Question

A die is rolled 120 times to see if it is fair. The table below shows the frequencies for each of the six possible outcomes. Use a level of significance of αα = 0.05.

1. Complete the rest of the table by filling in the expected frequencies (enter your answers in fraction form):

   Frequency of Dice Values

   Outcome	Frequency	Expected Frequency
   1	17
   2	20
   3	17
   4	27
   5	19
   6	20

Data: 17,20,17,27,19,20

Excel data

Expected Frequency for Outcome 1

Expected Frequency for Outcome 2  

Expected Frequency for Outcome 3

Expected Frequency for Outcome 4

Expected Frequency for Outcome 5  

Expected Frequency for Outcome 6

1. What is the correct statistical test to use?
   Select an answer Goodness-of-Fit Paired t-test Independence Homogeneity  
2. What are the null and alternative hypotheses?
   H0:H0: Select an answer The distribution of die rolls is not uniform. The die rolls are dependent. The distribution of die rolls is uniform. The die rolls are independent.  
   
   
   H1:H1: Select an answer The distribution of die rolls is uniform. The die rolls are independent. The distribution of die rolls is not uniform. The die rolls are dependent.
3. The degrees of freedom =   
   
4. The test-statistic for this data =  (Please show your answer to three decimal places.)  
   
5. The p-value for this sample = (Please show your answer to four decimal places.)  
   
6. The p-value is Select an answer less than (or equal to) greater than  αα  
   
7. Based on this, we should Select an answer reject the null fail to reject the null accept the null  
8. Thus, the conclusion is... Select an answer At the 5% significance level, the data provides sufficient evidence to conclude that the distribution of die rolls is not uniform. At the 5% significance level, the data provides insufficient evidence to conclude that the distribution of die rolls is not uniform.

Answer 1

expected frequency,E

20.00

20.00

20.00

20.00

20.00

20.00

Goodness-of-Fit

H0:The die rolls are independent.

H1:The die rolls are dependent

c)

observed frequencey, O	expected proportion	expected frequency,E	(O-E)		(O-E)²/E
17	0.167	20.00	-3.00		0.450
20	0.167	20.00	0.00		0.000
17	0.167	20.00	-3.00		0.450
27	0.167	20.00	7.00		2.450
19	0.167	20.00	-1.00		0.050
20	0.167	20.000	0.00		0.000

chi square test statistic,X² = Σ(O-E)²/E =   3.400
  
Degree of freedom=k-1=   6   -   1   =   5
                  
P value =   0.6386   [ excel function: =chisq.dist.rt(test-stat,df) ]          

p value > alpha

At the 5% significance level, the data provides insufficient evidence to conclude that the distribution of die rolls is not uniform.

Please let me know in case of any doubt.

Thanks in advance!

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