Question

Are phone calls equally likely to occur any day of the week? The day of the week for each of 392 randomly selected phone calls was observed. The results are displayed in the table below. Use an αα = 0.10 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	55
   Monday	54
   Tuesday	64
   Wednesday	50
   Thursday	58
   Friday	57
   Saturday	54

2. What is the correct statistical test to use?
   Select an answer Paired t-test Independence Goodness-of-Fit Homogeneity  
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.
   • Phone calls and days of the week are dependent.
   • The distribution of phone calls is not uniform over the days of the week.
   
   
   
   H1:H1:
   • The distribution of phone calls is uniform over the days of the week.
   • Phone calls and days of the week are dependent.
   • The distribution of phone calls is not uniform over the days of the week.
   • Phone calls and days of the week are independent.
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.

Hint: Helpful Video [+]
Help

Answer 1

Solution:

Here, we have to use chi square test for goodness of fit.

Null hypothesis: H₀: Phone calls are equally likely to occur any day of the week.

Alternative hypothesis: H_a: Phone calls are not equally likely to occur any day of the week.

We are given level of significance = α = 0.10

Test statistic formula is given as below:

Chi square = ∑[(O – E)^2/E]

Where, O is observed frequencies and E is expected frequencies.

We are given

N = 7

Degrees of freedom = df = N – 1 = 7 – 1 = 6

α = 0.10

Critical value = 10.645

(by using Chi square table or excel)

Calculation tables for test statistic are given as below:

Day	O	E	(O - E)^2/E
Sunday	55	56	0.017857143
Monday	54	56	0.071428571
Tuesday	64	56	1.142857143
Wednesday	50	56	0.642857143
Thursday	58	56	0.071428571
Friday	57	56	0.017857143
Saturday	54	56	0.071428571
Total	392	392	2.035714286

Chi square = ∑[(O – E)^2/E] = 2.035714286

Chi square test statistic = 2.036

P-value = 0.9164

(By using Chi square table or excel)

P-value > α = 0.10

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that Phone calls are equally likely to occur any day of the week.

What is the correct statistical test to use?

Answer: Goodness-of-Fit

What are the null and alternative hypotheses?

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

.

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

The degrees of freedom =  df = N – 1 = 7 – 1 = 6

The test-statistic for this data = 2.036

The p-value for this sample = 0.9164

The p-value is greater than α.

Based on this, we should fail to reject the null

Thus, the final conclusion is

• There is sufficient evidence to conclude that the distribution of phone calls is uniform over the days of the week.