| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a straightforward application of a one-tailed Poisson hypothesis test with standard structure. Students need to set up H₀: λ=50 vs H₁: λ<50, find P(X≤30) using tables/calculator, compare to 1%, and interpret. Parts (b) and (c) test standard understanding of Type I errors and model assumptions. While it requires knowledge of hypothesis testing framework, it's a textbook example with no novel problem-solving required, making it slightly easier than average for Further Maths Statistics. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \lambda = 50\); \(H_1: \lambda < 50\) | B1 | Both hypotheses in correct language |
| \(X \sim \text{Po}(50)\); \(P(X \leq 30) = 0.002\); \(P(X \leq 33) = 0.007\); \(P(X \leq 34) = 0.0108\) | M1 | Selects and uses Poisson with \(\lambda=50\) to find relevant tail probability |
| AWRT \(P(X \leq 30) = 0.002\) or AWRT \(P(X \leq 33) = 0.007\) or AWRT \(P(X \leq 34) = 0.0108\) | A1 | |
| \(p\)-value \(= 0.002 < 0.01\) (or \(30 < 34\) = critical value) | R1 | Evaluates by comparing \(p\)-value with \(0.01\) or 30 with critical value |
| Reject \(H_0\) in favour of \(H_1\) | E1 | Infers \(H_0\) rejected |
| Significant evidence that mean number of vehicles passing per minute has reduced | E1 | Concludes in context; not definite |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Type I error is to conclude that mean number of vehicles passing per minute has reduced when it has not. | E1 | States meaning in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The rate at which vehicles pass is unlikely to be constant over time; or vehicles may not pass independently (e.g. convoy) | E1 | Must be in context and consistent |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 50$; $H_1: \lambda < 50$ | B1 | Both hypotheses in correct language |
| $X \sim \text{Po}(50)$; $P(X \leq 30) = 0.002$; $P(X \leq 33) = 0.007$; $P(X \leq 34) = 0.0108$ | M1 | Selects and uses Poisson with $\lambda=50$ to find relevant tail probability |
| AWRT $P(X \leq 30) = 0.002$ or AWRT $P(X \leq 33) = 0.007$ or AWRT $P(X \leq 34) = 0.0108$ | A1 | |
| $p$-value $= 0.002 < 0.01$ (or $30 < 34$ = critical value) | R1 | Evaluates by comparing $p$-value with $0.01$ or 30 with critical value |
| Reject $H_0$ in favour of $H_1$ | E1 | Infers $H_0$ rejected |
| Significant evidence that mean number of vehicles passing per minute has reduced | E1 | Concludes in context; not definite |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Type I error is to conclude that mean number of vehicles passing per minute has reduced when it has not. | E1 | States meaning in context |
## Question 7(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The rate at which vehicles pass is unlikely to be constant over time; or vehicles may not pass independently (e.g. convoy) | E1 | Must be in context and consistent |7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute.
After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute.
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\begin{enumerate}[label=(\alph*)]
\item Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute.
Investigate Xander's claim, using a suitable test at the $1 \%$ level of significance.\\
7
\item For your test carried out in part (a) state, in context, the meaning of a Type 1 error.
7
\item Explain why the model used in part (a) might be invalid.
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2018 Q7 [8]}}