Suppose we want to test to see if a die is fair When the die

Suppose we want to test to see if a die is \"fair.\" When the die is tossed 240 times the following results are obtained:

Value of 1; Frequency of 36

Value of 2; Frequency of 47

Value of 3; Frequency of 35

Value of 4; Frquency of 42

Value of 5; Frequency of 44

Value of 6; Frequency of 36

a) find the expected frequencies if the die is fair

b) calculate the chi-squared statistics

c) find the 5% critical value

d) test the hypothesis that the die is fair

Solution

a)

If the die is fair, then we expect all the frequencies to be equal, that is, 240/6 = 40. [ANSWER]

So,

1:40
2:40
3:40
4:40
5:40
6:40

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B)

Doing an observed/expected value table,          
O   E   (O - E)^2/E  
36   40   0.4  
47   40   1.225  
35   40   0.625  
42   40   0.1  
44   40   0.4  
36   40   0.4  
          
Using chi^2 = Sum[(O - E)^2/E],          
          
chi^2 =    3.15   [ANSWER]  

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C)

As df = a - 1,           
          
a =    6      
df = a - 1 =    5      
          
Then, the critical chi^2 value is          
          
significance level =    0.05      
chi^2(crit) =    11.07049769 [ANSWER]      

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D.

Also, the p value is          
          
p =    0.676872882      
          
Thus, comparing chi^2 and chi^2(crit) [or, p and significance level], we   FAIL TO REJECT THE NULL HYPOTHESIS.      
          
Thus, there is no significant evidence that the die is not fair. [CONCLUSION]