| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of scaled variable |
| Difficulty | Moderate -0.8 This question tests standard properties of linear transformations of normal distributions (scaling by constant, adding independent normals). Part (i) requires direct application of formulas: μ_new = 1.5μ, σ_new = 1.5σ. Part (ii) involves finding the distribution of X + Y where Y = 1.5X (though they're independent samples), then a routine normal probability calculation. Straightforward application of well-practiced techniques with no conceptual challenges. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{mean} = 155.1\) | B1 | |
| \(\text{var} = 1.5^2 \times 10.2 \;(= 22.95)\), \(\text{sd} = \sqrt{"22.95"}\) | M1 | or \(1.5 \times \sqrt{10.2}\) |
| \(= 4.79\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{mean} = 103.4 + "155.1" (= 258.5)\); \(\text{var} = 10.2 + "22.95" (= 33.15)\) | B1ft | Both. ft their 155.1 and 22.95. Accept sd. |
| \(\frac{250 - "258.5"}{\sqrt{"33.15"}} \;(= -1.476)\) | M1 | Standardising – no sd/var mix. Their mean/sd must be from an attempt at combination |
| \(1 - \phi(-1.476) = \phi(1.476)\) | M1 | For area consistent with their working |
| \(= 0.930\) (3 sf) | A1 | Allow 0.93 |
**Question 4(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{mean} = 155.1$ | B1 | |
| $\text{var} = 1.5^2 \times 10.2 \;(= 22.95)$, $\text{sd} = \sqrt{"22.95"}$ | M1 | or $1.5 \times \sqrt{10.2}$ |
| $= 4.79$ | A1 | |
**Question 4(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{mean} = 103.4 + "155.1" (= 258.5)$; $\text{var} = 10.2 + "22.95" (= 33.15)$ | B1ft | Both. ft their 155.1 and 22.95. Accept sd. |
| $\frac{250 - "258.5"}{\sqrt{"33.15"}} \;(= -1.476)$ | M1 | Standardising – no sd/var mix. Their mean/sd must be from an attempt at combination |
| $1 - \phi(-1.476) = \phi(1.476)$ | M1 | For area consistent with their working |
| $= 0.930$ (3 sf) | A1 | Allow 0.93 |
---4 The volume, in millilitres, of a small cup of coffee has the distribution $\mathrm { N } ( 103.4,10.2 )$. The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.\\
(i) Find the mean and standard deviation of the volume of a large cup of coffee.\\
(ii) Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .\\
\hfill \mbox{\textit{CAIE S2 2018 Q4 [7]}}