| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find median or percentiles |
| Difficulty | Standard +0.3 This is a straightforward continuous probability distribution question requiring standard techniques: integrating the pdf to find a percentile (solving ∫f(t)dt = 0.9), and computing E(T) and Var(T) using standard formulas. The pdf is simple (power function), the integration is routine, and all steps follow directly from learned procedures with no problem-solving insight required. Slightly easier than average due to the simple functional form. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\dfrac{1}{2}\int_4^t \dfrac{1}{\sqrt{t}}\,dt = 0.9\) or \(\dfrac{1}{2}\int_t^9 \dfrac{1}{\sqrt{t}}\,dt = 0.1\) | M1 | Attempt integrating \(f(t)\) with unknown limit and 0.9/0.1 |
| \(\left[\sqrt{t}\right]_4^t = 0.9\) or \(\left[\sqrt{t}\right]_t^9 = 0.1\) | A1 | Correct integration & limits \(= 0.9\) or \(0.1\) |
| \(((\sqrt{t} - 2) = 0.9\) or \((3 - \sqrt{t}) = 0.1)\); \(t = 8.41\) (mins) (3 sf) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\dfrac{1}{2}\int_4^9 \dfrac{t}{\sqrt{t}}\,dt\) oe | M1 | Attempt integrating \(t\,f(t)\). Ignore limits |
| \(\dfrac{1}{2}\left[\dfrac{t^{1.5}}{1.5}\right]_4^9\) oe \(= \dfrac{19}{3}\) | A1, A1 | Correct integration & limits |
| \(\dfrac{1}{2}\int_4^9 \dfrac{t^2}{\sqrt{t}}\,dt\) oe | M1 | Attempt integrating \(t^2 f(t)\). Ignore limits |
| \(\left(= \dfrac{1}{2}\left[\dfrac{t^{2.5}}{2.5}\right]_4^9 = \dfrac{211}{5}\right)\); \(\dfrac{211}{5} - \left(\dfrac{19}{3}\right)^2\) | M1 | \(\int t^2 f(t) - \left(\int t\,f(t)\right)^2\) attempted |
| \(= \dfrac{94}{45}\) or \(2.09\) (3 sf) | A1 [6] |
## Question 6:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{1}{2}\int_4^t \dfrac{1}{\sqrt{t}}\,dt = 0.9$ or $\dfrac{1}{2}\int_t^9 \dfrac{1}{\sqrt{t}}\,dt = 0.1$ | M1 | Attempt integrating $f(t)$ with unknown limit and 0.9/0.1 |
| $\left[\sqrt{t}\right]_4^t = 0.9$ or $\left[\sqrt{t}\right]_t^9 = 0.1$ | A1 | Correct integration & limits $= 0.9$ or $0.1$ |
| $((\sqrt{t} - 2) = 0.9$ or $(3 - \sqrt{t}) = 0.1)$; $t = 8.41$ (mins) (3 sf) | A1 [3] | |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\dfrac{1}{2}\int_4^9 \dfrac{t}{\sqrt{t}}\,dt$ oe | M1 | Attempt integrating $t\,f(t)$. Ignore limits |
| $\dfrac{1}{2}\left[\dfrac{t^{1.5}}{1.5}\right]_4^9$ oe $= \dfrac{19}{3}$ | A1, A1 | Correct integration & limits |
| $\dfrac{1}{2}\int_4^9 \dfrac{t^2}{\sqrt{t}}\,dt$ oe | M1 | Attempt integrating $t^2 f(t)$. Ignore limits |
| $\left(= \dfrac{1}{2}\left[\dfrac{t^{2.5}}{2.5}\right]_4^9 = \dfrac{211}{5}\right)$; $\dfrac{211}{5} - \left(\dfrac{19}{3}\right)^2$ | M1 | $\int t^2 f(t) - \left(\int t\,f(t)\right)^2$ attempted |
| $= \dfrac{94}{45}$ or $2.09$ (3 sf) | A1 [6] | |
---6 The time in minutes taken by people to read a certain booklet is modelled by the random variable $T$ with probability density function given by
$$f ( t ) = \begin{cases} \frac { 1 } { 2 \sqrt { } t } & 4 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
(i) Find the time within which $90 \%$ of people finish reading the booklet.\\
(ii) Find $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
\hfill \mbox{\textit{CAIE S2 2013 Q6 [9]}}