8 A market gardener records the masses of a random sample of 100 of this year's crop of plums. The table shows his results.
| \(m < 25\) | \(25 \leq m < 35\) | \(35 \leq m < 45\) | \(45 \leq m < 55\) | \(55 \leq m < 65\) | \(65 \leq m < 75\) | \(m \geq 75\) |
| 0 | 3 | 29 | 36 | 30 | 2 | 0 |
- Explain why the normal distribution might be a reasonable model for this distribution.
The market gardener models the distribution of masses by \(\mathrm { N } \left( 47.5,10 ^ { 2 } \right)\).
- Find the number of plums in the sample that this model would predict to have masses in the range:
- \(35 \leq m < 45\)
- \(m < 25\).
- Use your answers to parts (b)(i) and (b)(ii) to comment on the suitability of this model.
The market gardener plans to use this model to predict the distribution of the masses of next year's crop of plums.
- Comment on this plan.