| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a straightforward S2 Poisson distribution question requiring (a) a standard probability calculation P(X≥2) = 1 - P(X≤1) and (b) a hypothesis test with clearly defined steps. Both parts follow textbook procedures with no novel insight required, though the hypothesis testing component adds slight complexity beyond pure recall. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.02i 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 modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | let \(X =\) no. of tries per match \(\therefore X \sim \text{Po}(0.4)\) | M1 |
| \(P(X \geq 2) = 1 - P(X \leq 1)\) | M1 | |
| \(= 1 - e^{-0.4}(1 + 0.4)\) | M1 A1 | |
| \(= 1 - 0.9384 = 0.0616 \text{ (3sf)}\) | A1 | |
| (b) | let \(Y =\) no. of tries per 5 matches \(\therefore Y \sim \text{Po}(2)\) | M1 |
| \(H_0: \lambda = 2\) \(H_1: \lambda > 2\) | B1 | |
| \(P(Y \geq 6) = 1 - P(Y \leq 5) = 1 - 0.9834 = 0.0166\) | M1 A1 | |
| less than 5% \(\therefore\) significant, evidence of increase | A1 | (10) |
(a) | let $X =$ no. of tries per match $\therefore X \sim \text{Po}(0.4)$ | M1 |
| $P(X \geq 2) = 1 - P(X \leq 1)$ | M1 |
| $= 1 - e^{-0.4}(1 + 0.4)$ | M1 A1 |
| $= 1 - 0.9384 = 0.0616 \text{ (3sf)}$ | A1 |
(b) | let $Y =$ no. of tries per 5 matches $\therefore Y \sim \text{Po}(2)$ | M1 |
| $H_0: \lambda = 2$ $H_1: \lambda > 2$ | B1 |
| $P(Y \geq 6) = 1 - P(Y \leq 5) = 1 - 0.9834 = 0.0166$ | M1 A1 |
| less than 5% $\therefore$ significant, evidence of increase | A1 | (10)A rugby player scores an average of 0.4 tries per match in which he plays.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that he scores 2 or more tries in a match. [5 marks]
\end{enumerate}
The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [10]}}