A die is thought to be fair ie if X denotes the random outco

A die is thought to be fair; i.e. if X denotes the random outcome of a throw of the die, then P(X = j) = 1/6 for each j = 1,..., 6. One hundred and twenty throws of the die are made and the number of times we got 1,2,3,4,5,6, were 16,18,23,24,20,19, respectively. Test for whether this assumption of fairness is justified.

Solution

Null hypothesis: the die is fair

Alternate hypothesis: the die is not fair

If the die is fair expect the frequency of 20 for each numbers.

Goodness of Fit Test

observed

expected

O - E

(O - E)² / E

1

16

20.

-4.000

0.800

2

18

20.

-2.000

0.200

3

23

20.

3.000

0.450

4

24

20.

4.000

0.800

5

20

20.

0.000

0.000

6

19

20.

-1.000

0.050

Total

120

120.

0.000

2.300

2.30

chi-square

5

df

.8063

p-value

Calculated chi square =2.30

Table value for chi square with 5Df at 0.05 level =11.07

Calculated value 2.3 < 11.07, the table value.

The null hypothesis is not rejected.

We conclude that the die is fair.

Goodness of Fit Test
	observed	expected	O - E	(O - E)² / E
1	16	20.	-4.000	0.800
2	18	20.	-2.000	0.200
3	23	20.	3.000	0.450
4	24	20.	4.000	0.800
5	20	20.	0.000	0.000
6	19	20.	-1.000	0.050
Total	120	120.	0.000	2.300
	2.30	chi-square
	5	df
	.8063	p-value