| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find median or percentiles |
| Difficulty | Standard +0.8 This S2 question requires multiple skills: verifying the pdf integrates to 1 (involving piecewise integration), finding the median (solving a quadratic equation from cumulative distribution), sketching a piecewise function, and critically evaluating a statistical model. While each component is standard S2 material, the combination of techniques and the need for statistical reasoning about model validity makes this moderately challenging, above average difficulty. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks |
|---|---|
| (a) Graph sketched: \(\frac{1}{2} \times 2c \times 5 = 1\), so \(c = \frac{1}{5}\) | B3 M1 A1 |
| (b) Median \(m\) has \(P(4 < X < m) = \frac{1}{2} - \frac{2}{5} = \frac{1}{10}\) | M1 A1 |
| \(\frac{2}{15}\int_m^7 t-1 \, dt = \frac{1}{10}\) leading to \(7m - \frac{1}{2}m^2 - 20 = \frac{3}{4}\) | M1 A1 M1 A1 |
| \(2m^2 - 28m + 83 = 0\), so \(m = 4.26\) | A1 M1 A1 |
| (c) Sections of graph will not be precise straight lines in reality; some people will take longer than 7 hours | B1; B1 |
(a) Graph sketched: $\frac{1}{2} \times 2c \times 5 = 1$, so $c = \frac{1}{5}$ | B3 M1 A1 |
(b) Median $m$ has $P(4 < X < m) = \frac{1}{2} - \frac{2}{5} = \frac{1}{10}$ | M1 A1 |
$\frac{2}{15}\int_m^7 t-1 \, dt = \frac{1}{10}$ leading to $7m - \frac{1}{2}m^2 - 20 = \frac{3}{4}$ | M1 A1 M1 A1 |
$2m^2 - 28m + 83 = 0$, so $m = 4.26$ | A1 M1 A1 |
(c) Sections of graph will not be precise straight lines in reality; some people will take longer than 7 hours | B1; B1 |
**Total: 16 marks**7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable $T$ with the probability density function
$$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4 \\ \frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the function $\mathrm { f } ( t )$, and show that $c = \frac { 1 } { 5 }$.
\item Calculate the median value of $T$.
\item Make two critical comments about the model.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q7 [16]}}