| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | All components random including container |
| Difficulty | Moderate -0.8 This is a straightforward application of standard results for linear combinations of independent normal random variables. Part (i) requires only direct use of formulas for mean and variance of sums (E[aX+bY]=aE[X]+bE[Y] and Var[aX+bY]=a²Var[X]+b²Var[Y]), while part (ii) involves a routine inverse normal calculation using tables or calculator. No problem-solving insight or novel reasoning is required—purely mechanical application of learned techniques. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 28 | B1 | |
| \(4 \times 0.1^2 + 3 \times 0.2^2\) | M1 | Not \(4^2\), \(3^2\) |
| 0.16 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{x - \text{"28"}}{\sqrt{\text{"0.16"}}}\) | M1 | |
| \(= 2.326\) | B1 | |
| 28.9 | A1 |
## Question 1:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 28 | B1 | |
| $4 \times 0.1^2 + 3 \times 0.2^2$ | M1 | Not $4^2$, $3^2$ |
| 0.16 | A1 | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{x - \text{"28"}}{\sqrt{\text{"0.16"}}}$ | M1 | |
| $= 2.326$ | B1 | |
| 28.9 | A1 | |
---1 A laminate consists of 4 layers of material $C$ and 3 layers of material $D$. The thickness of a layer of material $C$ has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material $D$ has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.\\
(i) Find the mean and variance of the total thickness of the laminate.\\
(ii) What total thickness is exceeded by $1 \%$ of the laminates?
\hfill \mbox{\textit{OCR S3 2015 Q1 [6]}}