Question

A new casino game involves rolling a die. The winnings are directly proportional to the result of the number rolled. Suppose a gambler plays the game 108 times, with the following observed counts:

Observed Number, Observed Frequencies

1 6

2 12

3 24

4 27

5 9

6 30

The casino becomes suspicious of the gambler and wishes to determine whether the die is fair. What would you conclude at = 0:05? Is the test you adopt an exact test or an asymptotic test?

Answer 1

To test if the die is fair, we will conduct Chi square test for Goodness of Fit. This is an exact test.

Step 1:

Ho : The die is fair. Hence proportions of getting all nos (1,2,....6) is equal = 1/6

Ha: Die is not fair. Some of the population proportions differ from the values stated in the null hypothesis.

Step 2: Test statistics:

We will create below table for expected frequencies and squared distances:

fo= observed frequencies

expected frequencies = fe = Total count * proportions

squared distances = (fo - fe)2/ fe

No on dice	Observed values (fo)	Expected Proportions	Expected values (fe)	(fo-fe)2/ fe
1	6	0.167	18.00	8.00
2	12	0.167	18.00	2.00
3	24	0.167	18.00	2.00
4	27	0.167	18.00	4.50
5	9	0.167	18.00	4.50
6	30	0.167	18.00	8.00
Total	108	1.000	108	29.00

= 29 (sum of last coloumn of the above table)

Step 3:

Chi square critical = CHISQ.INV.RT(probability,df) = CHISQ.INV.RT(0.05, 5)= 11.07

As the (29) is greater than critical (11.07), we reject the Null hypothesis.

Hence we have sufficient evidence to belive that the die is not fair and some of the population proportions differ from the values stated in the null hypothesis.