| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.3 This is a straightforward application of linear combinations of independent normal variables. Part (a) requires finding P(X > Y) by considering X - Y ~ N(μ_X - μ_Y, σ_X² + σ_Y²), and part (b) requires finding probabilities for 0.8X + 0.85Y. Both are standard textbook exercises requiring only routine application of formulas with no novel insight or complex problem-solving. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X - Y) = 10700 - 13400 \quad [= -2700]\) | B1 | Oe, e.g. \((Y - X)\) |
| \(\text{Var}(X - Y) = 950^2 + 1210^2 \quad [= 2366600]\) | M1 | |
| \(\frac{0 - (their\ {-2700})}{\sqrt{their\ 2366600}} \quad [= 1.755]\) | M1 | For standardising with *their* E and Var |
| \(1 - \Phi(their\ {1.755})\) | M1 | For area consistent with *their* values |
| \(= 0.0396\) or \(0.0397\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(\text{Total}) = 10700 \times 0.8 + 13400 \times 0.85 \quad [= 19950]\) | B1 | |
| \(\text{Var}(\text{Total}) = 950^2 \times 0.8^2 + 1210^2 \times 0.85^2 \quad [= 1635412.25]\) | M1 | |
| \(\frac{22000 - their\ 19950}{\sqrt{their\ 1635412.25}}\ [= 1.603]\) or \(\frac{20000 - their\ 19950}{\sqrt{their\ 1635412.25}}\ [= 0.0391]\) | M1 | For one standardisation with *their* E and Var |
| \(\Phi(their\ 1.603) - \Phi(their\ 0.0391) = 0.9455 - 0.5156\) | M1 | For area consistent with *their* values |
| \(= 0.43[0]\) (3 sf) | A1 |
## Question 4(a):
| $E(X - Y) = 10700 - 13400 \quad [= -2700]$ | B1 | Oe, e.g. $(Y - X)$ |
|---|---|---|
| $\text{Var}(X - Y) = 950^2 + 1210^2 \quad [= 2366600]$ | M1 | |
| $\frac{0 - (their\ {-2700})}{\sqrt{their\ 2366600}} \quad [= 1.755]$ | M1 | For standardising with *their* E and Var |
| $1 - \Phi(their\ {1.755})$ | M1 | For area consistent with *their* values |
| $= 0.0396$ or $0.0397$ (3 sf) | A1 | |
---
## Question 4(b):
| $E(\text{Total}) = 10700 \times 0.8 + 13400 \times 0.85 \quad [= 19950]$ | B1 | |
|---|---|---|
| $\text{Var}(\text{Total}) = 950^2 \times 0.8^2 + 1210^2 \times 0.85^2 \quad [= 1635412.25]$ | M1 | |
| $\frac{22000 - their\ 19950}{\sqrt{their\ 1635412.25}}\ [= 1.603]$ or $\frac{20000 - their\ 19950}{\sqrt{their\ 1635412.25}}\ [= 0.0391]$ | M1 | For one standardisation with *their* E and Var |
| $\Phi(their\ 1.603) - \Phi(their\ 0.0391) = 0.9455 - 0.5156$ | M1 | For area consistent with *their* values |
| $= 0.43[0]$ (3 sf) | A1 | |
---4 Each year a transport firm uses $X$ litres of gasoline and $Y$ litres of diesel fuel, where $X$ and $Y$ have the independent distributions $X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }$ and $Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.\\
The costs per litre of gasoline and diesel fuel are \$0.80 and \$0.85 respectively.
\item Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between $\$ 20000$ and $\$ 22000$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q4 [10]}}