| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Mixed sum threshold probability |
| Difficulty | Standard +0.3 This question tests standard linear combinations of independent normal variables with straightforward coefficient manipulation. Part (i) requires forming 2X + Y and finding a probability; part (ii) requires X + 1.5Y. Both are direct applications of the formula for combining normal distributions with no conceptual challenges beyond recognizing the setup and calculating variance correctly (σ² terms, not σ). Slightly above average due to the two-part structure and careful arithmetic with decimals, but remains a textbook exercise. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(E(\text{Tot}) = 2 \times 36 + 55\) \((= 127)\) | B1 | (Or \(\pm 13\)) |
| \(\text{Var}(\text{Tot}) = 2 \times 3.5^2 + 5.2^2\) \((= 51.54)\) | B1 | |
| \(\frac{140 - 127}{\sqrt{51.54}}\) \((= 1.811)\) | M1 | For standardising with their mean and var. Allow without \(\sqrt{}\) or with false cc, but their mean and variance must involve parameters from both given distributions |
| \(\Phi(1.811)\) | ||
| \(= 0.965\) (3 s.f.) | A1 | [4] |
| (ii) \(E(RM) = 36 + 1.5 \times 55\) \((= 118.5)\) | B1 | (Or \(\pm 18.5\)) |
| \(\text{Var}(RM) = 3.5^2 + 1.5^2 \times 5.2^2\) \((= 73.09)\) | B1 | |
| \(\frac{100 - 1185}{\sqrt{73.09}}\) \((= -2.164)\) | M1 | For standardising with their mean and var. Allow without \(\sqrt{}\) or with false cc, but their mean and variance must involve parameters from both given distributions |
| \(1 - \Phi(-2.164) = \Phi(2.164)\) | ||
| \(0.985\) (3 s.f.) | A1 | [4] |
**(i)** $E(\text{Tot}) = 2 \times 36 + 55$ $(= 127)$ | B1 | (Or $\pm 13$)
$\text{Var}(\text{Tot}) = 2 \times 3.5^2 + 5.2^2$ $(= 51.54)$ | B1 |
$\frac{140 - 127}{\sqrt{51.54}}$ $(= 1.811)$ | M1 | For standardising with their mean and var. Allow without $\sqrt{}$ or with false cc, but their mean and variance must involve parameters from both given distributions
$\Phi(1.811)$ | |
$= 0.965$ (3 s.f.) | A1 | [4]
**(ii)** $E(RM) = 36 + 1.5 \times 55$ $(= 118.5)$ | B1 | (Or $\pm 18.5$)
$\text{Var}(RM) = 3.5^2 + 1.5^2 \times 5.2^2$ $(= 73.09)$ | B1 |
$\frac{100 - 1185}{\sqrt{73.09}}$ $(= -2.164)$ | M1 | For standardising with their mean and var. Allow without $\sqrt{}$ or with false cc, but their mean and variance must involve parameters from both given distributions
$1 - \Phi(-2.164) = \Phi(2.164)$ | |
$0.985$ (3 s.f.) | A1 | [4]Ranjit goes to mathematics lectures and physics lectures. The length, in minutes, of a mathematics lecture is modelled by the variable $X$ with distribution N(36, 3.5²). The length, in minutes, of a physics lecture is modelled by the independent variable $Y$ with distribution N(55, 5.2²).
\begin{enumerate}[label=(\roman*)]
\item Find the probability that the total length of two mathematics lectures and one physics lecture is less than 140 minutes. [4]
\item Ranjit calculates how long he will need to spend revising the content of each lecture as follows. Each minute of a mathematics lecture requires 1 minute of revision and each minute of a physics lecture requires 1½ minutes of revision. Find the probability that the total revision time required for one mathematics lecture and one physics lecture is more than 100 minutes. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2011 Q6 [8]}}