| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unknown distribution, CLT applied |
| Difficulty | Standard +0.3 This is a straightforward CLT application with a Poisson distribution. Students need to recognize that for n=100, the sample mean is approximately normal with mean 2.3 and variance 2.3/100, then perform a standard normal calculation. While it requires understanding of CLT, the execution is mechanical with no conceptual subtleties—slightly easier than average since it's a direct textbook-style application with clear parameters given. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02i Poisson distribution: random events model |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Po(2.3)\), \(n=100\), \(\mu = 2.3\), \(\sigma^2 = 2.3\); CLT \(\Rightarrow \bar{X} \approx N\!\left(2.3, \frac{2.3}{100}\right)\) | M1, A1 | M1: realising need to use CLT; may be implied by using correct normal distribution. A1: fully correct normal stated or used |
| \(P(\bar{X} > 2.5) = P\!\left(Z > \frac{2.5 - 2.3}{\sqrt{0.023}}\right)\) | M1 | Use of normal model; can be awarded for \(\frac{2.5-2.3}{\sqrt{0.023}}\) or awrt 1.32 |
| \(= P(Z > 1.318\ldots) = 0.09632\ldots\) | A1 | awrt 0.0963 |
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Po(2.3)$, $n=100$, $\mu = 2.3$, $\sigma^2 = 2.3$; CLT $\Rightarrow \bar{X} \approx N\!\left(2.3, \frac{2.3}{100}\right)$ | M1, A1 | M1: realising need to use CLT; may be implied by using correct normal distribution. A1: fully correct normal stated or used |
| $P(\bar{X} > 2.5) = P\!\left(Z > \frac{2.5 - 2.3}{\sqrt{0.023}}\right)$ | M1 | Use of normal model; can be awarded for $\frac{2.5-2.3}{\sqrt{0.023}}$ or awrt 1.32 |
| $= P(Z > 1.318\ldots) = 0.09632\ldots$ | A1 | awrt 0.0963 |
---\begin{enumerate}
\item A random sample of 100 observations is taken from a Poisson distribution with mean 2.3
\end{enumerate}
Estimate the probability that the mean of the sample is greater than 2.5
\hfill \mbox{\textit{Edexcel FS1 Q4 [4]}}