| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Difficulty | Standard +0.3 This is a standard application of linear combinations of normal distributions requiring knowledge that sums scale variance by n and that aX+bY follows a normal distribution. Part (a) is routine (sum of 6 normals), part (b) requires setting up L < 4S as L - 4S < 0, which is slightly less routine but still a textbook exercise. The calculations are straightforward once the distributions are identified. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(240,\ 6 \times 1.5^2)\) or \(N(240,\ 13.5)\) | M1 | |
| \(\dfrac{245 - \text{"240"}}{\sqrt{\text{"13.5"}}}\) \((= 1.361)\) | M1 | |
| \(1 - \Phi(\text{"1.361"})\) | M1 | |
| \(0.0867\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use of \(L - 4S\) or similar | M1 | |
| \(E(L - 4S) = -8\) | B1 | |
| \(\text{Var}(L - 4S) = 1.5^2 + 16 \times 0.7^2\) or \(10.09\) | B1 | |
| \(\dfrac{0 - (\text{"}-8\text{"}) }{\sqrt{\text{"10.09"}}}\) \((= 2.519)\) | M1 | |
| \(\Phi(\text{"2.519"})\) | M1 | |
| \(0.994\) (3 sf) | A1 |
## Question 3:
**Part 3(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(240,\ 6 \times 1.5^2)$ or $N(240,\ 13.5)$ | M1 | |
| $\dfrac{245 - \text{"240"}}{\sqrt{\text{"13.5"}}}$ $(= 1.361)$ | M1 | |
| $1 - \Phi(\text{"1.361"})$ | M1 | |
| $0.0867$ (3 sf) | A1 | |
**Part 3(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Use of $L - 4S$ or similar | M1 | |
| $E(L - 4S) = -8$ | B1 | |
| $\text{Var}(L - 4S) = 1.5^2 + 16 \times 0.7^2$ or $10.09$ | B1 | |
| $\dfrac{0 - (\text{"}-8\text{"}) }{\sqrt{\text{"10.09"}}}$ $(= 2.519)$ | M1 | |
| $\Phi(\text{"2.519"})$ | M1 | |
| $0.994$ (3 sf) | A1 | |
---3 The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions $\mathrm { N } \left( 40,1.5 ^ { 2 } \right)$ and $\mathrm { N } \left( 12,0.7 ^ { 2 } \right)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg .
\item Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q3 [10]}}