| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Measurement error modeling |
| Difficulty | Moderate -0.8 This is a straightforward application of continuous uniform distribution properties requiring only basic recall and simple calculations: finding the normalizing constant k (1/10), the mean (1), P(X>0) by integration (0.6), and P(|X|>3.5) by considering two intervals (0.25). All parts are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = 0.1\) | B1 (1 mark) | OE |
| Answer | Marks |
|---|---|
| \(E(X) = 1\) | B1 (1 mark) |
| Answer | Marks |
|---|---|
| \(P(X > 0) = 6 \times 0.1 = 0.6\) | M1, A1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P( | X | > 3.5) = 1 - P( |
| Answer | Marks |
|---|---|
| \(P(X < -3.5) + P(X > 3.5) = \frac{0.5}{10} + \frac{2.5}{10} = \frac{3}{10}\) | (M1), (A1), (A1) |
## Question 4:
### Part (a)
$k = 0.1$ | B1 (1 mark) | OE
### Part (b)
$E(X) = 1$ | B1 (1 mark)
### Part (c)
$P(X > 0) = 6 \times 0.1 = 0.6$ | M1, A1 (2 marks)
### Part (d)
$P(|X| > 3.5) = 1 - P(|X| < 3.5) = 1 - 0.7 = 0.3$ | M1, A1, A1 (3 marks)
**Alternative:**
$P(X < -3.5) + P(X > 3.5) = \frac{0.5}{10} + \frac{2.5}{10} = \frac{3}{10}$ | (M1), (A1), (A1)
---4 The error, $X$ millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function.
$$f ( x ) = \begin{cases} k & - 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item State the value of $k$.
\item Write down the value of $\mathrm { E } ( X )$.
\item Calculate $\mathrm { P } ( X > 0 )$.
\item The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2005 Q4 [7]}}