| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Single sum threshold probability |
| Difficulty | Standard +0.3 This question tests standard linear combinations of normal distributions with straightforward applications. Part (i) requires summing 10 independent normals (routine variance scaling), and part (ii) involves forming L - 2S and standardizing. Both are textbook exercises requiring only formula application and normal table lookup, making it slightly easier than average. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| mean \(= 3250\), var. \(= 61\) | B1 | Or mean \(= 325\), var. \(= \frac{6.1}{10}\) |
| \(\frac{3240 - 3250}{\sqrt{61}} = -1.280\) | M1 | Standardise with their values (no mixed methods) |
| \(\phi(-1.280) = 1 - \phi(1.280)\) | M1 | Area consistent with their figures |
| \(0.100\) | A1 | Allow 0.1 |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(D) = 325 - 2 \times 167 = -9\) | B1 | Accept \(\pm 9\) |
| \(\text{Var}(D) = 6.1 + 2^2 \times 5.6 = 28.5\) | B1 | |
| \(\frac{0-(-9)}{\sqrt{28.5}} = 1.686\) | M1 | Standardising with *their* values. Must have a combination attempt on denominator and \(\sqrt{\phantom{x}}\) |
| \(1 - \phi(1.686)\) | M1 | Area consistent with their figures |
| \(0.0459\) | A1 | |
| 5 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| mean $= 3250$, var. $= 61$ | B1 | Or mean $= 325$, var. $= \frac{6.1}{10}$ |
| $\frac{3240 - 3250}{\sqrt{61}} = -1.280$ | M1 | Standardise with their values (no mixed methods) |
| $\phi(-1.280) = 1 - \phi(1.280)$ | M1 | Area consistent with their figures |
| $0.100$ | A1 | Allow 0.1 |
| | **4** | |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(D) = 325 - 2 \times 167 = -9$ | B1 | Accept $\pm 9$ |
| $\text{Var}(D) = 6.1 + 2^2 \times 5.6 = 28.5$ | B1 | |
| $\frac{0-(-9)}{\sqrt{28.5}} = 1.686$ | M1 | Standardising with *their* values. Must have a combination attempt on denominator and $\sqrt{\phantom{x}}$ |
| $1 - \phi(1.686)$ | M1 | Area consistent with their figures |
| $0.0459$ | A1 | |
| | **5** | |5 The masses, in grams, of large boxes of chocolates and small boxes of chocolates have the distributions $\mathrm { N } ( 325,6.1 )$ and $\mathrm { N } ( 167,5.6 )$ respectively.\\
(i) Find the probability that the total mass of 10 randomly chosen large boxes of chocolates is less than 3240 g .\\
(ii) Find the probability that the mass of a randomly chosen large box of chocolates is more than twice the mass of a randomly chosen small box of chocolates.\\
\hfill \mbox{\textit{CAIE S2 2019 Q5 [9]}}