5. The weights, in grams, of a random sample of 48 broad beans are summarised in the table.
| Weight in grams ( \(\boldsymbol { x }\) ) | Frequency (f) | Class midpoint (y) |
| \(0.9 < x \leqslant 1.1\) | 9 | 1.0 |
| \(1.1 < x \leqslant 1.3\) | 12 | 1.2 |
| \(1.3 < x \leqslant 1.5\) | 11 | 1.4 |
| \(1.5 < x \leqslant 1.7\) | 8 | 1.6 |
| \(1.7 < x \leqslant 1.9\) | 3 | 1.8 |
| \(1.9 < x \leqslant 2.1\) | 3 | 2.0 |
| \(2.1 < x \leqslant 2.7\) | 2 | 2.4 |
(You may assume \(\sum \mathrm { fy } { } ^ { 2 } = 101.56\) )
A histogram was drawn to represent these data. The \(2.1 < x \leqslant 2.7\) class was represented by a bar of width 1.5 cm and height 1 cm .
- Find the width and height of the \(0.9 < x \leqslant 1.1\) class.
- Give a reason to justify the use of a histogram to represent these data.
- Estimate the mean and the standard deviation of the weights of these broad beans.
- Use linear interpolation to estimate the median of the weights of these broad beans.
One of these broad beans is selected at random.
- Estimate the probability that its weight lies between 1.1 grams and 1.6 grams.
One of these broad beans having a recorded weight of 0.95 grams was incorrectly weighed. The correct weight is 1.4 grams.
- State, giving a reason, the effect this would have on your answers to part (c). Do not carry out any further calculations.