| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find or specify CDF |
| Difficulty | Standard +0.3 This is a straightforward S3 question requiring integration of a sine function to find the CDF (with the answer given), then using expectation with a piecewise constant function. The integration is routine, and part (ii) involves basic probability and solving a simple equation. Slightly easier than average due to the guided nature and standard techniques. |
| Spec | 1.08d Evaluate definite integrals: between limits5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use of \(\int_{20}^{a} f(t)dt\) | M1 | With limits and \(f(t)\) substituted |
| \(\left[-\frac{2}{3}\cos\frac{\pi t}{60}\right]_{20}^a\) | A1 | |
| AG | A1 | 3 |
| (ii) \(3 \times (1) + 2 \times (1-(1))\) | M1 | Idea of expectation |
| A1 | All correct | |
| Equate to 2.80 and attempt to solve | M1 | From equation in a, 2 or 3 |
| \(a=44.8\) | A1 | 4 |
| SR: \(\frac{1}{3}(1-2\cos....) = 0.8\) give max3/4 |
(i) Use of $\int_{20}^{a} f(t)dt$ | M1 | With limits and $f(t)$ substituted
$\left[-\frac{2}{3}\cos\frac{\pi t}{60}\right]_{20}^a$ | A1 |
AG | A1 | 3 | Properly obtained
(ii) $3 \times (1) + 2 \times (1-(1))$ | M1 | Idea of expectation
| A1 | All correct
Equate to 2.80 and attempt to solve | M1 | From equation in a, 2 or 3
$a=44.8$ | A1 | 4 | Accept 45
SR: $\frac{1}{3}(1-2\cos....) = 0.8$ give max3/4 | | |
---3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable $T$ with probability density function given by
$$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60 \\ 0 & \text { otherwise. } \end{cases}$$
(i) Given that $20 \leqslant a \leqslant 60$, show that $\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)$.
There is a delivery charge of $\pounds 3$ but this is reduced to $\pounds 2$ if the delivery time exceeds a minutes.\\
(ii) Find the value of $a$ for which the expected value of the delivery charge for a home-delivery is £2.80.
\hfill \mbox{\textit{OCR S3 2006 Q3 [7]}}