| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Distribution of sample mean |
| Difficulty | Moderate -0.8 This is a straightforward application of the sampling distribution of the mean from a normal population. Part (a) requires recalling that the sample mean is normally distributed with variance σ²/n, and part (b) is a routine probability calculation using standardization. No problem-solving or insight required—pure textbook application. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(12.5, \ldots)\) | B1 | |
| Variance \(= 0.4096\) | B1 | Accept \(0.410\) (3sf), condone \(\frac{10.24}{25}\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{13 - 12.5}{\sqrt{0.4096}} = 0.781\) | M1 | For standardising with *their* values. Accept standardising with \(12\). |
| \(\phi(0.781) - (1 - \Phi(0.781))\) | M1 | For attempting to find *their* central area. |
| \(0.565\) (3sf) | A1 | |
| Total: 3 |
**Question 1:**
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(12.5, \ldots)$ | B1 | |
| Variance $= 0.4096$ | B1 | Accept $0.410$ (3sf), condone $\frac{10.24}{25}$ |
| | **Total: 2** | |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{13 - 12.5}{\sqrt{0.4096}} = 0.781$ | M1 | For standardising with *their* values. Accept standardising with $12$. |
| $\phi(0.781) - (1 - \Phi(0.781))$ | M1 | For attempting to find *their* central area. |
| $0.565$ (3sf) | A1 | |
| | **Total: 3** | |
---1 It is known that the height $H$, in metres, of trees of a certain kind has the distribution $\mathrm { N } ( 12.5,10.24 )$. A scientist takes a random sample of 25 trees of this kind and finds the sample mean, $\bar { H }$, of the heights.
\begin{enumerate}[label=(\alph*)]
\item State the distribution of $\bar { H }$, giving the values of any parameters.
\item Find $\mathrm { P } ( 12 < \bar { H } < 13 )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q1 [5]}}