| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Total journey time probabilities |
| Difficulty | Standard +0.3 Part (i) requires forming a linear combination T₁+T₂ and standardizing—a direct application of the sum of independent normals. Part (ii) requires forming T₂-1.2T₁ and finding its distribution, which is slightly less routine but still a standard S2 technique. Both parts are straightforward applications of normal distribution properties with no novel problem-solving required. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(T_1 + T_2 \sim N(5,\ 0.4^2 + 0.5^2)\) | B1 | or \(N(5,\ 0.41)\) |
| \(\frac{6 - 5}{\sqrt{0.41'}}\ (= 1.562)\) | M1 | Allow cc |
| \(\Phi(1.562')\) | M1 | Correct area consistent with their working |
| \(= 0.941\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(T_2 - 1.2T_1) = 0.5^2 + 1.2^2 \times 0.4^2\ (= 0.4804)\) | B1 | Or similar using \(1.2T_1 - T_2\) |
| \(T_2 - 1.2T_1 \sim N(0.16,\ 0.4804)\) | B1 ft | Only ft attempt at combination. no ft for neg var. |
| \(\frac{0 - 0.16'}{\sqrt{0.4804'}}\ (= -0.231)\) | M1 | Standardise with their mean and variance. Allow cc |
| \(P(T_2 - 1.2T_1) > 0\) | ||
| \(= \Phi(0.231')\) | M1 | Correct area consistent with their working |
| \(= 0.591\) (3 sfs) | A1 | |
| Total: 5 |
## Question 5:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $T_1 + T_2 \sim N(5,\ 0.4^2 + 0.5^2)$ | B1 | or $N(5,\ 0.41)$ |
| $\frac{6 - 5}{\sqrt{0.41'}}\ (= 1.562)$ | M1 | Allow cc |
| $\Phi(1.562')$ | M1 | Correct area consistent with their working |
| $= 0.941$ | A1 | |
| **Total: 4** | | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(T_2 - 1.2T_1) = 0.5^2 + 1.2^2 \times 0.4^2\ (= 0.4804)$ | B1 | Or similar using $1.2T_1 - T_2$ |
| $T_2 - 1.2T_1 \sim N(0.16,\ 0.4804)$ | B1 ft | Only ft attempt at combination. no ft for neg var. |
| $\frac{0 - 0.16'}{\sqrt{0.4804'}}\ (= -0.231)$ | M1 | Standardise with their mean and variance. Allow cc |
| $P(T_2 - 1.2T_1) > 0$ | | |
| $= \Phi(0.231')$ | M1 | Correct area consistent with their working |
| $= 0.591$ (3 sfs) | A1 | |
| **Total: 5** | | |5 The times, in months, taken by a builder to build two types of house, $P$ and $Q$, are represented by the independent variables $T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)$ and $T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)$ respectively.\\
(i) Find the probability that the total time taken to build one house of each type is less than 6 months.\\
(ii) Find the probability that the time taken to build a type $Q$ house is more than 1.2 times the time taken to build a type $P$ house.\\
\hfill \mbox{\textit{CAIE S2 2018 Q5 [9]}}