Question

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.

Answer 1

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