2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, \(\pounds x\) per square foot. His results are given in the table below.
| Cost (£ \(\boldsymbol { x }\) ) | Frequency (f) | Midpoint (£y) |
| \(20 \leqslant x < 40\) | 12 | 30 |
| \(40 \leqslant x < 45\) | 13 | 42.5 |
| \(45 \leqslant x < 50\) | 25 | 47.5 |
| \(50 \leqslant x < 60\) | 32 | 55 |
| \(60 \leqslant x < 80\) | 8 | 70 |
A histogram is drawn for these data and the bar representing \(50 \leqslant x < 60\) is 2 cm wide and 8 cm high.
- Calculate the width and height of the bar representing \(20 \leqslant x < 40\)
- Use linear interpolation to estimate the median cost.
- Estimate the mean cost of office space for these data.
- Estimate the standard deviation for these data.
- Describe, giving a reason, the skewness.
Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean \(\pounds 50\) and standard deviation \(\pounds 10\)
- With reference to your answer to part (e), comment on Rika's suggestion.
- Use Rika's model to estimate the 80th percentile of the cost of office space in London.