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Q8. The table shows the waiting time, [tex]\( T \)[/tex], of people at a health center.

[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time waited, } T & \text{Number of people} \\
\hline
0 \ \textless \ T \leq 10 & 42 \\
\hline
10 \ \textless \ T \leq 20 & 30 \\
\hline
20 \ \textless \ T \leq 30 & 8 \\
\hline
\text{More than 30} & 0 \\
\hline
\end{array}
\][/tex]

(a) Work out an estimate of the mean time waited per person.



Answer :

GinnyAnswer
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.

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