2 The engine sizes \(x \mathrm {~cm} ^ { 3 }\) of a sample of 80 cars are summarised in the table below.
| Engine size \(x\) | \(500 \leqslant x \leqslant 1000\) | \(1000 < x \leqslant 1500\) | \(1500 < x \leqslant 2000\) | \(2000 < x \leqslant 3000\) | \(3000 < x \leqslant 5000\) |
| Frequency | 7 | 22 | 26 | 18 | 7 |
- Draw a histogram to illustrate the distribution.
- A student claims that the midrange is \(2750 \mathrm {~cm} ^ { 3 }\). Discuss briefly whether he is likely to be correct.
- Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers are only estimates.
- Hence investigate whether there are any outliers in the sample.
- A vehicle duty of \(\pounds 1000\) is proposed for all new cars with engine size greater than \(2000 \mathrm {~cm} ^ { 3 }\). Assuming that this sample of cars is representative of all new cars in Britain and that there are 2.5 million new cars registered in Britain each year, calculate an estimate of the total amount of money that this vehicle duty would raise in one year.
- Why in practice might your estimate in part (v) turn out to be too high?