Question

A new casino game involves rolling 3 dice. The winnings are directly proportional to the total number of sixes rolled. Suppose a gambler brings his own dice and plays the game 100 times, with the following observed results (number of sixes, number of rolls): (0, 48), (1, 35), (2, 15), (3, 3). The casino becomes suspicious of the gambler and asks you to determine whether the dice are fair. If the dice are fair, you would expect the probability of rolling a 6 on any given toss to be 1/6. Assuming that the number of sixes in the three rolls are independent, the number of sixes in three rolls should follow a Binomial Distribution with n=3 and p=1/6. Using a level of significance of 0.01, what is the calculated value of the statistic for goodness of fit? You must use the Binomial Distribution to calculate the number of expected occurrences for each value of x. Remember to group observed and expected frequencies, if necessary. Round answer to 3 significant figures.

Answer 1

Here it is a problem of tesing Goodness of fit and we have to apply test of Goodness of fit at 1% level of significance (l.s)

Here the null hypothesis H0: The number of sixes in three rolls follow a Binomial Distribution with n=3 and p=1/6. Alternate Hypothesis: H1: The number of sixes in three rolls does not follow a Binomial Distribution with n=3 and p=1/6.

Here Test Statistic   = with 3 degrees of freedom(d.f)

Number Six of of 100 throws	Oi	Ei	(Oi - Ei )2	(Oi - Ei )2 / Ei
i=0	48	98.618	2562.182	25.9809
i=1	35	1.376	1130.5734	821.638
i=2	15	0.0064	224.808	35126.256
i=3	2	0.000009923	3.99996	403099.9
Total	100	100	Total	439073.7749

The sum of should have been 100 as total 100 times the dice has been thrown, to make it 100, I am reducing the outcome all three six to "2" . Ei = , i = 0,1,2,3.

The calculated value of test statistic 439073.7749 is falling inside the critical region as it is a right tailed test. (The Critical Value 0.01,3 is 11.345).

Hence the Null Hypothesis: The number of sixes in three rolls follow a Binomial Distribution with n=3 and p=1/6 is rejected.