 11 3 22 4 30 5 40 6 45 7 64 8 45 9 40 10 30 11 22 12 11

Explanation / Answer

for dice probabilty of sum P(X=2)=1/36

P(X=3)=2/36

P(X=4)=3/36

P(X=5)=4/36

P(X=6)=5/36

P(X=7)=6/36

P(X=8)=5/36

P(X=9)=4/36

P(X=10|)=3/36

P(X=11)=2/36

P(X=12)=1/36

hence we will apply chi square goodness of fit test on above:

for above chi stat =1.8667

degree of freedom =n-1 =11-1=10

for 10 df and above chi stat p value =0.9973

as p vlaue is very high we can not reject the null hypothesis that die is fair.

observed Expected Chi square category Probability O E=total*p =(O-E)^2/E 2 1/36 11.00 10.00 0.10 3 1/18 22.00 20.00 0.20 4 1/12 30.00 30.00 0.00 5 1/9 40.00 40.00 0.00 6 5/36 45.00 50.00 0.50 7 1/6 64.00 60.00 0.27 8 5/36 45.00 50.00 0.50 9 1/9 40.00 40.00 0.00 10 1/12 30.00 30.00 0.00 11 1/18 22.00 20.00 0.20 12 1/36 11.00 10.00 0.10 360 360 1.8667

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