| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Expectation and variance with context application |
| Difficulty | Standard +0.3 This question tests standard application of normal distribution properties: scaling by constants for part (i) and linear combinations for part (ii). Both parts require straightforward formula application (mean/variance of sums, linear transformations) with no conceptual challenges beyond recognizing independence and applying E(aX+bY) and Var(aX+bY) rules. The context is clear and the calculations are routine for S2 level. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| 6080 (litres) | B1 | |
| 106 (litres) | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(21Y - 2X) = 635\) | B1 | |
| \(\text{Var}(21Y - 2X) = 21^2 \times 12^2 + 2^2 \times 53^2\) | B1 | correct expression or result or sd \(= 273\) seen |
| \((= 74740)\) | ||
| \(\frac{0 - 635}{\sqrt{74740}}\) \((= -2.323)\) | M1 | no sd/var mixes |
| \(1 - \Phi(-2.323) = \Phi(2.323)\) | M1 | Area consistent with their working |
| \(= 0.99(0)\) (3 sf) | A1 [5] | No errors seen |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| 6080 (litres) | B1 | |
| 106 (litres) | B1 [2] | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(21Y - 2X) = 635$ | B1 | |
| $\text{Var}(21Y - 2X) = 21^2 \times 12^2 + 2^2 \times 53^2$ | B1 | correct expression or result or sd $= 273$ seen |
| $(= 74740)$ | | |
| $\frac{0 - 635}{\sqrt{74740}}$ $(= -2.323)$ | M1 | no sd/var mixes |
| $1 - \Phi(-2.323) = \Phi(2.323)$ | M1 | Area consistent with their working |
| $= 0.99(0)$ (3 sf) | A1 [5] | No errors seen |
---4 Each week a farmer sells $X$ litres of milk and $Y \mathrm {~kg}$ of cheese, where $X$ and $Y$ have the independent distributions $\mathrm { N } \left( 1520,53 ^ { 2 } \right)$ and $\mathrm { N } \left( 175,12 ^ { 2 } \right)$ respectively.\\
(i) Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks.
During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is $( 21 Y - 2 X )$ cents.\\
(ii) Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.
\hfill \mbox{\textit{CAIE S2 2016 Q4 [7]}}