Question

Are phone calls equally likely to occur any day of the week? The day of the week for each of 525 randomly selected phone calls was observed. The results are displayed in the table below. Use an αα = 0.05 significance level.

1. Complete the rest of the table by filling in the expected frequencies:

   Frequencies of Phone Calls for Each Day of the Week

   Outcome	Frequency	Expected Frequency
   Sunday	80
   Monday	73
   Tuesday	69
   Wednesday	71
   Thursday	80
   Friday	77
   Saturday	75

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

Answer 1

The correct statistical test is Goodness of fit test.

H0: the distribution of phone calls is uniform over the days of week.

H1: the distribution of phone calls is not uniform over the days of week.

Categories Observed Expected Sun 80 525*0.142857=75 Mon 73 525*0.142857=75 Tue 69 525*0.142857=75 Wed 71 525*0.142857=75 Thurs 80 525*0.142857=75 Fri 77 525*0.142857=75 Sat 75 525*0.142857=75 Sum 525 525

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: = 0.142857, P3 = 0.142857, P4 = 0.142857, p5 = 0.142857, po = H. : P1 = 0.142857, P2 0.142857, P7 = 0.142857 Ha: 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. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, the number of degrees of freedom is df = 7–1 = 6, so then the rejection region for this test is R= {x? : x² > 12.592}.

(3) Test Statistics The Chi-Squared statistic is computed as follows: n x (O; – E;) = Σ = = 0.333 + 0.053 + 0.48 +0.213 + 0.333 + 0.053 +0= 1.467 Ei i=1 (4) Decision about the null hypothesis Since it is observed that x? = 1.467 < xã = 12.592, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the a = 0.05 significance level.

Ans: there is sufficient evidence to conclude that the distribution of phone calls is uniform over the days of the week.

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