| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Expectation and variance with context application |
| Difficulty | Standard +0.3 This question tests standard linear transformations of normal variables (part i) and forming/calculating probabilities for linear combinations (part ii). While it requires understanding of E(aX+b) and Var(aX+b) rules plus combining independent normals, these are core S2 techniques applied in a straightforward context with no conceptual surprises—slightly easier than average A-level. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(E(\text{cost to Stella}) = 600 + 5.52 \times 500 = 3360\) | M1 | For multiplying by 5.52 and adding 600 |
| A1 | Correct mean | |
| \(\text{Var}(\text{cost to Stella}) = 5.52^2 \times 7.1^2 = 1540(1536)\) | M1 | For mult \(7.1^2/7.1^2/50.41\) by \(5.52^2\) |
| M1 | For \(5.52^{(2)} \times 7.1^{(2)}\) or \(50.41^2\) with no addition/subtraction | |
| A1 | 5 | For correct answer |
| (ii) \(P(D > 2S) = P(D - 2S > 0)\) | M1 | For attempt \((D-2S)\) (or equiv) either \(<\) or \(>\) |
| \(D - 2S \sim N(-120, 421 + 4 \times 1536) \sim N(-120, 6565)\) | B1 | For correct mean (seen or implied) |
| A1 ft | For correct unsimplified variance | |
| \(P(D - 2S > 0) = P\left(z > \frac{120}{\sqrt{6565}}\right) = P(z > 1.481) = 0.0693\) | M1 | For standardising attempt |
| A1 | 5 | For correct answer, accept 0.069 |
**(i)** $E(\text{cost to Stella}) = 600 + 5.52 \times 500 = 3360$ | M1 | For multiplying by 5.52 and adding 600
| A1 | Correct mean
$\text{Var}(\text{cost to Stella}) = 5.52^2 \times 7.1^2 = 1540(1536)$ | M1 | For mult $7.1^2/7.1^2/50.41$ by $5.52^2$
| M1 | For $5.52^{(2)} \times 7.1^{(2)}$ or $50.41^2$ with **no addition/subtraction**
| A1 | 5 | For correct answer
**(ii)** $P(D > 2S) = P(D - 2S > 0)$ | M1 | For attempt $(D-2S)$ (or equiv) either $<$ or $>$
$D - 2S \sim N(-120, 421 + 4 \times 1536) \sim N(-120, 6565)$ | B1 | For correct mean (seen or implied)
| A1 ft | For correct unsimplified variance
$P(D - 2S > 0) = P\left(z > \frac{120}{\sqrt{6565}}\right) = P(z > 1.481) = 0.0693$ | M1 | For standardising attempt
| A1 | 5 | For correct answer, accept 0.069
---4 The cost of electricity for a month in a certain town under scheme $A$ consists of a fixed charge of 600 cents together with a charge of 5.52 cents per unit of electricity used. Stella uses scheme $A$. The number of units she uses in a month is normally distributed with mean 500 and variance 50.41.\\
(i) Find the mean and variance of the total cost of Stella's electricity in a randomly chosen month.
Under scheme $B$ there is no fixed charge and the cost in cents for a month is normally distributed with mean 6600 and variance 421. Derek uses scheme $B$.\\
(ii) Find the probability that, in a randomly chosen month, Derek spends more than twice as much as Stella spends.
\hfill \mbox{\textit{CAIE S2 2007 Q4 [10]}}