To find an estimate of the mean distance travelled, we will use the midpoints of the distance intervals along with the corresponding frequencies. Here is the step-by-step solution:
1. Identify the distance intervals and their respective frequencies:
- [tex]\(0 < x \leq 20\)[/tex] has a frequency of [tex]\(17\)[/tex].
- [tex]\(20 < x \leq 40\)[/tex] has a frequency of [tex]\(0\)[/tex].
- [tex]\(40 < x \leq 60\)[/tex] has a frequency of [tex]\(8\)[/tex].
2. Calculate the midpoint (the average value) of each distance interval:
- For [tex]\(0 < x \leq 20\)[/tex], the midpoint is [tex]\((0 + 20) / 2 = 10\)[/tex].
- For [tex]\(20 < x \leq 40\)[/tex], the midpoint is [tex]\((20 + 40) / 2 = 30\)[/tex].
- For [tex]\(40 < x \leq 60\)[/tex], the midpoint is [tex]\((40 + 60) / 2 = 50\)[/tex].
3. Tabulate the midpoints and frequencies:
- Midpoints: [tex]\([10, 30, 50]\)[/tex]
- Frequencies: [tex]\([17, 0, 8]\)[/tex]
4. Calculate the total frequency:
[tex]\[
\text{Total frequency} = 17 + 0 + 8 = 25
\][/tex]
5. Calculate the sum of the products of midpoints and frequencies:
[tex]\[
\text{Sum of products} = (10 \times 17) + (30 \times 0) + (50 \times 8) = 170 + 0 + 400 = 570
\][/tex]
6. Calculate the estimate of the mean distance:
[tex]\[
\text{Mean distance} = \frac{\text{Sum of products}}{\text{Total frequency}} = \frac{570}{25} = 22.8 \, \text{km}
\][/tex]
Thus, the estimate of the mean distance travelled is [tex]\(22.8\)[/tex] kilometres.
