To calculate an estimate of the mean travelling time, we follow these steps:
1. Determine the midpoint of each class interval:
- For the interval \(0 < t \leq 10\): midpoint \( = \frac{0 + 10}{2} = 5\)
- For the interval \(10 < t \leq 20\): midpoint \( = \frac{10 + 20}{2} = 15\)
- For the interval \(20 < t \leq 30\): midpoint \( = \frac{20 + 30}{2} = 25\)
- For the interval \(30 < t \leq 40\): midpoint \( = \frac{30 + 40}{2} = 35\)
- For the interval \(40 < t \leq 50\): midpoint \( = \frac{40 + 50}{2} = 45\)
2. List the frequencies:
- The frequencies are \(5, 15, 13, 10, \) and \(7\) corresponding to each class interval respectively.
3. Calculate the sum of the frequencies:
[tex]\[
\text{Total frequency} = 5 + 15 + 13 + 10 + 7 = 50
\][/tex]
4. Calculate the weighted sum of the midpoints:
- Multiply each midpoint by its respective frequency and sum up the results:
[tex]\[
(5 \times 5) + (15 \times 15) + (25 \times 13) + (35 \times 10) + (45 \times 7)
\][/tex]
- Calculate each term:
[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]
- Sum these values:
[tex]\[
25 + 225 + 325 + 350 + 315 = 1240
\][/tex]
5. Calculate the mean traveling time:
- Divide the weighted sum of the midpoints by the total frequency:
[tex]\[
\text{Mean travelling time} = \frac{1240}{50} = 24.8
\][/tex]
Therefore, the estimate of the mean travelling time is [tex]\(24.8\)[/tex] minutes.
