| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a straightforward Poisson hypothesis test with standard structure: state variance (trivial recall that variance = mean for Poisson), perform a one-tailed test at 5% level (routine application of critical value method or p-value), and define Type II error in context. All components are textbook exercises requiring no novel insight, though the multi-step nature and hypothesis testing framework place it slightly above average difficulty. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.02i Poisson distribution: random events model5.02m Poisson: mean = variance = lambda5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Variance \(= 42\) | B1 | Obtains the correct value of the variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \lambda = 42\); \(H_1: \lambda > 42\) | B1 | States both hypotheses using correct language |
| \(Y \sim \text{Po}(42)\) | M1 | Uses Poisson model with \(\lambda = 42\) to calculate any Poisson probability |
| \(P(Y \geq 53) = 0.057\) | A1 | AWRT 0.057 |
| \(0.057 > 0.05\), Accept \(H_0\) | R1 | Evaluates the Poisson model by correctly comparing their probability with 0.05 |
| Insufficient evidence to suggest that the mean number of computers sold per day has increased | E1F | Infers \(H_0\) not rejected; FT their comparison using a Poisson model |
| Conclusion must refer to the mean number of computers (conclusion must not be definite) | E1F | Concludes in context; FT their incorrect rejection of \(H_0\) if stated or their comparison if not |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Type II error is to conclude that the mean number of computers sold per day has not increased when it has | E1 | States the meaning in context of a Type II error; condone missing "mean" or "per day"; condone "changed" for "increased" |
## Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Variance $= 42$ | B1 | Obtains the correct value of the variance |
## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 42$; $H_1: \lambda > 42$ | B1 | States both hypotheses using correct language |
| $Y \sim \text{Po}(42)$ | M1 | Uses Poisson model with $\lambda = 42$ to calculate any Poisson probability |
| $P(Y \geq 53) = 0.057$ | A1 | AWRT 0.057 |
| $0.057 > 0.05$, Accept $H_0$ | R1 | Evaluates the Poisson model by correctly comparing their probability with 0.05 |
| Insufficient evidence to suggest that the mean number of computers sold per day has increased | E1F | Infers $H_0$ not rejected; FT their comparison using a Poisson model |
| Conclusion must refer to the mean number of computers (conclusion must not be definite) | E1F | Concludes in context; FT their incorrect rejection of $H_0$ if stated or their comparison if not |
## Question 6(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Type II error is to conclude that the mean number of computers sold per day has not increased when it has | E1 | States the meaning in context of a Type II error; condone missing "mean" or "per day"; condone "changed" for "increased" |6 The number of computers sold per day by a shop can be modelled by the random variable $Y$ where $Y \sim \operatorname { Po } ( 42 )$
6
\begin{enumerate}[label=(\alph*)]
\item State the variance of $Y$
6
\item One month ago, the shop started selling a new model of computer.\\
On a randomly chosen day in the last month, the shop sold 53 computers.\\
Carry out a hypothesis test, at the $5 \%$ level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.\\[0pt]
[6 marks]\\
6
\item Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2022 Q6 [8]}}