| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unknown distribution, CLT applied |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring CLT application to a binomial distribution with careful handling of parameters. Students must recognize that X~B(400,0.64) has mean 256 and variance 92.16, then apply CLT to the sample mean of 100 observations, requiring σ_X̄ = √(92.16/100) = 0.9608. The threshold of 257 is close to the mean (256), requiring precise calculation. While the CLT mechanics are standard, the binomial context, parameter extraction, and precision needed elevate this above typical A-level questions. |
| Spec | 5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{X} \approx N(256,...)\) | M1 | 3.1a - For realising the need to use the CLT with correct mean |
| \(\bar{X} \approx N(256, 0.9216)\) | A1 | 1.1b - For a correct normal stated |
| \(P(\bar{X} > 257) = P\left(Z > \dfrac{257-256}{\sqrt{0.9216}}\right)\) [= awrt 1.04] | dM1 | 3.4 - Dep on previous M mark. Use of normal model to find \(P(\bar{X}>257)\). If final answer incorrect, must see standardisation using their \(\sigma\) |
| \(p = 0.1492...\) | A1 | 1.1b - awrt 0.149 (0.14878... from calculator). Allow awrt 0.148 if continuity correction used |
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{X} \approx N(256,...)$ | M1 | 3.1a - For realising the need to use the CLT with correct mean |
| $\bar{X} \approx N(256, 0.9216)$ | A1 | 1.1b - For a correct normal stated |
| $P(\bar{X} > 257) = P\left(Z > \dfrac{257-256}{\sqrt{0.9216}}\right)$ [= awrt 1.04] | dM1 | 3.4 - Dep on previous M mark. Use of normal model to find $P(\bar{X}>257)$. If final answer incorrect, must see standardisation using their $\sigma$ |
| $p = 0.1492...$ | A1 | 1.1b - awrt 0.149 (0.14878... from calculator). Allow awrt 0.148 if continuity correction used |
---\begin{enumerate}
\item A courier delivers parcels. The random variable $X$ represents the number of parcels delivered successfully each day by the courier where $X \sim \mathrm {~B} ( 400,0.64 )$
\end{enumerate}
A random sample $X _ { 1 } , X _ { 2 } , \ldots X _ { 100 }$ is taken.\\
Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257
\hfill \mbox{\textit{Edexcel FS1 2021 Q3 [4]}}