The table shows the traveling time to work of 50 people.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Traveling time [tex]$(t)$[/tex] \\
in minutes
\end{tabular} & Frequency \\
\hline [tex]$0\ \textless \ t \leqslant 10$[/tex] & 5 \\
\hline [tex]$10\ \textless \ t \leqslant 20$[/tex] & 15 \\
\hline [tex]$20\ \textless \ t \leqslant 30$[/tex] & 13 \\
\hline [tex]$30\ \textless \ t \leqslant 40$[/tex] & 10 \\
\hline [tex]$40\ \textless \ t \leqslant 50$[/tex] & 7 \\
\hline
\end{tabular}

Calculate an estimate of the mean traveling time.



Answer :

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.