Question

5. A dle was rolled 90 times. The results were: Observed # Frequency 1 20 2 22 3 10 4 8 5 13 6 17 Find the test- statistic that would be used to determine whether the die is a goodness of fit or is loaded.

Answer 1

Step 1:

Ho: The dice is fair. i.e. proportion of each no = 1/6

Ho: Some of the population proportions differ from the values stated in the null hypothesis

This corresponds to a Chi-Square test for Goodness of Fit.

Step 2: Test statistics

Data	Observed values (fo)	Expected Proportions	Expected values (fe)	(fo-fe)2/ fe
1	20	0.167	15.00	1.667
2	22	0.167	15.00	3.267
3	10	0.167	15.00	1.667
4	8	0.167	15.00	3.267
5	13	0.167	15.00	0.267
6	17	0.167	15.00	0.267
Total	90	1.000	90.00	10.400

$\chi^{2}$ = 10.40

Step 3:

df = 6-1 = 5

As the level of significance is not given, we will take alpha = 0.05

$\chi^{2}$ critical = 11.070

As $\chi^{2}$ (10.40) is less than $\chi^{2}$ critical, we fail to reject the Null hypothesis.

Hence at alpah = 0.05, we do not have sufficient evidence to believe that the dice is loaded.