| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Deriving sampling distribution |
| Difficulty | Moderate -0.8 This is a straightforward S2 question requiring basic probability calculations and enumeration of a small sample space. Part (a) involves routine mean/variance calculations from a discrete distribution, while parts (b) and (c) require listing only 6 possible samples and computing their probabilities—mechanical work with no conceptual challenge or connection to the central limit theorem despite the topic tag. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{X}\): \(1, 2, 5\) with \(P(X=x)\): \(\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\) | ||
| \(\text{Mean} = 1\times\frac{1}{2} + 2\times\frac{1}{3} + 5\times\frac{1}{6} = 2\) | M1A1 | \(\sum x \cdot p(x)\); need \(\frac{1}{2}\) and \(\frac{1}{3}\) |
| \(\text{Variance} = 1^2\times\frac{1}{2} + 2^2\times\frac{1}{3} + 5^2\times\frac{1}{6} - 2^2 = 2\) | M1A1 | \(\sum x^2 \cdot p(x) - \lambda^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Pairs listed: \((1,1)\); \((1,2)\) and \((2,1)\); \((1,5)\) and \((5,1)\) | B2 | LHS \(-1\) |
| \((2,2)\); \((2,5)\) and \((5,2)\); \((5,5)\) | B1 | repeat of "theirs" on RHS |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x}\): \(1, 1.5, 2, 3, 3.5, 5\) with probabilities \(\frac{1}{4}, \cdot, \frac{1}{9}, \frac{1}{6}, \cdot, \frac{1}{36}\) | M1A1 | \(\frac{1}{4}\) correct |
| All remaining probabilities correct | M1A2 | 1.5+, \(-1\) each error |
# Question 6:
## Part (a)
| $\bar{X}$: $1, 2, 5$ with $P(X=x)$: $\frac{1}{2}, \frac{1}{3}, \frac{1}{6}$ | | |
| $\text{Mean} = 1\times\frac{1}{2} + 2\times\frac{1}{3} + 5\times\frac{1}{6} = 2$ | M1A1 | $\sum x \cdot p(x)$; need $\frac{1}{2}$ and $\frac{1}{3}$ |
| $\text{Variance} = 1^2\times\frac{1}{2} + 2^2\times\frac{1}{3} + 5^2\times\frac{1}{6} - 2^2 = 2$ | M1A1 | $\sum x^2 \cdot p(x) - \lambda^2$ |
## Part (b)
| Pairs listed: $(1,1)$; $(1,2)$ and $(2,1)$; $(1,5)$ and $(5,1)$ | B2 | LHS $-1$ |
| $(2,2)$; $(2,5)$ and $(5,2)$; $(5,5)$ | B1 | repeat of "theirs" on RHS |
## Part (c)
| $\bar{x}$: $1, 1.5, 2, 3, 3.5, 5$ with probabilities $\frac{1}{4}, \cdot, \frac{1}{9}, \frac{1}{6}, \cdot, \frac{1}{36}$ | M1A1 | $\frac{1}{4}$ correct |
| All remaining probabilities correct | M1A2 | 1.5+, $-1$ each error |6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of the value of the coins.
A random sample of 2 coins is chosen from the bag.
\item List all the possible samples that can be drawn.
\item Find the sampling distribution of the mean value of these samples.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q6 [13]}}