Answer :
To estimate the mean traveling time from the given frequency table, we follow these steps:
1. Identify the midpoints of each class interval: The midpoint is the average of the lower and upper bounds of each interval.
2. Calculate the midpoint for each interval:
- For [tex]\( 0 < t \leq 10 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{0 + 10}{2} = 5 \][/tex]
- For [tex]\( 10 < t \leq 20 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{10 + 20}{2} = 15 \][/tex]
- For [tex]\( 20 < t \leq 30 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{20 + 30}{2} = 25 \][/tex]
- For [tex]\( 30 < t \leq 40 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{30 + 40}{2} = 35 \][/tex]
- For [tex]\( 40 < t \leq 50 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{40 + 50}{2} = 45 \][/tex]
3. Multiply each midpoint by the corresponding frequency to find the weighted midpoint sum:
- For [tex]\( 0 < t \leq 10 \)[/tex]:
[tex]\[ 5 \times 5 = 25 \][/tex]
- For [tex]\( 10 < t \leq 20 \)[/tex]:
[tex]\[ 15 \times 15 = 225 \][/tex]
- For [tex]\( 20 < t \leq 30 \)[/tex]:
[tex]\[ 25 \times 13 = 325 \][/tex]
- For [tex]\( 30 < t \leq 40 \)[/tex]:
[tex]\[ 35 \times 10 = 350 \][/tex]
- For [tex]\( 40 < t \leq 50 \)[/tex]:
[tex]\[ 45 \times 7 = 315 \][/tex]
4. Sum these products to get the total weighted sum of midpoints:
[tex]\[ 25 + 225 + 325 + 350 + 315 = 1240 \][/tex]
5. Calculate the total frequency (total number of people), which is 50 in this case.
6. Divide the total weighted sum of midpoints by the total frequency to find the estimate of the mean travelling time:
[tex]\[ \text{Mean travelling time} = \frac{1240}{50} = 24.8 \text{ minutes} \][/tex]
By following these steps, we obtain that the estimated mean traveling time is 24.8 minutes.
1. Identify the midpoints of each class interval: The midpoint is the average of the lower and upper bounds of each interval.
2. Calculate the midpoint for each interval:
- For [tex]\( 0 < t \leq 10 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{0 + 10}{2} = 5 \][/tex]
- For [tex]\( 10 < t \leq 20 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{10 + 20}{2} = 15 \][/tex]
- For [tex]\( 20 < t \leq 30 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{20 + 30}{2} = 25 \][/tex]
- For [tex]\( 30 < t \leq 40 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{30 + 40}{2} = 35 \][/tex]
- For [tex]\( 40 < t \leq 50 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{40 + 50}{2} = 45 \][/tex]
3. Multiply each midpoint by the corresponding frequency to find the weighted midpoint sum:
- For [tex]\( 0 < t \leq 10 \)[/tex]:
[tex]\[ 5 \times 5 = 25 \][/tex]
- For [tex]\( 10 < t \leq 20 \)[/tex]:
[tex]\[ 15 \times 15 = 225 \][/tex]
- For [tex]\( 20 < t \leq 30 \)[/tex]:
[tex]\[ 25 \times 13 = 325 \][/tex]
- For [tex]\( 30 < t \leq 40 \)[/tex]:
[tex]\[ 35 \times 10 = 350 \][/tex]
- For [tex]\( 40 < t \leq 50 \)[/tex]:
[tex]\[ 45 \times 7 = 315 \][/tex]
4. Sum these products to get the total weighted sum of midpoints:
[tex]\[ 25 + 225 + 325 + 350 + 315 = 1240 \][/tex]
5. Calculate the total frequency (total number of people), which is 50 in this case.
6. Divide the total weighted sum of midpoints by the total frequency to find the estimate of the mean travelling time:
[tex]\[ \text{Mean travelling time} = \frac{1240}{50} = 24.8 \text{ minutes} \][/tex]
By following these steps, we obtain that the estimated mean traveling time is 24.8 minutes.
