| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| 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 standard application of linear combinations of normal variables requiring students to form F+J and F-2J, find their distributions using standard formulas (mean and variance rules), then perform routine normal probability calculations. While it requires careful setup of part (ii), the techniques are textbook exercises with no novel insight needed. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(F + J \sim N(24,\ 2.8^2 + 2.6^2)\) | B1 | or \(N(24, 14.6)\) for correct mean and variance |
| \(\frac{30-24}{\sqrt{14.6}}\ (= 1.570)\) | M1 | Allow without \(\sqrt{}\) (ignore false cc) |
| \(P(F + J < 30) = \Phi(1.570)\) | M1 | Correct area consistent with their working |
| \(0.942\) (3 sfs) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \(F - 2J \sim N(-11.4,\ 2.8^2 + 4 \times 2.6^2)\) | B1 | or \(N(-11.4, 34.88)\) for correct mean and variance |
| \(\frac{0-(-11.4)}{\sqrt{34.88}}\ (= 1.930)\) | M1 | Allow without \(\sqrt{}\) (ignore false cc) |
| \(P(F - 2J > 0) = 1 - \Phi(1.930)\) | M1 | Correct area consistent with their working or similar scheme using \(2J - F\) |
| \(= 0.0268\) (3 sfs) | A1 [4] |
## Question 5:
**(i)**
$F + J \sim N(24,\ 2.8^2 + 2.6^2)$ | B1 | or $N(24, 14.6)$ for correct mean and variance
$\frac{30-24}{\sqrt{14.6}}\ (= 1.570)$ | M1 | Allow without $\sqrt{}$ (ignore false cc)
$P(F + J < 30) = \Phi(1.570)$ | M1 | Correct area consistent with their working
$0.942$ (3 sfs) | A1 [4] |
**(ii)**
$F - 2J \sim N(-11.4,\ 2.8^2 + 4 \times 2.6^2)$ | B1 | or $N(-11.4, 34.88)$ for correct mean and variance
$\frac{0-(-11.4)}{\sqrt{34.88}}\ (= 1.930)$ | M1 | Allow without $\sqrt{}$ (ignore false cc)
$P(F - 2J > 0) = 1 - \Phi(1.930)$ | M1 | Correct area consistent with their working or similar scheme using $2J - F$
$= 0.0268$ (3 sfs) | A1 [4] |
---5 Fiona and Jhoti each take one shower per day. The times, in minutes, taken by Fiona and Jhoti to take a shower are represented by the independent variables $F \sim \mathrm {~N} \left( 12.2,2.8 ^ { 2 } \right)$ and $J \sim \mathrm {~N} \left( 11.8,2.6 ^ { 2 } \right)$ respectively. Find the probability that, on a randomly chosen day,\\
(i) the total time taken to shower by Fiona and Jhoti is less than 30 minutes,\\
(ii) Fiona takes at least twice as long as Jhoti to take a shower.
\hfill \mbox{\textit{CAIE S2 2012 Q5 [8]}}