Answer :
### Part (a) Calculate the Mean, Median, and Modal Values of the Length of Calls
To determine the mean, median, and modal values, we will use the given data:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Length of call (nearest minute)} & \text{No. of calls} \\ \hline 10 \text{ and under } 12 & 40 \\ 12 \text{ and under } 14 & 109 \\ 14 \text{ and under } 16 & 54 \\ 16 \text{ and under } 18 & 22 \\ 18 \text{ and under } 20 & 15 \\ \hline \end{tabular} \][/tex]
1. Mean Length of Calls:
The mean length can be calculated using the sum of the midpoints of each interval multiplied by the number of calls in each interval:
- Midpoints: (11, 13, 15, 17, 19)
- Calls: (40, 109, 54, 22, 15)
[tex]\[ \text{Mean Length} = \frac{\sum (\text{Midpoint} \times \text{No. of calls})}{\sum \text{No. of calls}} = \frac{(11 \times 40) + (13 \times 109) + (15 \times 54) + (17 \times 22) + (19 \times 15)}{40 + 109 + 54 + 22 + 15} \][/tex]
[tex]\[ = \frac{440 + 1417 + 810 + 374 + 285}{240} \][/tex]
[tex]\[ = \frac{3326}{240} \approx 13.86 \][/tex]
So, the mean length of calls is approximately 13.86 minutes.
2. Median Length of Calls:
To find the median:
- Total number of calls: 240
- The median occurs around the 120th call.
The cumulative frequency helps in identifying the median interval:
- Cumulative calls: 40, 149, 203, 225, 240
Since 120 falls within the second interval (12 to under 14):
[tex]\[ \text{Median Length} = \text{Midpoint of the second interval} = 13 \text{ minutes} \][/tex]
So, the median length of calls is 13 minutes.
3. Modal Length of Calls:
The mode is the length of calls corresponding to the highest frequency. The highest frequency is 109 calls, which is for the interval 12 and under 14 minutes.
So, the modal length of calls is 12 to under 14 minutes.
### Part (b) Drawing the Histogram and Estimating the Mode
To draw the histogram, plot the length intervals on the x-axis and the number of calls on the y-axis. The highest bar (interval with the highest number of calls) shows the mode: 12 to under 14 minutes.
#### But, since drawing is not supported by this medium, here we will just describe:
- The first interval (10 to under 12) has 40 calls.
- The second interval (12 to under 14) has 109 calls.
- The third interval (14 to under 16) has 54 calls.
- The fourth interval (16 to under 18) has 22 calls.
- The fifth interval (18 to under 20) has 15 calls.
Estimated Mode: From the histogram, the highest bar represents the mode, which confirms the modal length as 12 to under 14 minutes.
### Part (c) Drawing the 'Less Than' Cumulative Frequency Polygon and Estimating the Median
#### For cumulative frequency polygon:
- Plot cumulative frequencies at the upper boundary of each interval.
- Connect these points with a straight line.
Cumulative frequencies:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Upper Boundary} & \text{Cumulative Frequency} \\ \hline 12 & 40 \\ 14 & 149 \\ 16 & 203 \\ 18 & 225 \\ 20 & 240 \\ \hline \end{tabular} \][/tex]
Connecting these points will illustrate that the median corresponds to 120 calls, which falls in the interval 12 to under 14 minutes.
Estimated Median: This confirms the median as approximately 13 minutes.
### Part (d) Estimation of Total Telephone Charges
To calculate the total telephone charges, we need to use the mean length of calls, as it provides an average duration.
[tex]\[ \text{Mean Length} = 13.86 \text{ minutes} \][/tex]
[tex]\[ \text{Charge per Minute} = \$0.30 \][/tex]
[tex]\[ \text{Total Charges} = \text{Mean Length} \times \text{Total Number of Calls} \times \text{Charge per Minute} \][/tex]
[tex]\[ = 13.86 \times 240 \times 0.30 = 72.048 \approx 72 \text{ dollars} \][/tex]
So, the total telephone charges are estimated to be \$72 based on the mean length of calls. The mean is most suitable for this estimation because it accounts for all values in the data set, providing a comprehensive average charge.
To determine the mean, median, and modal values, we will use the given data:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Length of call (nearest minute)} & \text{No. of calls} \\ \hline 10 \text{ and under } 12 & 40 \\ 12 \text{ and under } 14 & 109 \\ 14 \text{ and under } 16 & 54 \\ 16 \text{ and under } 18 & 22 \\ 18 \text{ and under } 20 & 15 \\ \hline \end{tabular} \][/tex]
1. Mean Length of Calls:
The mean length can be calculated using the sum of the midpoints of each interval multiplied by the number of calls in each interval:
- Midpoints: (11, 13, 15, 17, 19)
- Calls: (40, 109, 54, 22, 15)
[tex]\[ \text{Mean Length} = \frac{\sum (\text{Midpoint} \times \text{No. of calls})}{\sum \text{No. of calls}} = \frac{(11 \times 40) + (13 \times 109) + (15 \times 54) + (17 \times 22) + (19 \times 15)}{40 + 109 + 54 + 22 + 15} \][/tex]
[tex]\[ = \frac{440 + 1417 + 810 + 374 + 285}{240} \][/tex]
[tex]\[ = \frac{3326}{240} \approx 13.86 \][/tex]
So, the mean length of calls is approximately 13.86 minutes.
2. Median Length of Calls:
To find the median:
- Total number of calls: 240
- The median occurs around the 120th call.
The cumulative frequency helps in identifying the median interval:
- Cumulative calls: 40, 149, 203, 225, 240
Since 120 falls within the second interval (12 to under 14):
[tex]\[ \text{Median Length} = \text{Midpoint of the second interval} = 13 \text{ minutes} \][/tex]
So, the median length of calls is 13 minutes.
3. Modal Length of Calls:
The mode is the length of calls corresponding to the highest frequency. The highest frequency is 109 calls, which is for the interval 12 and under 14 minutes.
So, the modal length of calls is 12 to under 14 minutes.
### Part (b) Drawing the Histogram and Estimating the Mode
To draw the histogram, plot the length intervals on the x-axis and the number of calls on the y-axis. The highest bar (interval with the highest number of calls) shows the mode: 12 to under 14 minutes.
#### But, since drawing is not supported by this medium, here we will just describe:
- The first interval (10 to under 12) has 40 calls.
- The second interval (12 to under 14) has 109 calls.
- The third interval (14 to under 16) has 54 calls.
- The fourth interval (16 to under 18) has 22 calls.
- The fifth interval (18 to under 20) has 15 calls.
Estimated Mode: From the histogram, the highest bar represents the mode, which confirms the modal length as 12 to under 14 minutes.
### Part (c) Drawing the 'Less Than' Cumulative Frequency Polygon and Estimating the Median
#### For cumulative frequency polygon:
- Plot cumulative frequencies at the upper boundary of each interval.
- Connect these points with a straight line.
Cumulative frequencies:
[tex]\[ \begin{tabular}{|c|c|} \hline \text{Upper Boundary} & \text{Cumulative Frequency} \\ \hline 12 & 40 \\ 14 & 149 \\ 16 & 203 \\ 18 & 225 \\ 20 & 240 \\ \hline \end{tabular} \][/tex]
Connecting these points will illustrate that the median corresponds to 120 calls, which falls in the interval 12 to under 14 minutes.
Estimated Median: This confirms the median as approximately 13 minutes.
### Part (d) Estimation of Total Telephone Charges
To calculate the total telephone charges, we need to use the mean length of calls, as it provides an average duration.
[tex]\[ \text{Mean Length} = 13.86 \text{ minutes} \][/tex]
[tex]\[ \text{Charge per Minute} = \$0.30 \][/tex]
[tex]\[ \text{Total Charges} = \text{Mean Length} \times \text{Total Number of Calls} \times \text{Charge per Minute} \][/tex]
[tex]\[ = 13.86 \times 240 \times 0.30 = 72.048 \approx 72 \text{ dollars} \][/tex]
So, the total telephone charges are estimated to be \$72 based on the mean length of calls. The mean is most suitable for this estimation because it accounts for all values in the data set, providing a comprehensive average charge.
