To calculate the estimate of the mean waiting time per person using the given data, we will follow these steps:
1. Identify the midpoints of the intervals:
- For the interval [tex]\(0 < T \leq 10\)[/tex], the midpoint is [tex]\( \frac{0 + 10}{2} = 5 \)[/tex].
- For the interval [tex]\(10 < T \leq 20\)[/tex], the midpoint is [tex]\( \frac{10 + 20}{2} = 15 \)[/tex].
- For the interval [tex]\(20 < T \leq 30\)[/tex], the midpoint is [tex]\( \frac{20 + 30}{2} = 25 \)[/tex].
2. List the midpoints and the corresponding frequencies:
- Midpoints: [tex]\( [5, 15, 25] \)[/tex]
- Frequencies: [tex]\( [42, 30, 8] \)[/tex]
3. Calculate the total number of people (sum of frequencies):
[tex]\[
\text{Total number of people} = 42 + 30 + 8 = 80
\][/tex]
4. Calculate the sum of the product of midpoints and frequencies:
[tex]\[
\text{Total time waited} = (5 \times 42) + (15 \times 30) + (25 \times 8)
\][/tex]
Let’s break this down:
[tex]\[
5 \times 42 = 210
\][/tex]
[tex]\[
15 \times 30 = 450
\][/tex]
[tex]\[
25 \times 8 = 200
\][/tex]
Summing these:
[tex]\[
210 + 450 + 200 = 860
\][/tex]
5. Estimate the mean time waited per person:
[tex]\[
\text{Mean time waited} = \frac{\text{Total time waited}}{\text{Total number of people}} = \frac{860}{80} = 15.75
\][/tex]
Thus, the estimated mean time waited per person is [tex]\( 15.75 \)[/tex] minutes.
