- A random sample of 100 carrots is taken from a farm and their lengths, \(L \mathrm {~cm}\), recorded. The data are summarised in the following table.
| Length, \(L\) cm | Frequency, f | Class mid point, \(\boldsymbol { x } \mathbf { c m }\) |
| \(5 \leqslant L < 8\) | 5 | 6.5 |
| \(8 \leqslant L < 10\) | 13 | 9 |
| \(10 \leqslant L < 12\) | 16 | 11 |
| \(12 \leqslant L < 15\) | 25 | 13.5 |
| \(15 \leqslant L < 20\) | 30 | 17.5 |
| \(20 \leqslant L < 28\) | 11 | 24 |
A histogram is drawn to represent these data.
The bar representing the class \(5 \leqslant L < 8\) is 1.5 cm wide and 1 cm high.
- Find the width and height of the bar representing the class \(15 \leqslant L < 20\)
- Use linear interpolation to estimate the median length of these carrots.
- Estimate
- the mean length of these carrots,
- the standard deviation of the lengths of these carrots.
A supermarket will only buy carrots with length between 9 cm and 22 cm .
- Estimate the proportion of carrots from the farm that the supermarket will buy.
Any carrots that the supermarket does not buy are sold as animal feed.
The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
- Find an estimate of the mean profit per carrot made by the farm.