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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}
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.
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