| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Challenging +1.2 This is a structured Poisson hypothesis test with clearly defined parts: stating hypotheses (routine), applying a given critical region (straightforward comparison), and calculating power (standard formula application with λ=7). While power calculations are less common than basic hypothesis tests, all steps follow standard procedures with no novel problem-solving required, making it moderately above average difficulty for Further Maths Statistics. |
| Spec | 5.02i Poisson distribution: random events model5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \lambda = 8\), \(H_1: \lambda \neq 8\) | B1 | States both hypotheses using correct language |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| 3 is not in the critical region | R1 | Evaluates Poisson model by comparing sample with critical region |
| Accept \(H_0\) | E1F | Infers \(H_0\) not rejected; FT 'their' comparison |
| There is no significant evidence to suggest that the average number of runners per minute passing the shop is not 8 | E1F | Concludes in context; conclusion must not be definite; FT incorrect rejection of \(H_0\) if stated or 'their' comparison if not |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses Poisson model with \(\lambda = 7\) to calculate a cumulative probability | M1 | |
| \(P(X \leq 2) = 0.0296\), \(P(X \geq 14) = 0.0128\); or \(P(3 \leq X \leq 13) = 0.958\) | A1 | AWRT 0.0296 and AWRT 0.0128 |
| Power \(= 0.0296 + 0.0128 = 0.0424\) | A1F | AWRT 0.0424 |
## Question 5:
### Part 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \lambda = 8$, $H_1: \lambda \neq 8$ | B1 | States both hypotheses using correct language |
### Part 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| 3 is not in the critical region | R1 | Evaluates Poisson model by comparing sample with critical region |
| Accept $H_0$ | E1F | Infers $H_0$ not rejected; FT 'their' comparison |
| There is no significant evidence to suggest that the average number of runners per minute passing the shop is not 8 | E1F | Concludes in context; conclusion must not be definite; FT incorrect rejection of $H_0$ if stated or 'their' comparison if not |
### Part 5(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses Poisson model with $\lambda = 7$ to calculate a cumulative probability | M1 | |
| $P(X \leq 2) = 0.0296$, $P(X \geq 14) = 0.0128$; or $P(3 \leq X \leq 13) = 0.958$ | A1 | AWRT 0.0296 and AWRT 0.0128 |
| Power $= 0.0296 + 0.0128 = 0.0424$ | A1F | AWRT 0.0424 |
---5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8
Emily conducts a hypothesis test to investigate her claim.\\
5
\begin{enumerate}[label=(\alph*)]
\item State the hypotheses for Emily's test.
5
\item Emily counts the number of runners, $X$, passing the shop in a randomly chosen minute.
The critical region for Emily's test is $X \leq 2$ or $X \geq 14$\\
During a randomly chosen minute, Emily counts 3 runners passing the shop.\\
Determine the outcome of Emily's hypothesis test.\\
5
\item The actual average number of runners per minute passing the shop is 7
Find the power of Emily's hypothesis test, giving your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2020 Q5 [7]}}