| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Moderate -0.3 This is a standard S2 hypothesis testing question with routine Poisson calculations. Part (a) tests recall of conditions, parts (b)-(c) are direct calculator work with λ=8.5, and part (d) follows the standard one-tailed test template. While it requires multiple steps, all techniques are textbook exercises with no novel problem-solving or insight required, making it slightly easier than average. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Vehicles pass at random / one at a time / independently / at a constant rate | Any 2&context B1B1dep | (2 marks) |
| Answer | Marks |
|---|---|
| \(X \sim \text{Po}\left(\frac{51}{60} \times 10\right) = \text{Po}(8.5)\) | Implied Po(8.5) B1 |
| \(P(X = 6) = \frac{8.5^6 e^{-8.5}}{6!} = 0.1066\) (or 0.2562-0.1496=0.1066) | Clear attempt using 6, .4dp M1A1 |
| Answer | Marks |
|---|---|
| \(P(X \geq 9) = 1 - P(X \leq 8) = 0.4769\) | Require 1 minus and correct inequality M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \lambda = 8.5\), \(H_1: \lambda < 8.5\) | One tailed test only for all hyp B1J, B1J | |
| \(P(X \leq 4 | \lambda = 8.5) = 0.0744, > 0.05\) | \(X \leq 4\) for method, 0.0744 M1, A1 |
| (Or \(P(X \leq 3 | \lambda = 8.5) = 0.0301, < 0.05\) so CR \(X \leq 3\) | correct CR M1, A1 |
| Insufficient evidence to reject \(H_0\), | 'Accept' M1 | |
| so no evidence to suggest number of vehicles has decreased. | Context A1J |
## Part (a)
Vehicles pass at random / one at a time / independently / at a constant rate | Any 2&context B1B1dep | (2 marks)
## Part (b)
$X$ is the number of vehicles passing in a 10 minute interval.
$X \sim \text{Po}\left(\frac{51}{60} \times 10\right) = \text{Po}(8.5)$ | Implied Po(8.5) B1 |
$P(X = 6) = \frac{8.5^6 e^{-8.5}}{6!} = 0.1066$ (or 0.2562-0.1496=0.1066) | Clear attempt using 6, .4dp M1A1 |
(3 marks)
## Part (c)
$P(X \geq 9) = 1 - P(X \leq 8) = 0.4769$ | Require 1 minus and correct inequality M1A1 |
(2 marks)
## Part (d)
$H_0: \lambda = 8.5$, $H_1: \lambda < 8.5$ | One tailed test only for all hyp B1J, B1J |
$P(X \leq 4 | \lambda = 8.5) = 0.0744, > 0.05$ | $X \leq 4$ for method, 0.0744 M1, A1 |
(Or $P(X \leq 3 | \lambda = 8.5) = 0.0301, < 0.05$ so CR $X \leq 3$ | correct CR M1, A1 |)
Insufficient evidence to reject $H_0$, | 'Accept' M1 |
so no evidence to suggest number of vehicles has decreased. | Context A1J |
(6 marks)
**Total 13 Marks**
---Vehicles pass a particular point on a road at a rate of 51 vehicles per hour.
\begin{enumerate}[label=(\alph*)]
\item Give two reasons to support the use of the Poisson distribution as a suitable model for the number of vehicles passing this point. [2]
\end{enumerate}
Find the probability that in any randomly selected 10 minute interval
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item exactly 6 cars pass this point, [3]
\item at least 9 cars pass this point. [2]
\end{enumerate}
After the introduction of a roundabout some distance away from this point it is suggested that the number of vehicles passing it has decreased. During a randomly selected 10 minute interval 4 vehicles pass the point.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Test, at the 5\% level of significance, whether or not there is evidence to support the suggestion that the number of vehicles has decreased. State your hypotheses clearly. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2004 Q5 [13]}}