To estimate the mean travel time, we will use the midpoint method. Here is a step-by-step solution:
1. Identify the travel time intervals and their corresponding frequencies:
| Travelling Time [tex]\((t)\)[/tex] (in minutes) | Frequency |
| ------------------------------------- | --------- |
| [tex]\(0 < t \leq 10\)[/tex] | 5 |
| [tex]\(10 < t \leq 20\)[/tex] | 15 |
| [tex]\(20 < t \leq 30\)[/tex] | 13 |
| [tex]\(30 < t \leq 40\)[/tex] | 10 |
| [tex]\(40 < t \leq 50\)[/tex] | 7 |
2. Calculate the midpoint for each interval:
The midpoint of an interval [tex]\((a, b)\)[/tex] is [tex]\(\frac{a + b}{2}\)[/tex].
[tex]\[
\text{Midpoints} = \left[\frac{0 + 10}{2}, \frac{10 + 20}{2}, \frac{20 + 30}{2}, \frac{30 + 40}{2}, \frac{40 + 50}{2}\right]
\][/tex]
[tex]\[
\text{Midpoints} = [5, 15, 25, 35, 45]
\][/tex]
3. Multiply each midpoint by its corresponding frequency:
[tex]\[
5 \times 5 = 25
\][/tex]
[tex]\[
15 \times 15 = 225
\][/tex]
[tex]\[
25 \times 13 = 325
\][/tex]
[tex]\[
35 \times 10 = 350
\][/tex]
[tex]\[
45 \times 7 = 315
\][/tex]
4. Calculate the total sum of these products:
[tex]\[
\text{Total Midpoint Frequency} = 25 + 225 + 325 + 350 + 315 = 1240
\][/tex]
5. Calculate the total frequency:
[tex]\[
\text{Total Frequency} = 5 + 15 + 13 + 10 + 7 = 50
\][/tex]
6. Calculate the estimated mean travel time:
The estimated mean travel time is calculated by dividing the total sum of the midpoint frequencies by the total frequency:
[tex]\[
\text{Estimated Mean} = \frac{\text{Total Midpoint Frequency}}{\text{Total Frequency}} = \frac{1240}{50} = 24.8 \, \text{minutes}
\][/tex]
Therefore, the estimated mean travelling time is [tex]\(24.8\)[/tex] minutes.
