Chapter 24, Problem 31RE

(a)

To determine

To find: The two-way table of time of day versus answered or not and also the percentage of calls answered in each time period.

Answer to Problem 31RE

Solution: The result obtained is as follows:

	Weekday morning	Weekday evening	Total
Answered	1313	1840	3153
Unanswered	991	614	1605
Total	2304	2454	Grand Total = 4758

The percentage of calls answered on weekday morning and weekday evening is 57.0% and 75.0%, respectively.

Explanation of Solution

Calculation: Compute the row and column total as follows:

	Weekday morning	Weekday evening	Total
Answered	1313	1840	1313+1840 = 3153
Unanswered	2304-1313 = 991	2454-1840 = 614	991+614 = 1605
Total	2304	2454	Grand Total = 4758

The number of answered calls on weekday morning is 1313 and on weekday evening is 1840. The number of unanswered calls on weekday morning is 991 and on weekday evening is 614. The total answered calls on weekday morning and weekday evening is 3153 and total unanswered calls on weekday morning and weekday evening is 1605. The grand total is 4758. Now, compute the percentage of answered call in weekday morning as shown below:

$\begin{array}{c}\left(\begin{array}{l}\text{ Percentage of answered}\\ \text{calls on weekday morning}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of calls answered}\\ \text{ on weekday morning}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of call}\\ \text{on weekday morning}\end{array}\right)}\\ =\frac{1313}{2304}\\ =0.5699\\ \approx 57.0\%\end{array}$

Similarly, compute the percentage of answered call in weekday evening as shown below:

$\begin{array}{c}\left(\begin{array}{l}\text{ Percentage of answered}\\ \text{calls on weekday evening}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of calls answered}\\ \text{ on weekday evening}\end{array}\right)}{\left(\begin{array}{l}\text{Total number of call}\\ \text{on weekday evening}\end{array}\right)}\\ =\frac{1840}{2454}\\ =0.7498\\ \approx 75.0\%\end{array}$

Interpretation: Therefore, it can be concluded number of calls answered on weekday morning is less in comparison to the number of calls answered on weekday evening.

(b)

To determine

To explain: Why there should be significant relationship between time of day and answering the call.

Answer to Problem 31RE

Solution: The relationship will be significant because there is a huge difference in the percentage of calls answered in morning and evening.

Explanation of Solution

The percentage of calls answered on weekday morning and weekday evening obtained in section 2 or part (a) is 57.0% and 75.0%, respectively.

It can be seen that there is a big difference in the percentage of calls answered. The most of the calls were answered in the weekday evening which infers that the time of the calls matters. Thus, the percent of calls answered in the evening versus morning are very different.

(c)

To determine

To test: The chi-square test on the referred data.

Answer to Problem 31RE

Solution: The result obtained after carrying out chi-square test is as follows:

$\begin{array}{c}\text{Test statistics}\left({\chi }^2\right)=172.08\\ \text{Degrees of freedom}\left(df\right)=1\\ p-\text{value}=0.000\end{array}$

Yes, it can be said that there is a significant relationship between time of the day and number of calls answered.

Explanation of Solution

Calculation: The two-way table obtained in section 1 of part (a) is as follows:

	Weekday morning	Weekday evening	Total
Answered	1313	1840	3153
Unanswered	991	614	1605
Total	2304	2454	Grand Total = 4758

The null hypothesis $\left(H_0\right)$ and the alternative hypothesis $\left(H_a\right)$ is framed as shown below:

$H_0$: There is no significant relationship between the time of the day and the number of calls answered.

$H_a$: There is significant relationship between the time of the day and the number of calls answered.

Now, perform a chi-square test using Minitab as follows:

Step 1: Enter only the number of answered and unanswered call on weekday morning and evening in Minitab.

Step 2: Click on Stat $\to$ Tables $\to$ Chi-square test.

Step 3: Double click on “Weekday morning” and “Weekday evening” to move it to variable column.

Step 4: Click “OK” to obtain the result.

The result obtained is as follows:

$\begin{array}{c}\text{Test statistics}\left({\chi }^2\right)=172.08\\ \text{Degrees of freedom}\left(df\right)=1\\ p-\text{value}=0.000\end{array}$

For 5% level of significance, the decision rule states that, if the P-value is less than 0.05, then the null hypothesis is rejected.

Conclusion: Since the P-value $\left(0.000\right)$ is less than the significance level 0.05, the decision is to reject the null hypothesis. Therefore, it can be said that there is enough evidence to conclude that there is a significant relationship between the time of the day and the number of calls answered.