| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sampling distribution theory |
| Difficulty | Easy -1.2 This is a pure recall question testing knowledge of the Central Limit Theorem's basic statements. Part (i) requires stating standard formulas (μ and σ²/n), while part (ii) asks for textbook statements about CLT conditions. No calculations, problem-solving, or application required—just direct reproduction of theory. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Mean \(\mu\) | B1 | |
| Variance \(\sigma^2/n\) | B1 | 2 marks total |
| (ii) Normal | B1 | 1 mark |
| (iii) Unknown, or normal if the pop is normal | B1 | Accept either |
**(i)** Mean $\mu$ | B1 |
Variance $\sigma^2/n$ | B1 | 2 marks total
**(ii)** Normal | B1 | 1 mark
**(iii)** Unknown, or normal if the pop is normal | B1 | Accept either | 1 mark
---2 (i) Write down the mean and variance of the distribution of the means of random samples of size $n$ taken from a very large population having mean $\mu$ and variance $\sigma ^ { 2 }$.\\
(ii) What, if anything, can you say about the distribution of sample means
\begin{enumerate}[label=(\alph*)]
\item if $n$ is large,
\item if $n$ is small?
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2006 Q2 [4]}}