| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Multiple stage process probability |
| Difficulty | Challenging +1.2 This question requires understanding of linear combinations of normal random variables and the distribution of sums/differences of normals. Part (i) involves finding P(X₂ < 0.5X₁), requiring the distribution of 2X₂ - X₁. Part (ii) is more straightforward, summing independent normals. While conceptually demanding for S3 level, the calculations are standard once the correct distributions are identified. The independence question in (iii) is routine recall. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(T_1 \sim N(2.2, 0.75^2)\), \(T_2 \sim N(1.8, 0.70^2)\); Use \(T_2 - \frac{1}{2}T_1\) normal | M1 | Or \(\frac{1}{2}T_1 - T_2\) |
| \(\mu = 0.7\) | A1 | |
| \(\sigma^2 = 0.7^2 + \frac{1}{4} \times 0.75^2\ (0.630625)\) | A1 | |
| \((0 - \mu)/\sigma\) | M1 | From reasonable \(\sigma^2\) not just sum |
| \(-0.881\) | A1 | \(+\) or \(-\) |
| Probability \(0.189\) | A1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use sum of 5 \(T\)s | M1 | |
| \(\mu = 9.4\) | A1 | |
| \(\sigma^2 = 2.5225\) | A1 | |
| \(z = (10 - \mu)/\sigma\) | M1 | Standardising, must be \(\sigma\) |
| Probability \(0.6473\), \(0.647\) | A1 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Calculation of variance | B1 | 1 |
# Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T_1 \sim N(2.2, 0.75^2)$, $T_2 \sim N(1.8, 0.70^2)$; Use $T_2 - \frac{1}{2}T_1$ normal | M1 | Or $\frac{1}{2}T_1 - T_2$ |
| $\mu = 0.7$ | A1 | |
| $\sigma^2 = 0.7^2 + \frac{1}{4} \times 0.75^2\ (0.630625)$ | A1 | |
| $(0 - \mu)/\sigma$ | M1 | From reasonable $\sigma^2$ not just sum |
| $-0.881$ | A1 | $+$ or $-$ |
| Probability $0.189$ | A1 | **6** | |
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# Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use sum of 5 $T$s | M1 | |
| $\mu = 9.4$ | A1 | |
| $\sigma^2 = 2.5225$ | A1 | |
| $z = (10 - \mu)/\sigma$ | M1 | Standardising, must be $\sigma$ |
| Probability $0.6473$, $0.647$ | A1 | **5** | |
---
# Question 7(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Calculation of variance | B1 | **1** | |
---7 A queue of cars has built up at a set of traffic lights which are at red. When the lights turn green, the time for the first car to start to move has a normal distribution with mean 2.2 s and standard deviation 0.75 s . This time is the reaction time for the first car. For each subsequent car the reaction time is the time taken for it to start to move after the car in front starts to move. These reaction times have identical normal distributions with mean 1.8 s and standard deviation 0.70 s . It may be assumed that all reaction times are independent.\\
(i) Calculate the probability that the reaction time for the second car in the queue is less than half of the reaction time for the first car.\\
(ii) Calculate the probability that the fifth car in the queue starts to move less than 10 seconds after the lights turn green.\\
(iii) State where, in part (i), independence is required.
\hfill \mbox{\textit{OCR S3 2006 Q7 [12]}}