| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| 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 the continuous uniform distribution to a familiar context (rounding error). Part (a) requires explaining why the pdf has the given form (routine reasoning about rounding), part (b) is a direct probability calculation using the rectangular distribution, and part (c) involves recalling and applying standard formulas for mean and variance of a uniform distribution. All parts are textbook-standard with no novel problem-solving required. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}\) with \((-0.05, 0.05)\) | B1 | explain error \(\pm 0.05\) |
| \(\frac{1}{b-a} = \frac{1}{0.05-(-0.05)} = \frac{1}{0.1} = 10\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(-0.01 < X < 0.02) = 0.03 \times 10 = 0.3\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mean \(= 0\) | B1 | CAO |
| Standard deviation \(= 0.0289\) | B1 | \(\frac{1}{20\sqrt{3}}\) OE |
# Question 4(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$ with $(-0.05, 0.05)$ | B1 | explain error $\pm 0.05$ |
| $\frac{1}{b-a} = \frac{1}{0.05-(-0.05)} = \frac{1}{0.1} = 10$ | M1, A1 | |
# Question 4(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(-0.01 < X < 0.02) = 0.03 \times 10 = 0.3$ | M1, A1 | |
# Question 4(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mean $= 0$ | B1 | CAO |
| Standard deviation $= 0.0289$ | B1 | $\frac{1}{20\sqrt{3}}$ OE |
---4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.
\begin{enumerate}[label=(\alph*)]
\item Given that the rounding error, $X$ metres, in these measurements has a rectangular distribution, explain why its probability density function is
$$f ( x ) = \left\{ \begin{array} { c c }
10 & - 0.05 < x \leqslant 0.05 \\
0 & \text { otherwise }
\end{array} \right.$$
\item Calculate $\mathrm { P } ( - 0.01 < X < 0.02 )$.
\item Find the mean and the standard deviation of $X$.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q4 [7]}}