The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.
| Population |
100–800 |
900–1200 |
1300–2000 |
2100–3200 |
3300–4800 |
| Number of villages |
8 |
12 |
50 |
48 |
32 |
(a) Draw a histogram to represent this information.
(b) Write down the class interval which contains the median for this information.
(c) Find the greatest possible value of the interquartile range for the populations of the 150 villages.
Solution
(a) To draw the histogram, calculate the frequency density for each class interval using the formula:
\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)
For each interval:
- 100 – 800: \(\frac{8}{800} = 0.01\)
- 900 – 1200: \(\frac{12}{400} = 0.03\)
- 1300 – 2000: \(\frac{50}{800} = 0.0625\)
- 2100 – 3200: \(\frac{48}{1200} = 0.04\)
- 3300 – 4800: \(\frac{32}{1600} = 0.02\)
Draw bars with these heights on the histogram.
(b) The median is the 75th value. Cumulative frequencies are:
- 100 – 800: 8
- 900 – 1200: 20
- 1300 – 2000: 70
- 2100 – 3200: 118
- 3300 – 4800: 150
The median falls in the 2100 – 3200 interval.
(c) To find the interquartile range, calculate the lower quartile (LQ) and upper quartile (UQ):
LQ is the 37.5th value, which is in the 1300 – 2000 interval.
UQ is the 112.5th value, which is in the 2100 – 3200 interval.
Greatest possible value of IQR is \(3200 - 1250 = 1999\).
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