| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question covering Poisson distribution identification, mean/variance calculation, and a one-tailed test. All parts follow textbook procedures with no novel problem-solving required. The calculations are straightforward, and the context is typical for this module. Slightly easier than average due to the routine nature of all three parts. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05b Unbiased estimates: of population mean and variance |
| Number of bicycles | 0 | 1 | 2 | 3 | 4 | 5 | 6 or more |
| Frequency | 7 | 14 | 10 | 2 | 1 | 2 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Poisson; e.g. reasonable to suggest bicycles passing will occur singly, at random and at constant rate | B1, B2 | |
| (b) \(n = 36\), \(\Sigma fx = 54\), \(\therefore\) mean \(= \frac{54}{36} = 1.5\); \(\Sigma fx^2 = 0 + 14 + 40 + 18 + 16 + 50 = 138\); variance \(= \frac{138}{36} - 1.5^2 = 1.58\) (3sf); values support Poisson as expect mean = variance | A1, M1, A1, B1 | |
| (c) let \(X\) = no. of bicycles passing per 30-mins \(\therefore X \sim \text{Po}(9)\); \(H_0: \lambda = 9\) \(H_1: \lambda > 9\); \(P(X \geq 16) = 1 - P(X \leq 15) = 1 - 0.9780 = 0.0220\); less than 5% \(\therefore\) significant, evidence of more bicycles | M1, B1, M1, A1, A1 | (14 marks) |
**(a)** Poisson; e.g. reasonable to suggest bicycles passing will occur singly, at random and at constant rate | B1, B2 |
**(b)** $n = 36$, $\Sigma fx = 54$, $\therefore$ mean $= \frac{54}{36} = 1.5$; $\Sigma fx^2 = 0 + 14 + 40 + 18 + 16 + 50 = 138$; variance $= \frac{138}{36} - 1.5^2 = 1.58$ (3sf); values support Poisson as expect mean = variance | A1, M1, A1, B1 |
**(c)** let $X$ = no. of bicycles passing per 30-mins $\therefore X \sim \text{Po}(9)$; $H_0: \lambda = 9$ $H_1: \lambda > 9$; $P(X \geq 16) = 1 - P(X \leq 15) = 1 - 0.9780 = 0.0220$; less than 5% $\therefore$ significant, evidence of more bicycles | M1, B1, M1, A1, A1 | (14 marks)
---
**Total: (75 marks)**A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.
\begin{enumerate}[label=(\alph*)]
\item Suggest, with reasons, a suitable distribution for modelling this situation. [3]
\end{enumerate}
The student's data is shown in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Number of bicycles & 0 & 1 & 2 & 3 & 4 & 5 & 6 or more \\
\hline
Frequency & 7 & 14 & 10 & 2 & 1 & 2 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). [6]
\end{enumerate}
An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Stating your hypotheses clearly, test at the 5\% level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q7 [14]}}