S3 June 2013 (R) Q8
8. A farmer supplies both duck eggs and chicken eggs. The weights of duck eggs, \(D\) grams, and chicken eggs, \(C\) grams, are such that
\[D \sim \mathrm{N}(54, 1.2^2) \text{ and } C \sim \mathrm{N}(44, 0.8^2).\]Eggs are packed in boxes which contain either 6 randomly selected duck eggs or 6 randomly selected chicken eggs. The weight of an empty box has distribution \(\mathrm{N}\left(28, \sqrt{5}^{\,2}\right)\).
| Scheme | Marks |
|---|---|
| Let \(W = D_1 - D_2\) | M1 |
| \(W \sim \mathrm{N}(0, 2.88)\) | A1, A1 |
| \(\mathrm{P}(|W| \gt 3) = 2 \times \mathrm{P}(W \gt 3)\) | M1 |
| \(= 2 \times \mathrm{P}\left(Z \gt \dfrac{3 - 0}{\sqrt{2.88}}\right)\) | dM1 |
| \(= 2 \times \mathrm{P}(Z \gt 1.76776\ldots)\) \(= 2 \times (1 - 0.9616)\) \(= 0.0768\) awrt 0.077 | A1 |
| (6) |
Notes
Award full marks in each part for a correct answer with no incorrect working seen.
1st M1 for explicitly defining a suitable \(W\) and attempt to find the distribution of \(W\).
May be implied by sight of \(\mathrm{N}(0, 2.88)\)
1st A1 for normal and mean of 0, 2nd A1 for variance of 2.88. Award M1A1A1 for \(\mathrm{N}(0, 2.88)\) seen.
2nd M1 for realising need \(2 \times \mathrm{P}(W \gt 3)\)
3rd dM1 Dep on 1st M1 for standardising with 3, 0 and their s.d. Must lead to \(\mathrm{P}(Z \gt +\text{ve})\) (o.e.)
| Scheme | Marks |
|---|---|
| Let \(T = 5C - 4D\) or \(4D - 5C\) or \(C - \tfrac{4}{5}D\) or \(\tfrac{4}{5}D - C\) | M1 |
| \(T \sim \mathrm{N}(\pm 4, 39.04)\) or \(\mathrm{N}(\pm 0.8, 1.5616)\) | A1 A1 |
| \(\mathrm{P}(T \lt 0) = \mathrm{P}\left(Z \lt \dfrac{0 - 4}{\sqrt{39.04}}\right)\) or \(\mathrm{P}\left(Z \lt \dfrac{0 - 0.8}{\sqrt{1.5616}}\right)\) | M1 |
| \(= \mathrm{P}(Z \lt -0.64018\ldots)\) \(= (1 - 0.7389)\) \(= 0.2611\) awrt 0.261 | A1 |
| (5) |
Notes
1st M1 for explicitly defining a suitable \(T\) but may be implied by sight of one of these normals
1st A1 for normal and correct mean, 2nd A1 for correct variance. Accept awrt 3sf i.e. 39.0, 1.56
2nd M1 for standardising with 0 and their mean and their s.d. Must lead to \(\mathrm{P}(Z \lt -\text{ve})\) (o.e.)
| Scheme | Marks |
|---|---|
| Let \(P = D + D + D + D + D + D + B\) Let \(Q = C + C + C + C + C + C + B\) | M1 |
| \(P \sim \mathrm{N}(352, 13.64)\) and \(Q \sim \mathrm{N}(292, 8.84)\) | A1, A1 |
| [Let \(R = P - Q\)] \(R \sim \mathrm{N}(\pm 60, 22.48)\) | M1 |
| \(\mathrm{P}(R \gt 50) = \mathrm{P}\left(Z \gt \dfrac{50 - 60}{\sqrt{22.48}}\right)\) | dM1 |
| \(= \mathrm{P}(Z \gt -2.10\ldots)\) \(= 0.9821\) awrt 0.982 ~ 0.983 | A1 |
| (6) | |
| (17 marks) |
Notes
1st M1 for explicitly defining a correct \(P\) or \(Q\). May be implied by a correct distribution for \(P\) or \(Q\)
1st A1 for a correct distribution for \(P\) 2nd A1 for a correct distribution for \(Q\)
2nd M1 for attempting \(R\) and obtaining its distribution- ft their \(P\) and \(Q\) means and variances
3rd dM1 for attempting \(\mathrm{P}(R \gt 50)\) and standardising with 50 and their \(\mathrm{E}(R)\) and their \(\sqrt{\mathrm{Var}(R)}\)
Dependent on 2nd M1. Must lead to a \(\mathrm{P}(Z \gt -\text{ve})\) (o.e.)