To find the mean waiting time from the given frequency distribution, we need to follow a systematic approach. Let's break down the steps:
1. Identify Interval Midpoints:
For each interval, we need to calculate the midpoint, which represents the average waiting time for that interval. The midpoint is calculated as the average of the lower and upper boundaries of each interval.
- [tex]\(0-3\)[/tex]: midpoint is [tex]\((0 + 3)/2 = 1.5\)[/tex]
- [tex]\(4-7\)[/tex]: midpoint is [tex]\((4 + 7)/2 = 5.5\)[/tex]
- [tex]\(8-11\)[/tex]: midpoint is [tex]\((8 + 11)/2 = 9.5\)[/tex]
- [tex]\(12-15\)[/tex]: midpoint is [tex]\((12 + 15)/2 = 13.5\)[/tex]
- [tex]\(16-19\)[/tex]: midpoint is [tex]\((16 + 19)/2 = 17.5\)[/tex]
- [tex]\(20-23\)[/tex]: midpoint is [tex]\((20 + 23)/2 = 21.5\)[/tex]
- [tex]\(24-27\)[/tex]: midpoint is [tex]\((24 + 27)/2 = 25.5\)[/tex]
2. Multiply Each Midpoint by Its Corresponding Frequency:
This step helps us find the weighted sum of all midpoints based on their frequencies.
[tex]\[
\begin{align*}
1.5 \times 14 & = 21 \\
5.5 \times 13 & = 71.5 \\
9.5 \times 14 & = 133 \\
13.5 \times 8 & = 108 \\
17.5 \times 6 & = 105 \\
21.5 \times 3 & = 64.5 \\
25.5 \times 3 & = 76.5 \\
\end{align*}
\][/tex]
3. Find the Total Sum of Midpoint-Frequency Products:
Adding all the products from step 2:
[tex]\[
21 + 71.5 + 133 + 108 + 105 + 64.5 + 76.5 = 579.5
\][/tex]
4. Calculate the Total Number of Customers:
Sum of all frequencies:
[tex]\[
14 + 13 + 14 + 8 + 6 + 3 + 3 = 61
\][/tex]
5. Calculate the Mean Waiting Time:
The mean waiting time is obtained by dividing the total sum of the midpoint-frequency products by the total number of customers.
[tex]\[
\text{Mean Waiting Time} = \frac{579.5}{61} = 9.5 \text{ minutes}
\][/tex]
Therefore, the mean waiting time is [tex]\(9.5\)[/tex] minutes.
