Question

1. You have a set of three dice.(12 marks, 4 marks each)
(a) Calculate the probability of getting 0, 1, 2 and 3 sixes if the
dice are fair.
(b) Suppose you roll the three dice 100 times. The results are
# of 6s # of times
0 47
1 35
2 15
3 3
Test the null hypthesis that the dice are fair at α = .01 significance level.

Answer 1

1)

Let the random variable X is the number of 6 in three dice.

The random variable, X follows binomial distribution with n = 3 and p = 1/6

For calculation purpose, probabilities are obtained in excel as shown below,

X	P(X)	Excel function
0	0.578704	BINOM.DIST(0,3,1/6,FALSE)
1	0.347222	BINOM.DIST(1,3,1/6,FALSE)
2	0.069444	BINOM.DIST(2,3,1/6,FALSE)
3	0.00463	BINOM.DIST(3,3,1/6,FALSE)

2)

If three dice roll 100 times, expected number of 6s are obtained by multiplying probability by 100

X	Probability	Expected value
0	0.5787	57.8704 58
1	0.3472	34.7222 35
2	0.0694	6.9444 7
3	0.0046	0.4630 0

The observed values are

X	Observed value
0	47
1	35
2	15
3	3

The Chi-Square goodness of fit test is used here to test whether the sample are from same population or we can say whether the observed value of frequency are significantly different from the expected value.

The Null hypothesis is defined as,

Null hypothesis: There is no significant difference between the observed proportion and the expected proportion.

The Chi-Square statistic is,

X	Observed ,	Expected ,
0	47	57.87	-11	118	2.04189
1	35	34.72	0	0	0.002222
2	15	6.94	8	65	9.344444
3	3	0.46	3	6	13.90296
					25.29152

The chi square critical value is obtained from chi square distribution table for = k - 1 = 4 - 1 = 3

Since,

at 1% significance level. it can be concluded that the null hypothesis is rejected.