The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.
| Length of car \( x \) (metres) |
\( 2.80 \le x < 3.00 \) |
\( 3.00 \le x < 3.10 \) |
\( 3.10 \le x < 3.20 \) |
\( 3.20 \le x < 3.40 \) |
| Frequency |
17 |
24 |
19 |
8 |
| Frequency density |
85 |
240 |
190 |
\( a \) |
(i) Find the value of \( a \).
(ii) Draw a histogram on graph paper to represent the data.
Solution
(i) To find the frequency density, use the formula:
\(\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}\)
For the class 3.20 ≤ x < 3.40, the class width is \(3.40 - 3.20 = 0.20\).
Given frequency is 8, so:
\(a = \frac{8}{0.20} = 40\)
(ii) To draw the histogram:
- Use uniform linear scales from at least 2.8 to 3.4 on the x-axis and 0 to 240 on the y-axis.
- Label both axes, with 'length (m)' on the x-axis and 'frequency density' on the y-axis.
- Draw bars with the following heights based on frequency density:
- 2.80 ≤ x < 3.00: height = 85
- 3.00 ≤ x < 3.10: height = 240
- 3.10 ≤ x < 3.20: height = 190
- 3.20 ≤ x < 3.40: height = 40
- Ensure correct widths with no gaps between bars.
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