| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Find quantiles from CDF |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard CDF operations: finding quantiles by solving F(t)=0.9, differentiating to get the PDF, and computing mean/variance using integration. All steps are routine applications of formulas with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable $T$ having the cumulative distribution function
$$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 \\ \frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6 \\ 1 & t > 6 \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
\item Find the probability density function of $T$ for all values of $t$.
\item Calculate the mean and the variance of $T$.
\item Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2013 Q6 [16]}}