| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | State distribution of sample mean |
| Difficulty | Standard +0.3 This is a straightforward S3 question testing standard techniques: finding percentiles from a CDF (inverse function), differentiating to get PDF, integration for expectation, and applying the Central Limit Theorem. All parts follow routine procedures with no novel problem-solving required. The calculus involved (differentiating e^{-1/t} and integrating te^{-1/t}) is moderately technical but well within A-level scope. Slightly easier than average due to the step-by-step structure and given variance. |
| 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 integration5.03d E(g(X)): general expectation formula |
A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable $T$ with the following cumulative distribution function.
$$F(t) = \begin{cases}
0 & t \leq 0 \\
1 - e^{-\frac{1}{t}} & t > 0
\end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item Find the median delay and the 90th percentile delay. [5]
\item Derive the probability density function of $T$. Hence use calculus to find the mean delay. [5]
\item Find the probability that a delay lasts longer than the mean delay. [2]
\end{enumerate}
You are given that the variance of $T$ is 9.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Let $\overline{T}$ denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of $\overline{T}$. [3]
\item A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S3 2006 Q1 [18]}}