June 2023 Paper 3 Q16
16 A farm supplies apples to a supermarket.
The diameters of the apples, \(D\) centimetres, are normally distributed with mean 6.5 and standard deviation 0.73
(a)
(i) Find \(\mathrm{P}(D \lt 5.2)\) [1 mark]
(ii) Find \(\mathrm{P}(D \gt 7)\) [1 mark]
(iii) The supermarket only accepts apples with diameters between 5 cm and 8 cm.
Find the proportion of apples that the supermarket accepts. [1 mark]
(b) The farm also supplies plums to the supermarket.
These plums have diameters that are normally distributed.
It is found that 60% of these plums have a diameter less than 5.9 cm.
It is found that 20% of these plums have a diameter greater than 6.1 cm.
Find the mean and standard deviation of the diameter, in centimetres, of the plums supplied by the farm. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains correct probability AWFW [0.037, 0.038] Ignore incorrect rounding after correct probability seen | B1 | 1.1b |
| (1) | ||
| (ii) Obtains correct probability AWFW [0.246, 0.25] Ignore incorrect rounding after correct probability seen | B1 | 1.1b |
| (1) | ||
| (iii) Obtains correct probability AWFW [0.96, 0.9602] Ignore incorrect rounding after correct probability seen | B1 | 3.3 |
| (1) |
Typical solution
(i)
0.0375
(ii)
0.2467
(iii)
0.9601
| Scheme | Marks | AO |
|---|---|---|
| Obtains either \(z\)-value from inverse normal distribution AWFW [0.25, 0.26] or AWFW [0.84, 0.85] Ignore signs | B1 | 3.1b |
| Forms an equation with unknown \(\mu\) and \(\sigma\) using standardised result and \(z\)-value for 0.6 Accept \(z\) = AWFW [−4, 4] but do not allow 0, ±0.2, ±0.4, ±0.6 or ±0.8 Condone \(\mu - 5.9\) Must use 5.9 | M1 | 3.3 |
| Forms an equation with unknown \(\mu\) and \(\sigma\) using standardised result and \(z\)-value for 0.2 Accept \(z\) = AWFW [−4, 4] but do not allow 0, ±0.2, ±0.4, ±0.6 or ±0.8 Condone \(\mu - 6.1\) Must use 6.1 | M1 | 3.3 |
| Obtains both equations correctly | A1 | 1.1b |
| Obtains correct value of \(\mu\) AWFW [5.8, 5.82] ISW | A1 | 1.1b |
| Obtains correct value of \(\sigma\) AWFW [0.33, 0.35] ISW | A1 | 1.1b |
| (6) | ||
| (9 marks) |