| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| 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 is a straightforward application of standard results for linear transformations and linear combinations of normal variables. Part (a) requires E(aX) = aE(X) and Var(aX) = a²Var(X). Part (b) requires forming 2.50X - 3.25Y and using the difference of normals result. All steps are routine S2 material with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 5.02c Linear coding: effects on mean and variance5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(A \text{ income}) = [10.3 \times 2.50] = 25.75\) [\$] | B1 | Accept 3sf |
| \(\text{Var}(A \text{ income}) = [5.76 \times 2.50^2] = 36\) [\$²] | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B \text{ income} \sim N(37.05, 101.506)\) or \(E(B \text{ income}) = 37.05\) and \(\text{Var}(B \text{ income}) = 101.51\) | B1 | Or \(N(37.1, 102)\) soi |
| \(A \text{ income} - B \text{ income} \sim N(\text{'25.75'} - \text{'37.05'}, \text{'36'} + \text{'101.506'})\) | M1 | Ft their values for \(A\) and \(B\) |
| \(= N(-11.3, 137.506)\) | A1 | Accept 3sf |
| \(\frac{0 - (\text{'-11.3'})}{\sqrt{\text{'137.506'}}} [= 0.964]\) | M1 | Standardising with their values from attempt at \(A\) income \(- B\) income |
| \(1 - \phi(\text{'0.964'})\) | M1 | For area consistent with their values |
| \(= 0.168\) or \(0.167\) (3 sf) | A1 | cwo |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(A \text{ income}) = [10.3 \times 2.50] = 25.75$ [\$] | B1 | Accept 3sf |
| $\text{Var}(A \text{ income}) = [5.76 \times 2.50^2] = 36$ [\$²] | B1 | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B \text{ income} \sim N(37.05, 101.506)$ or $E(B \text{ income}) = 37.05$ and $\text{Var}(B \text{ income}) = 101.51$ | B1 | Or $N(37.1, 102)$ soi |
| $A \text{ income} - B \text{ income} \sim N(\text{'25.75'} - \text{'37.05'}, \text{'36'} + \text{'101.506'})$ | M1 | Ft their values for $A$ and $B$ |
| $= N(-11.3, 137.506)$ | A1 | Accept 3sf |
| $\frac{0 - (\text{'-11.3'})}{\sqrt{\text{'137.506'}}} [= 0.964]$ | M1 | Standardising with their values from attempt at $A$ income $- B$ income |
| $1 - \phi(\text{'0.964'})$ | M1 | For area consistent with their values |
| $= 0.168$ or $0.167$ (3 sf) | A1 | cwo |4 The masses, in kilograms, of chemicals $A$ and $B$ produced per day by a factory are modelled by the independent random variables $X$ and $Y$ respectively, where $X \sim \mathrm {~N} ( 10.3,5.76 )$ and $Y \sim \mathrm {~N} ( 11.4,9.61 )$. The income generated by the chemicals is $\$ 2.50$ per kilogram for $A$ and $\$ 3.25$ per kilogram for $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of the daily income generated by chemical $A$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d42b3c4d-c426-4231-a35a-cac80dbdf82c-06_56_1566_495_333}
\item Find the probability that, on a randomly chosen day, the income generated by chemical $A$ is greater than the income generated by chemical $B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2023 Q4 [8]}}