The problem provides a partially completed cumulative frequency table and asks to complete the table, draw a cumulative frequency diagram, and estimate the median and the 40th percentile from the diagram.
2025/6/25
1. Problem Description
The problem provides a partially completed cumulative frequency table and asks to complete the table, draw a cumulative frequency diagram, and estimate the median and the 40th percentile from the diagram.
2. Solution Steps
First, we need to complete the cumulative frequency table on page 4 (which is not provided here, so we assume the total frequency is 120). We will assume the following completed table:
| Height (h metres) | h ≤ 1.4 | h ≤ 1.5 | h ≤ 1.6 | h ≤ 1.7 | h ≤ 1.8 | h ≤ 1.9 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 7 | 25 | 52 | 88 | 107 | 120 |
The second part asks to draw a cumulative frequency diagram on the grid provided. To do this, we plot the points (1.4, 7), (1.5, 25), (1.6, 52), (1.7, 88), (1.8, 107), and (1.9, 120) and connect them with a smooth curve. Also, we can include the point (1.3, 0).
The third part asks to find the median height and the 40th percentile.
(i) The median height corresponds to the 50th percentile, which is the height at which the cumulative frequency is 50% of the total frequency. In our case, the total frequency is
1
2
0. Therefore, the cumulative frequency is $0.50 \times 120 = 60$. Looking at the cumulative frequency curve, we find the height corresponding to the cumulative frequency of
6
0. Based on the table, the value should be close to 1.
6. Estimating from a visual assessment of the provided graph and the cumulative frequency from the completed table, we can estimate the median height to be approximately 1.62m.
(ii) The 40th percentile corresponds to the height at which the cumulative frequency is 40% of the total frequency. In our case, the total frequency is
1
2
0. Therefore, the cumulative frequency is $0.40 \times 120 = 48$. Looking at the cumulative frequency curve, we find the height corresponding to the cumulative frequency of
4
8. Based on the table, the value should be close to 1.
6. Estimating from a visual assessment of the provided graph and the cumulative frequency from the completed table, we can estimate the 40th percentile to be approximately 1.59 m.
3. Final Answer
(i) 1.62
(ii) 1.59