| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Different variables, one observation each |
| Difficulty | Standard +0.3 This is a standard application of linear combinations of normal distributions requiring knowledge that X-Y and aX+bY are normally distributed with calculable parameters. The calculations are straightforward (finding means and variances, then using normal tables), though slightly above average difficulty due to requiring two independent applications and the 1.5 coefficient in part (ii). |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X-Y) = 56-43\) \((= 13)\) | B1 | |
| \(\text{Var}(X-Y) = 6^2 + 5^2\) \((= 61)\) | M1 | |
| \(\frac{0-13}{\sqrt{61}}\) \((= -1.664)\) | M1 | Ignore any attempted cc/no SD/var mixes. var must be attempt at a combination |
| \(1 - \phi(-1.664) = \phi(1.664)\) | M1 | For area consistent with their working |
| \(= 0.952\) (3 sf) | A1 | Similar scheme for use of \(Y - X\) |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(M) = 56 + 1.5(43)\) \((= 120.5)\) | B1 | |
| \(\text{Var}(M) = 6^2 + 1.5^2 \times 5^2\) \((= 92.25)\) | M1 | |
| \(\frac{135-120.5}{\sqrt{92.25}}\) \((= 1.510)\) | M1 | Ignore any attempted cc/no SD/var mixes. var must be attempt at a combination |
| \(1 - \phi(1.510)\) | M1 | For area consistent with their working |
| \(= 0.0655\) or \(0.0656\) or \(6.55\%\) or \(6.56\%\) (3 sf) As final answer | A1 | Allow \(6.6\%\) or \(6.5\%\) or \(7\%\) if correct working seen |
| 5 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X-Y) = 56-43$ $(= 13)$ | B1 | |
| $\text{Var}(X-Y) = 6^2 + 5^2$ $(= 61)$ | M1 | |
| $\frac{0-13}{\sqrt{61}}$ $(= -1.664)$ | M1 | Ignore any attempted cc/no SD/var mixes. var must be attempt at a combination |
| $1 - \phi(-1.664) = \phi(1.664)$ | M1 | For area consistent with their working |
| $= 0.952$ (3 sf) | A1 | Similar scheme for use of $Y - X$ |
| | **5** | |
---
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(M) = 56 + 1.5(43)$ $(= 120.5)$ | B1 | |
| $\text{Var}(M) = 6^2 + 1.5^2 \times 5^2$ $(= 92.25)$ | M1 | |
| $\frac{135-120.5}{\sqrt{92.25}}$ $(= 1.510)$ | M1 | Ignore any attempted cc/no SD/var mixes. var must be attempt at a combination |
| $1 - \phi(1.510)$ | M1 | For area consistent with their working |
| $= 0.0655$ or $0.0656$ or $6.55\%$ or $6.56\%$ (3 sf) As final answer | A1 | Allow $6.6\%$ or $6.5\%$ or $7\%$ if correct working seen |
| | **5** | |
---5 The marks in paper 1 and paper 2 of an examination are denoted by $X$ and $Y$ respectively, where $X$ and $Y$ have the independent continuous distributions $\mathrm { N } \left( 56,6 ^ { 2 } \right)$ and $\mathrm { N } \left( 43,5 ^ { 2 } \right)$ respectively.\\
(i) Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.\\
(ii) Each candidate's overall mark is $M$ where $M = X + 1.5 Y$. The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .\\
\hfill \mbox{\textit{CAIE S2 2017 Q5 [10]}}