| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Multiple observations or trials |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing basic uniform distribution probability calculations, followed by standard binomial probability (part c) and normal approximation to binomial (part d). All techniques are routine applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y = \text{no. of lengths with } | X | < 1.5\) \(\therefore Y \sim B(10, 0.3)\) |
| \(P(Y > 5) = 1 - P(Y \leq 5) = 1 - 0.9527 = 0.0473\) | M1, A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(R = \text{no. of lengths of piping rejected}\) | ||
| \(R \sim B(60, 0.08) \Rightarrow R \approx \text{Po}(4.8)\) | B1 | |
| \(P(R \leq 2) = e^{-4.8}\left[1 + 4.8 + \frac{(4.8)^2}{2!}\right]\) | M1, M1, A1 | Po and \(\leq 2\), formula |
| \(= 17.32 \times e^{-4.8} = 0.1425\ldots\) (accept awrt 0.143) | A1 cao | (5) |
## Part (c)
$Y = \text{no. of lengths with } |X| < 1.5$ $\therefore Y \sim B(10, 0.3)$ | M1 | |
$P(Y > 5) = 1 - P(Y \leq 5) = 1 - 0.9527 = 0.0473$ | M1, A1 | (3) |
## Part (d)
$R = \text{no. of lengths of piping rejected}$ | | |
$R \sim B(60, 0.08) \Rightarrow R \approx \text{Po}(4.8)$ | B1 | |
$P(R \leq 2) = e^{-4.8}\left[1 + 4.8 + \frac{(4.8)^2}{2!}\right]$ | M1, M1, A1 | Po and $\leq 2$, formula |
$= 17.32 \times e^{-4.8} = 0.1425\ldots$ (accept awrt 0.143) | A1 cao | (5) |
---2. The continuous random variable $X$ represents the error, in mm, made when a machine cuts piping to a target length. The distribution of $X$ is rectangular over the interval $[ - 5.0,5.0 ]$.
Find
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( X < - 4.2 )$,
\item $\mathrm { P } ( | X | < 1.5 )$.
A supervisor checks a random sample of 10 lengths of piping cut by the machine.
\item Find the probability that more than half of them are within 1.5 cm of the target length.\\
(3 marks)\\
If $X < - 4.2$, the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
\item Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.\\
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q2 [11]}}