A die is cast 600 times and the numbers 1 through 6 appear w

A die is cast 600 times and the numbers 1 through 6 appear with the frequencies recorded below.

1 2 3 4 5 6

100 94 103 89 110 104 At the level of significance = 0.1, test the fairness of the die

Solution

A die is fair if the distribution is uniform.

Thus, we expect that the numbers 1-6 have 100 occurrences each.

Thus,

Doing an observed/expected value table,          
O   E   (O - E)^2/E  
100   100   0  
94   100   0.36  
103   100   0.09  
89   100   1.21  
110   100   1  
104   100   0.16  
          
Using chi^2 = Sum[(O - E)^2/E],          
          
chi^2 =    2.82      
          
As df = a - 1,           
          
a =    5      
df = a - 1 =    4      
          
Then, the critical chi^2 value is          
          
significance level =    0.1      
chi^2(crit) =    7.77944034      
          
Also, the p value is          
          
p =    0.588385312      
          
Thus, comparing chi^2 and chi^2(crit) [or, p and significance level], we   FAIL TO REJECT THE NULL HYPOTHESIS.      

Thus, there is no sufficient evidence at 0.10 level that the die is not fair. [CONCLUSION]