| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Deriving sampling distribution |
| Difficulty | Moderate -0.8 This is a straightforward enumeration problem with a small sample size (n=3). Students simply list the 8 possible samples (555, 551, 515, etc.), calculate the range for each, and tabulate the probability distribution. It requires basic probability and understanding of 'range' but no complex reasoning or CLT application—significantly easier than typical A-level questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| Any two correct triples e.g. \((1,1,1), (5,5,5), (1,5,5), (1,5,1)\) | B1 | 1st B1 for any two of the triples |
| All 8 cases: \((1,1,1); (5,5,5); (1,5,5); (5,1,5); (5,5,1)(5,1,1); (1,5,1); (1,1,5)\) | B1 | 2nd B1 for all 8 cases. No incorrect extras – condone repeats. Allow \((1,5,5)(\times 3)\) and \((1,1,5)(\times 3)\) instead of writing all three cases |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(r: 0\) and \(4\) | B1 | For both values of \(r\) |
| \(P(R=0) = \frac{9}{27}\) or \(\frac{1}{3}\), \(P(R=4) = \frac{18}{27}\) or \(\frac{2}{3}\) | M1d A1 | M1 dependent on previous B1; attempt to evaluate one probability correctly e.g. \(\left(\frac{2}{3}\right)^3+\left(\frac{1}{3}\right)^3\) for \(r=0\); \(3\times\left(\frac{1}{3}\right)^2\times\left(\frac{2}{3}\right)+3\times\left(\frac{1}{3}\right)\times\left(\frac{2}{3}\right)^2\) for \(r=4\). Working must be shown. A1 for both values correct. Allow awrt 0.333 and 0.667. NB Correct answer with no working gains B1M0A0 |
## Question 1:
### Part (a)
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| Any two correct triples e.g. $(1,1,1), (5,5,5), (1,5,5), (1,5,1)$ | B1 | 1st B1 for any two of the triples |
| All 8 cases: $(1,1,1); (5,5,5); (1,5,5); (5,1,5); (5,5,1)(5,1,1); (1,5,1); (1,1,5)$ | B1 | 2nd B1 for all 8 cases. No incorrect extras – condone repeats. Allow $(1,5,5)(\times 3)$ and $(1,1,5)(\times 3)$ instead of writing all three cases |
### Part (b)
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $r: 0$ and $4$ | B1 | For both values of $r$ |
| $P(R=0) = \frac{9}{27}$ or $\frac{1}{3}$, $P(R=4) = \frac{18}{27}$ or $\frac{2}{3}$ | M1d A1 | M1 dependent on previous B1; attempt to evaluate one probability correctly e.g. $\left(\frac{2}{3}\right)^3+\left(\frac{1}{3}\right)^3$ for $r=0$; $3\times\left(\frac{1}{3}\right)^2\times\left(\frac{2}{3}\right)+3\times\left(\frac{1}{3}\right)\times\left(\frac{2}{3}\right)^2$ for $r=4$. **Working must be shown.** A1 for both values correct. Allow awrt 0.333 and 0.667. **NB** Correct answer with no working gains B1M0A0 |
---\begin{enumerate}
\item A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .
\end{enumerate}
A random sample of 3 counters is selected.\\
(a) List all possible samples.\\
(b) Find the sampling distribution for the range.\\
\hfill \mbox{\textit{Edexcel S2 2013 Q1 [5]}}