| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Direct comparison with scalar multiple (different variables) |
| Difficulty | Standard +0.3 This is a standard linear combinations of normal variables question requiring students to form new distributions (K+A and K-1.2A), calculate means and variances using independence, then find probabilities using normal tables. While it requires careful algebraic manipulation in part (ii), it follows a well-established procedure taught explicitly in S2 with no novel problem-solving insight needed. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(N(10.61, 0.1017)\) | B1 | o.e. Stated or implied (accept in un-simplified form) |
| \(\frac{11-10.61}{\sqrt{0.1017}}\) (\(= 1.223)\) | M1 | Allow without \(\sqrt{\phantom{x}}\) |
| \(\Phi(\){'}1.223\({'}\)) | M1 | For attempt to find correct area consistent with their working |
| \(= 0.889\) (3 s.f.) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(K - 1.24 > 0)\) | M1 | Or similar stated or implied |
| Var \(= 0.0576 + 1.2^2 \times 0.0441\) | ||
| \((= 0.121104)\) | B1 B1 | o.e. May be implied (accept in un-simplified form) |
| \(N(-0.324, 0.121104)\) | ||
| \(\frac{0-(-0.324)}{\sqrt{0.121104}}\) (= 0.931) | M1 | Allow without \(\sqrt{\phantom{x}}\) |
| \(1 - \Phi(\){'}0.931\({'}\))$ | M1 | For attempt to find correct area consistent with their working |
| \(= 0.176\) (3 s.f.) | A1 [6] |
**(i)**
$N(10.61, 0.1017)$ | B1 | o.e. Stated or implied (accept in un-simplified form)
$\frac{11-10.61}{\sqrt{0.1017}}$ ($= 1.223)$ | M1 | Allow without $\sqrt{\phantom{x}}$
$\Phi(${'}1.223${'}$) | M1 | For attempt to find correct area consistent with their working
$= 0.889$ (3 s.f.) | A1 [4] |
**(ii)**
$P(K - 1.24 > 0)$ | M1 | Or similar stated or implied
Var $= 0.0576 + 1.2^2 \times 0.0441$ | |
$(= 0.121104)$ | B1 B1 | o.e. May be implied (accept in un-simplified form)
$N(-0.324, 0.121104)$ | |
$\frac{0-(-0.324)}{\sqrt{0.121104}}$ (= 0.931) | M1 | Allow without $\sqrt{\phantom{x}}$
$1 - \Phi(${'}0.931${'}$)$ | M1 | For attempt to find correct area consistent with their working
$= 0.176$ (3 s.f.) | A1 [6] |7 Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables $K \sim \mathrm {~N} ( 5.64,0.0576 )$ and $A \sim \mathrm {~N} ( 4.97,0.0441 )$ respectively. They each make a jump and measure the length. Find the probability that\\
(i) the sum of the lengths of their jumps is less than 11 m ,\\
(ii) Kieran jumps more than 1.2 times as far as Andreas.
\hfill \mbox{\textit{CAIE S2 2013 Q7 [10]}}