| 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 | Single sum threshold probability |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal variables and forming composite distributions. Part (i) is straightforward (sum of 5 identical normals), but part (ii) requires the non-trivial insight to form E - 3L ~ N(μ, σ²) and correctly handle the variance of 3L, which goes beyond routine application and tests deeper conceptual understanding of transformations. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(N(5100,\ 5 \times 45^2)\) or \(N(5100, 10125)\) | B1 | Seen or implied |
| \(\frac{5200 - 5100}{\sqrt{10125}}\ (= 0.994)\) | M1 | Standardising with their new mean and new var; area consistent with their working with normal |
| \(\Phi(0.994)\) | M1 | |
| \(= 0.840\) (3 sf) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Use of \(E - 3L\) or similar | M1 | |
| \(E(E - 3L) = -260\) | B1 | \(2800 - 3 \times 1020\) |
| \(\text{Var}(E - 3L) = 52^2 + 9 \times 45^2\) or \(20929\) | B1 | |
| \(\frac{0 - (-260)}{\sqrt{20929}}\ (= 1.797)\) | M1 | With a positive var with \(45^2\) and \(52^2\) combined |
| \(1 - \Phi(1.797)\) | M1 | Consistent area, must clearly be \(\varphi\) |
| \(= 0.0361\) (3 sf) or \(0.0362\) | A1 | [6] \(P(3L - E < 0)\): similar scheme. SR: use of \(3E - L\), M1, 7380 B1, 26361 B1; stand 0 with these values M1, M0A0 max 4/6 |
## Question 6(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $N(5100,\ 5 \times 45^2)$ or $N(5100, 10125)$ | B1 | Seen or implied |
| $\frac{5200 - 5100}{\sqrt{10125}}\ (= 0.994)$ | M1 | Standardising with their new mean and new var; area consistent with their working with normal |
| $\Phi(0.994)$ | M1 | |
| $= 0.840$ (3 sf) | A1 | **[4]** |
## Question 6(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Use of $E - 3L$ or similar | M1 | |
| $E(E - 3L) = -260$ | B1 | $2800 - 3 \times 1020$ |
| $\text{Var}(E - 3L) = 52^2 + 9 \times 45^2$ or $20929$ | B1 | |
| $\frac{0 - (-260)}{\sqrt{20929}}\ (= 1.797)$ | M1 | With a positive var with $45^2$ and $52^2$ combined |
| $1 - \Phi(1.797)$ | M1 | Consistent area, must clearly be $\varphi$ |
| $= 0.0361$ (3 sf) or $0.0362$ | A1 | **[6]** $P(3L - E < 0)$: similar scheme. SR: use of $3E - L$, M1, 7380 B1, 26361 B1; stand 0 with these values M1, M0A0 max 4/6 |6 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions $\mathrm { N } \left( 1020,45 ^ { 2 } \right)$ and $\mathrm { N } \left( 2800,52 ^ { 2 } \right)$ respectively.\\
(i) Find the probability that the total of the lifetimes of 5 randomly chosen Longlive bulbs is less than 5200 hours.\\
(ii) Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least 3 times that of a randomly chosen Longlive bulb.
\hfill \mbox{\textit{CAIE S2 2013 Q6 [10]}}