| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Multiple observations or trials |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing basic continuous uniform distribution calculations and binomial probability. Parts (a) and (b) require simple probability calculations using the uniform distribution formula, while part (c) involves recognizing a binomial distribution and computing P(X ≥ 11) where X ~ B(20, 0.75). All techniques are standard and routine for S2, though the multi-step nature and 7 total marks place it slightly below average difficulty overall. |
| Spec | 2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(L < -2.6) = 1.4 \times \frac{1}{8} = \frac{7}{40}\) or 0.175 or equivalent | B1 | |
| (b) \(P(L < -3.0 \text{ or } L > 3.0) = 2 \times (1 \times \frac{1}{8}) = \frac{1}{4}\) | M1; A1 | M1 for 1/8 seen |
| (c) \(P(\text{within 3mm}) = 1 - \frac{1}{4} = 0.75\), \(B(20, 0.75)\) recognises binomial. Using \(B(20,p)\) | B1; M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 9 / p = 0.25)\) or \(1 - P(X \leq 10 / p = 0.75)\) | M1 | |
| \(= 0.9861\) | awrt 0.9861 | A1 |
**(a)** $P(L < -2.6) = 1.4 \times \frac{1}{8} = \frac{7}{40}$ or 0.175 or equivalent | B1 |
**(b)** $P(L < -3.0 \text{ or } L > 3.0) = 2 \times (1 \times \frac{1}{8}) = \frac{1}{4}$ | M1; A1 | M1 for 1/8 seen
**(c)** $P(\text{within 3mm}) = 1 - \frac{1}{4} = 0.75$, $B(20, 0.75)$ recognises binomial. Using $B(20,p)$ | B1; M1 |
Let $X$ represent number of rods within 3mm
$P(X \leq 9 / p = 0.25)$ or $1 - P(X \leq 10 / p = 0.75)$ | M1 |
$= 0.9861$ | awrt 0.9861 | A1 |The continuous random variable $L$ represents the error, in mm, made when a machine cuts rods to a target length. The distribution of $L$ is continuous uniform over the interval $[-4.0, 4.0]$.
Find
\begin{enumerate}[label=(\alph*)]
\item P$(L < -2.6)$, [1]
\item P$(L < -3.0 \text{ or } L > 3.0)$. [2]
\end{enumerate}
A random sample of 20 rods cut by the machine was checked.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the probability that more than half of them were within 3.0 mm of the target length. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q2 [7]}}