| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | State distribution of sample mean |
| Difficulty | Easy -1.2 This is a straightforward recall question about the Central Limit Theorem requiring students to state standard results (mean μ, SD σ/√n) and conditions for normality (population normal vs. large n). No calculations or problem-solving required, just bookwork knowledge of a core statistical theorem. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| \(7, \frac{3}{\sqrt{n}}\) | B1, B1 [2] | oe |
| Answer | Marks | Guidance |
|---|---|---|
| Pop is normal | B1 [1] | Allow \(X\) is normal |
| Answer | Marks | Guidance |
|---|---|---|
| Large sample | B1 [1] | or large \(n\) (can be implied by \(n \geq 30\)) |
## Question 2:
**(i)**
$7, \frac{3}{\sqrt{n}}$ | B1, B1 [2] | oe
**(ii)(a)**
Pop is normal | B1 [1] | Allow $X$ is normal
**(ii)(b)**
Large sample | B1 [1] | or large $n$ (can be implied by $n \geq 30$)
---2 A population has mean 7 and standard deviation 3. A random sample of size $n$ is chosen from this population.\\
(i) Write down the mean and standard deviation of the distribution of the sample mean.\\
(ii) Under what circumstances does the sample mean have
\begin{enumerate}[label=(\alph*)]
\item a normal distribution,
\item an approximately normal distribution?
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2012 Q2 [4]}}