| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Same variable, two observations |
| Difficulty | Standard +0.3 This question tests standard application of linear combinations of normal variables with straightforward calculations. Part (a) requires recognizing that the sum of 5 independent normals has variance 5σ² and standardizing; part (b) needs knowing that the difference of two normals has variance 2σ². Both are direct textbook applications requiring no problem-solving insight, though slightly above average difficulty due to being Further Maths content and requiring careful variance calculations. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Mean \(= 5 \times 18.3\) and Variance \(= 5 \times 2.5^2\) \([= N(91.5, 31.25)]\) | B1 | SOI |
| \(\frac{95 - '91.5'}{\sqrt{'31.25'}}\) \([= 0.626]\) | M1 | FT *their* mean and variance |
| \(1 - \Phi('0.626')\) | M1 | For finding area consistent with *their* values |
| \(0.266\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(D) = 0\) | B1 | Or \(E(D-1) = -1\) |
| \(\text{Var}(D) = 2.5^2 \times 2\) \([= 12.5]\) | B1 | |
| \(\frac{1-0}{\sqrt{'12.5'}} [= 0.283]\) or \(\frac{-1-0}{\sqrt{'12.5'}} [= -0.283]\) | M1 | FT *their* E and Var |
| \(\Phi('0.283') - (1 - \phi(0.283))\) \([= 0.6115 - 0.3885]\) | M1 | For finding area consistent with *their* values |
| \(0.223\) (3 sf) | A1 |
## Question 5:
### Part 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Mean $= 5 \times 18.3$ and Variance $= 5 \times 2.5^2$ $[= N(91.5, 31.25)]$ | B1 | SOI |
| $\frac{95 - '91.5'}{\sqrt{'31.25'}}$ $[= 0.626]$ | M1 | FT *their* mean and variance |
| $1 - \Phi('0.626')$ | M1 | For finding area consistent with *their* values |
| $0.266$ (3 sf) | A1 | |
### Part 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(D) = 0$ | B1 | Or $E(D-1) = -1$ |
| $\text{Var}(D) = 2.5^2 \times 2$ $[= 12.5]$ | B1 | |
| $\frac{1-0}{\sqrt{'12.5'}} [= 0.283]$ or $\frac{-1-0}{\sqrt{'12.5'}} [= -0.283]$ | M1 | FT *their* E and Var |
| $\Phi('0.283') - (1 - \phi(0.283))$ $[= 0.6115 - 0.3885]$ | M1 | For finding area consistent with *their* values |
| $0.223$ (3 sf) | A1 | |
---5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
\item Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q5 [9]}}