| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unknown distribution, CLT applied |
| Difficulty | Challenging +1.2 This is a Further Maths Statistics question requiring CLT application to a negative binomial distribution. Students must recognize X~NB(4, 1/6), find E(X)=24 and Var(X)=120, apply CLT to the sample mean with n=32, then calculate a standard normal probability. While conceptually more demanding than typical A-level questions due to the Further Maths content and multi-step reasoning, it follows a standard CLT template without requiring novel insight. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| \([X = \text{no. of rolls to 4 sixes}]\) \(X \sim \text{NegBin}(4, \frac{1}{6})\) | M1 | 3.3 |
| \(\mu = \left[\frac{r}{p}\right] = \frac{24}{1}\), \(\sigma^2 = \left[\frac{r(1-p)}{p^2}\right] = \frac{4 \times \frac{5}{6}}{1/36} = \frac{120}{1} = \mathbf{120}\) | A1, A1 | 1.1b(x2) |
| \([\bar{X} \approx \sim \text{N}\left(''24'', \sqrt{\frac{''120''}{32}}\right)]\) | M1 M1 | 2.1, 3.4 |
| \(\text{P}(\bar{X} < 27.2) = 0.95078\ldots\) awrt \(\mathbf{0.951}\) | A1 | 1.1b |
$[X = \text{no. of rolls to 4 sixes}]$ $X \sim \text{NegBin}(4, \frac{1}{6})$ | M1 | 3.3 |
$\mu = \left[\frac{r}{p}\right] = \frac{24}{1}$, $\sigma^2 = \left[\frac{r(1-p)}{p^2}\right] = \frac{4 \times \frac{5}{6}}{1/36} = \frac{120}{1} = \mathbf{120}$ | A1, A1 | 1.1b(x2) |
$[\bar{X} \approx \sim \text{N}\left(''24'', \sqrt{\frac{''120''}{32}}\right)]$ | M1 M1 | 2.1, 3.4 |
$\text{P}(\bar{X} < 27.2) = 0.95078\ldots$ awrt $\mathbf{0.951}$ | A1 | 1.1b | (6)
**Notes:**
1st M1 for selecting the correct negative binomial model. May be implied by correct mean or variance. NegBin on its own is M0
1st A1 for mean $= 24$
2nd A1 for variance $= 120$, $\sigma = 120$ is A0 unless recovered
2nd M1 for writing or using of normal with mean 24 (may be implied by correct answer) ft their mean which may come from any distribution
3rd M1 for writing or using normal with standard deviation $= \sqrt{\frac{120}{32}} \left[= \sqrt{3.75}\right]$ ft their $\sigma$ where $\sigma$ may come from any distribution (may be implied by correct answer)
2nd A1 for awrt 0.951 (correct answer scores 6 out of 6)
**(Total: 6 marks)**
---\begin{enumerate}
\item There are 32 students in a class.
\end{enumerate}
Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die.
Estimate the probability that the mean number of rolls for the class is less than 27.2
\hfill \mbox{\textit{Edexcel FS1 2023 Q4 [6]}}