| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Estimator properties and bias |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard S3 topics (finding pdf constant, moments, CLT application, confidence intervals). Parts (i)-(iii) involve routine integration and calculus; part (iv) is direct CLT recall; part (v) applies standard confidence interval formula. While it requires multiple techniques across 5 parts, each step follows textbook procedures without requiring novel insight or complex problem-solving, making it slightly easier than average for Further Maths Statistics. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.05d Confidence intervals: using normal distribution |
1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable $T$ with probability density function
$$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$
where $k$ is a constant.\\
(i) Show that $k = \frac { 5 } { 8 }$.\\
(ii) Find the modal time.\\
(iii) Find $\mathrm { E } ( T )$ and show that $\operatorname { Var } ( T ) = \frac { 8 } { 63 }$.\\
(iv) A large random sample of $n$ fireworks of this type is tested. Write down in terms of $n$ the approximate distribution of $\bar { T }$, the sample mean time.\\
(v) For a random sample of 100 such fireworks the times are summarised as follows.
$$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$
Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.
\hfill \mbox{\textit{OCR MEI S3 2007 Q1 [18]}}