| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find critical region |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question following a routine template: state assumptions, write hypotheses, find critical region for a two-tailed Poisson test, calculate actual significance level, and interpret. All steps are algorithmic with no novel problem-solving required, though it does require careful handling of the two-tailed test and discrete distribution. Slightly easier than average due to its highly structured nature and direct application of textbook methods. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02l Poisson conditions: for modelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Breakdowns are [rare], independent events occurring at a constant rate | B1 | Must include "independent" or "constant rate" or "singly". No context needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \lambda = 8 \quad H_1: \lambda \neq 8\) | B1 | Both hypotheses correct. Must be in terms of \(\lambda\) or \(\mu\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(X \sim \text{Po}(8)\) | ||
| \(P(X \leqslant 2) = 0.0138\) oe \(\quad P(X \leqslant 3) = 0.0424\) oe | M1 | Use of Po(8) for lower critical value. May be implied by 0.0138 or 0.0424 or \(X \leqslant 2\) (Calculator values: 0.01375... and 0.04238...) |
| \(P(X \geqslant 14) = 0.0342\) oe \(\quad P(X \geqslant 15) = 0.0173\) oe | M1 | Use of Po(8) for upper critical value. May be implied by 0.0342 or 0.0173 or 0.9658 or 0.9827 or \(X \geqslant 15\) (Calculator values: 0.03418... and 0.01725... and 0.96581... and 0.98274...) |
| \(X \leqslant 2 \cup X \geqslant 15\) oe | A1 | Allow \([0,2]\) or \([0,3)\) and \([15,\infty]\) or \([15,\infty)\) or \((14,\infty]\) or \((14,\infty)\). Do not allow as probability statements |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{"0.0138"} + \text{"0.0173"}\) | M1 | Adding the two probabilities for their critical region |
| \(= \text{"0.0311"}\) | A1ft | 0.0311. Allow 3.11 or awrt 3.1[0] or awrt 0.031[0] ft their critical region. NB: 3.11 or 0.0311 or awrt 3.1[0] or awrt 0.031[0] will score 2/2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| "[4 is] not in the critical region" | M1 | Correct statement ft their critical region e.g. Do not reject \(H_0\)/Accept \(H_0\)/not significant |
| So there is insufficient evidence that refurbishment has changed the mean breakdown rate | A1 | Correct conclusion in context. Must include rate/number of breakdown (Allow "decreased"). NB: Award M1 A1 for a correct contextual statement on its own |
# Question 3:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Breakdowns are [rare], independent events occurring at a constant rate | B1 | Must include "independent" or "constant rate" or "singly". No context needed |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda = 8 \quad H_1: \lambda \neq 8$ | B1 | Both hypotheses correct. Must be in terms of $\lambda$ or $\mu$ |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $X \sim \text{Po}(8)$ | | |
| $P(X \leqslant 2) = 0.0138$ oe $\quad P(X \leqslant 3) = 0.0424$ oe | M1 | Use of Po(8) for lower critical value. May be implied by 0.0138 or 0.0424 or $X \leqslant 2$ (Calculator values: 0.01375... and 0.04238...) |
| $P(X \geqslant 14) = 0.0342$ oe $\quad P(X \geqslant 15) = 0.0173$ oe | M1 | Use of Po(8) for upper critical value. May be implied by 0.0342 or 0.0173 or 0.9658 or 0.9827 or $X \geqslant 15$ (Calculator values: 0.03418... and 0.01725... and 0.96581... and 0.98274...) |
| $X \leqslant 2 \cup X \geqslant 15$ oe | A1 | Allow $[0,2]$ or $[0,3)$ and $[15,\infty]$ or $[15,\infty)$ or $(14,\infty]$ or $(14,\infty)$. Do not allow as probability statements |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{"0.0138"} + \text{"0.0173"}$ | M1 | Adding the two probabilities for their critical region |
| $= \text{"0.0311"}$ | A1ft | 0.0311. Allow 3.11 or awrt 3.1[0] or awrt 0.031[0] ft their critical region. NB: 3.11 or 0.0311 or awrt 3.1[0] or awrt 0.031[0] will score 2/2 |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| "[4 is] not in the critical region" | M1 | Correct statement ft their critical region e.g. Do not reject $H_0$/Accept $H_0$/not significant |
| So there is insufficient evidence that refurbishment has changed the mean breakdown rate | A1 | Correct conclusion in context. Must include rate/number of breakdown (Allow "decreased"). NB: Award M1 A1 for a correct contextual statement on its own |
---3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.
\begin{enumerate}[label=(\alph*)]
\item Give a reason why a Poisson distribution could be used to model the number of breakdowns.
The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
\item Write down suitable null and alternative hypotheses that the headteacher should use.
The refurbished photocopier was monitored for the first week after it was installed.
\item Using a $5 \%$ level of significance, find the critical region to test whether the rate of breakdowns has now changed.
\item Find the actual significance level of a test based on the critical region from part (c).
During the first week after it was installed there were 4 breakdowns.
\item Comment on this finding in the light of the critical region found in part (c).
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2022 Q3 [9]}}