Question

In: Statistics and Probability

A new casino game involves rolling 2 dice. The winnings are directly proportional to the total...

A new casino game involves rolling 2 dice. The winnings are directly proportional to the total number of sixes rolled. Suppose a gambler plays the game 100 times, with 0,1 and 2 sixes observed 40, 30, 30 times respectively. Do you reject the hypothesis H0: that the dice are fair at 5% level of significance? Use the fact that P(χ2^2>5.99) = 0.05.

Solutions

Expert Solution

here from binomial distribution with parameter n=2 and p=1/6

P(X=0 six rolled )=(2C0)*(1/6)^0*(5/6)^2=25/36 =0.6944

P(X=1 six rolled) =(2C1)*(1/6)^1*(5/6)^1=10/36 =0.2778

P(X=2 six rolled) =(2C2)*(1/6)^2*(5/6)^0=1/36 =0.0278

degree of freedom =categories-1= 2
for 0.05 level and 2 df :crtiical value X2 = 5.991
Decision rule: reject Ho if value of test statistic X2>5.991
applying chi square goodness of fit test:
           relative observed Expected residual Chi square
category frequency(p) Oi Ei=total*p R2i=(Oi-Ei)/√Ei R2i=(Oi-Ei)2/Ei
0 0.6944 40.0 69.44 -3.53 12.484
1 0.2778 30.0 27.78 0.42 0.178
2 0.0278 30.0 2.78 16.33 266.778
total 1.000 100 100 279.4400
since test statistic falls in rejection region we reject null hypothesis
we have sufficient evidence to conclude that dice are not fair at 5% level of significance

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