S3 June 2018 Q5
5. The weights, in kg, of cars may be assumed to follow the normal distribution \(\mathrm{N}(1000, 250^2)\). The weights, in kg, of lorries may be assumed to follow the normal distribution \(\mathrm{N}(2800, 650^2)\).
A lorry and a car are chosen at random.
A ferry carries vehicles across a river. The ferry is designed to carry a maximum weight of 20 000 kg.
| Scheme | Marks |
|---|---|
| (Let \(W =\))\(L - 3C\) | B1 |
| \(\mathrm{E}(W) = 2800 - 3 \times 1000 = -200\) | B1 |
| \(\mathrm{Var}(W) = 650^2 + 3^2 \times 250^2 = 985\,000\) | M1A1 |
| \(\mathrm{P}(W \gt 0) = \mathrm{P}\left(Z \gt \frac{200}{\sqrt{985000}}\right) = \mathrm{P}(Z \gt 0.20157\ldots),\ = 0.42015\) (calc) or 0.4207 (tables) | dM1 A1 |
| (6) |
Notes
1st B1 for forming a suitable variable. May be implied by correct variance.
2nd B1 for \(-200\) cao or 200 if their \(W = 3C - L\)
1st M1 for attempting \(\mathrm{Var}(W) = \mathrm{Var}(L) + 3^2 \times \mathrm{Var}(C)\). Condone swapping \(L\) and \(C\).
1st A1 for 985 000 cao
2nd M1 dependent upon first M1 for standardising with their \(-200\) and their 985000
2nd A1 awrt 0.420-0.421
| Scheme | Marks |
|---|---|
| \((F = C_1 + C_2 + \ldots + C_8 + L_1 + L_2 + L_3)\) | |
| \(\mathrm{E}(F) = 16400\) | B1 |
| \(\mathrm{Var}(F) = 8 \times 250^2 + 3 \times 650^2 = 1767500\) | M1A1 |
| \(\mathrm{P}(F \gt 20\,000) = \mathrm{P}\left(Z \gt \frac{20000 - 16400}{\sqrt{1767500}}\right) = \mathrm{P}(Z \gt 2.7078\ldots),\ = 0.003386\ldots\)(calc) or 0.0035 (tables) or 0.0034 (interpolation) | dM1,A1 |
| (5) |
Notes
1st B1 for 16400 cao
1st M1 for attempting \(\mathrm{Var}(F) = 8 \times \mathrm{Var}(C) + 3 \times \mathrm{Var}(L)\)
1st A1 for 1 767 500 cao
2nd M1 dependent upon first M1 for standardising with their 16400 and their 1767500
2nd A1 awrt 0.003-0.004
| Scheme | Marks |
|---|---|
| Assume selection of cars and lorries is random. Weights of cars and lorries are independent. | B1 |
| (1) | |
| (12 marks) |
Notes
B1 Either random selection or independent weights