| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| 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 Poisson hypothesis testing with clear context. Part (a) requires recognizing the Poisson model (standard for S2), part (b) is a routine one-tailed test with simple parameter conversion (36 per minute → 9 per 15s), and part (c) requires finding a critical value by testing values systematically. All techniques are standard S2 material with no novel insight required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Poisson | B1 | For Poisson or Po. Ignore their value for the mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: \mu = 9\) (or \(\lambda = 36\)) | B1 | 1st B1 for \(H_0: \mu/\lambda = 9\) or \(\mu/\lambda = 36\) |
| \(H_1: \mu > 9\) (or \(\lambda > 36\)) | B1 | 2nd B1 for \(H_1: \mu/\lambda > 9\) or \(\mu/\lambda > 36\) |
| \(X \sim Po(9)\) and \(P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.8030 = \underline{0.197}\) or \(P(X \leq 14) = 0.9585\), \(P(X \geq 15) = 0.0415\), \(CR\ X \geq 15\) | M1 | For writing or using \(1 - P(X \leq 11)\) or writing \(P(X \leq 14) = 0.9585\) or \(P(X \geq 15) = 0.0415\). May be implied by correct CR or probability \(= 0.197\) |
| \(0.197\) (A1 for \(0.197\) or correct CR) | A1 | Allow \(X > 14\). NB \(P(X \leq 11) = 0.8030\) on its own scores M1A1 |
| \((0.197 > 0.05)\) so not significant / accept \(H_0\) / Not in CR | M1d | Dependent on 1st M1. For correct statement based on table |
| He does not have evidence to switch on the speed restrictions | A1ft | Correct contextualised statement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let \(Y =\) number of vehicles in 10s, then \(Y \sim Po(6)\) | B1 | For identifying \(Po(6)\) — may be implied by use of correct tables |
| Tables: \(P(Y \leq 10) = 0.9574\) so \(P(Y \geq 11) = 0.0426\), so needs \(\underline{11}\) vehicles | M1 | Any one of the probs \(0.9574\) or \(0.0426\) or \(0.9799\) or \(0.0201\) may be implied by correct answer of 11 |
| A1 | cao — do not accept \(X \geq 11\). NB answer of 11 with no working gains all three marks |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Poisson | B1 | For Poisson or Po. Ignore their value for the mean |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 9$ (or $\lambda = 36$) | B1 | 1st B1 for $H_0: \mu/\lambda = 9$ or $\mu/\lambda = 36$ |
| $H_1: \mu > 9$ (or $\lambda > 36$) | B1 | 2nd B1 for $H_1: \mu/\lambda > 9$ or $\mu/\lambda > 36$ |
| $X \sim Po(9)$ and $P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.8030 = \underline{0.197}$ or $P(X \leq 14) = 0.9585$, $P(X \geq 15) = 0.0415$, $CR\ X \geq 15$ | M1 | For writing or using $1 - P(X \leq 11)$ or writing $P(X \leq 14) = 0.9585$ or $P(X \geq 15) = 0.0415$. May be implied by correct CR or probability $= 0.197$ |
| $0.197$ (A1 for $0.197$ or correct CR) | A1 | Allow $X > 14$. NB $P(X \leq 11) = 0.8030$ on its own scores M1A1 |
| $(0.197 > 0.05)$ so not significant / accept $H_0$ / Not in CR | M1d | Dependent on 1st M1. For correct statement based on table |
| He does not have evidence to switch on the speed restrictions | A1ft | Correct contextualised statement |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $Y =$ number of vehicles in 10s, then $Y \sim Po(6)$ | B1 | For identifying $Po(6)$ — may be implied by use of correct tables |
| Tables: $P(Y \leq 10) = 0.9574$ so $P(Y \geq 11) = 0.0426$, so needs $\underline{11}$ vehicles | M1 | Any one of the probs $0.9574$ or $0.0426$ or $0.9799$ or $0.0201$ may be implied by correct answer of 11 |
| | A1 | cao — do not accept $X \geq 11$. NB answer of 11 with no working gains all three marks |
---2. A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.
The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.
\item Stating your hypotheses clearly, and using a $5 \%$ level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.
\item Using a $5 \%$ level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2011 Q2 [10]}}