| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.8 This is a standard Further Maths Statistics question covering Poisson probability calculations and hypothesis testing with critical regions. Part (a) involves routine Poisson probability calculations. Parts (b) and (c) require understanding of two-tailed hypothesis tests, finding critical regions from tables, and calculating Type I error probability. The word count scaling (250 vs 100) adds a minor complication but is straightforward. This is typical FS1 material requiring multiple techniques but no novel insight, placing it moderately above average A-level difficulty. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) | \(0.20901\ldots\) awrt \(\mathbf{0.209}\) | B1 |
| (a)(ii) | \(0.30844\ldots\) awrt \(\mathbf{0.308}\) | B1 |
| (b) | \(H_0: \lambda = 2.4\) (or \(\mu = 6\)) \(\quad H_1: \lambda \neq 2.4\) (or \(\mu \neq 6\)) | B1 |
| \([E = \text{no. of errors}] E \sim \text{Po}(6)\) | M1 | 3.3 |
| \(P(E \leqslant 1) = 0.0174\) or \(P(E \leqslant 2) = 0.0620\) and \(P(E \leqslant 11) = 0.980\) or \(P(E \geqslant 12) = 0.0201\) | M1 | 3.4 |
| Critical region: \(E \leqslant 1\) or \(E \geqslant 12\) | A1 | 1.1b |
| (c) | \([P(\text{Type I error}) = 0.0174 + 0.0201 =] \mathbf{0.0375}\) (Calc gives: \(0.017351\ldots + 0.0200919\ldots = 0.037443\ldots\)) | B1ft |
(a)(i) | $0.20901\ldots$ awrt $\mathbf{0.209}$ | B1 | 3.4 |
(a)(ii) | $0.30844\ldots$ awrt $\mathbf{0.308}$ | B1 | 1.1b |
(b) | $H_0: \lambda = 2.4$ (or $\mu = 6$) $\quad H_1: \lambda \neq 2.4$ (or $\mu \neq 6$) | B1 | 2.5 |
| $[E = \text{no. of errors}] E \sim \text{Po}(6)$ | M1 | 3.3 |
| $P(E \leqslant 1) = 0.0174$ or $P(E \leqslant 2) = 0.0620$ and $P(E \leqslant 11) = 0.980$ or $P(E \geqslant 12) = 0.0201$ | M1 | 3.4 |
| Critical region: $E \leqslant 1$ or $E \geqslant 12$ | A1 | 1.1b |
(c) | $[P(\text{Type I error}) = 0.0174 + 0.0201 =] \mathbf{0.0375}$ (Calc gives: $0.017351\ldots + 0.0200919\ldots = 0.037443\ldots$) | B1ft | 1.2 |
**Notes:**
- (a) 1st B1 for awrt 0.209; 2nd B1 for awrt 0.308
- (b) B1 for both hypotheses correct in terms of $\lambda$ or $\mu$ (allow $\lambda = 6$ etc)
- (b) 1st M1 for selecting the correct model. Sight or use of Po(6)
- (b) 2nd M1 for use of the correct model with two probs correct to 2.s.f. (accept $P(E \geq 12) = 0.02$). Must see attempt at lower and upper limit. Probabilities may be seen in (c).
- (b) A1 for correct critical region (both parts). Allow $E \leq 1$ and $E \geq 12$ or $E \leq 1$, $E \geq 12$ etc. Writing CR as probability statements is A0.
- **NB: Completely correct CR implies M1M1A1**
- **SC: 1-tailed test**
- **B0** as hypotheses are incorrect
- **M1** for sight or use of Po(6)
- **M1** (dep on $H_1$) for sight of $P(E \leq 1) = 0.0174$ or $P(E \geq 11) = 0.0426$, in line with their $H_1$
- **A1** for CR: $E \leq 1$ or CR: $E \geq 11$, in line with their hypotheses
- (c) **B1ft** for 0.0375 or 0.0374 or summing their two appropriate probs (ft their CR)
- **NB: If candidate uses a 1-tailed test, this mark cannot be gained**
---\begin{enumerate}
\item The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.
\end{enumerate}
A 100-word piece of work completed by the secretary is selected at random.\\
(a) Find the probability that\\
(i) there are exactly 3 errors,\\
(ii) there are fewer than 2 errors.
After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.\\
(b) Stating your hypotheses clearly, and using a $5 \%$ level of significance, find the critical region for a suitable test.\\
(c) Find P (Type I error) for the test in part (b)
\hfill \mbox{\textit{Edexcel FS1 2024 Q2 [7]}}