| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2003 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Deriving sampling distribution |
| Difficulty | Standard +0.3 This is a structured, multi-part question that guides students through finding a sampling distribution from first principles with small sample size (n=2). While it requires careful enumeration of all possible samples and probability calculations, the steps are clearly signposted and the computations are straightforward. The verification in part (e) is a standard result. This is slightly easier than average as it's highly scaffolded with no conceptual leaps required. |
| Spec | 2.04a Discrete probability distributions5.05a Sample mean distribution: central limit theorem |
7. A bag contains a large number of coins of which $30 \%$ are 50 p coins, $20 \%$ are 10 p coins and the rest are 2 p coins.
\begin{enumerate}[label=(\alph*)]
\item Find the mean $\mu$ and the variance $\sigma ^ { 2 }$ of this population of coins.
A random sample of 2 coins is drawn from the bag one after the other.
\item List all possible samples that could be drawn.
\item Find the sampling distribution of $\bar { X }$, the mean of the coins drawn.
\item Find $\mathrm { P } ( 2 \leq \bar { X } < 7 )$.
\item Use the sampling distribution of $\bar { X }$ to verify $\mathrm { E } ( \bar { X } ) = \mu$ and $\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }$.
END
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2003 Q7 [17]}}