| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson with binomial combination |
| Difficulty | Standard +0.3 This is a straightforward two-part Poisson question requiring standard techniques: (a) binomial probability with Poisson probabilities, and (b) a routine hypothesis test. Both parts follow textbook procedures with no novel insight required, making it slightly easier than average for S2. |
| 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 probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a) | \(F \sim \text{Po}(4)\) | B1 |
| \(P(F \geq 6) = 1 - 0.7851 = 0.2149\) | M1 A1 | 1st M1 writing or using \(1 - P(F \leq 5)\); 1st A1 for awrt 0.215 |
| Let \(Y =\) the number of fishing trips where at least 6 fish are caught | B1 | |
| \(Y \sim B(5, 0.2149)\) | dM1 A1 | dM1 Dep. on 2nd B1 correct expression for \(P(Y = 3)\) using their value of \(p\); 2nd A1 awrt 0.0612 |
| \(P(Y = 3) = 10(0.2149)^3(0.7851)^2 = 0.06117291856\) | ||
| Part (b) | \(\text{H}_0: \lambda = 4\) (\(\mu = 8\)); \(\text{H}_1: \lambda \neq 4\) (\(\mu \neq 8\)) | B1 B1 |
| \(X \sim \text{Po}(8)\) | B1 | 2nd B1 writing or using Po(8) |
| \(P(X \geq 14)\) or CR | M1 | 1st M1 writing \(1 - P(X \leq 13)\) (may be implied by sight of \(1 - 0.9658\)) or for CR method: \(P(X \leq 13) = 0.9658\); \(P(X \leq 14) = 0.9827\); \(\text{CR: } [(X \leq 2)\ldots](X \geq 15)\) |
| \(= 1 - P(X \leq 13) = 1 - 0.9658 = 0.0342\) | A1 dM1 | 1st A1 for probability awrt 0.0342 or CR \(X > 14\) or \(X \geq 15\); 2nd dM1 Dep. on 1st B1. Non-contradictory statement which follows from their probability/CR. |
| \([0.0342 > 0.025]\) so do not reject \(\text{H}_0\) or 14 is not in the critical region or Not significant | A1 cso | 2nd A1 also correct contextual statement and fully correct solution with all other marks scored. |
| There is not enough evidence of a change in the mean number of fish caught or number of fish caught per hour or rate of fish caught. | ||
| Total (12) |
| **Part (a)** | $F \sim \text{Po}(4)$ | B1 | |
| | $P(F \geq 6) = 1 - 0.7851 = 0.2149$ | M1 A1 | 1st M1 writing or using $1 - P(F \leq 5)$; 1st A1 for awrt 0.215 |
| | Let $Y =$ the number of fishing trips where at least 6 fish are caught | B1 | |
| | $Y \sim B(5, 0.2149)$ | dM1 A1 | dM1 Dep. on 2nd B1 correct expression for $P(Y = 3)$ using their value of $p$; 2nd A1 awrt **0.0612** |
| | $P(Y = 3) = 10(0.2149)^3(0.7851)^2 = 0.06117291856$ | | |
| **Part (b)** | $\text{H}_0: \lambda = 4$ ($\mu = 8$); $\text{H}_1: \lambda \neq 4$ ($\mu \neq 8$) | B1 B1 | 1st B1 both hypotheses correct. Must use $\lambda$ or $\mu$ |
| | $X \sim \text{Po}(8)$ | B1 | 2nd B1 writing or using Po(8) |
| | $P(X \geq 14)$ or CR | M1 | 1st M1 writing $1 - P(X \leq 13)$ (may be implied by sight of $1 - 0.9658$) or for CR method: $P(X \leq 13) = 0.9658$; $P(X \leq 14) = 0.9827$; $\text{CR: } [(X \leq 2)\ldots](X \geq 15)$ |
| | $= 1 - P(X \leq 13) = 1 - 0.9658 = 0.0342$ | A1 dM1 | 1st A1 for probability awrt 0.0342 or CR $X > 14$ or $X \geq 15$; 2nd dM1 Dep. on 1st B1. Non-contradictory statement which follows from their probability/CR. |
| | $[0.0342 > 0.025]$ so do not reject $\text{H}_0$ or 14 is not in the critical region or Not significant | A1 cso | 2nd A1 also correct contextual statement and fully correct solution with all other marks scored. |
| | There is not enough evidence of a change in the mean number of fish caught or number of fish caught per hour or rate of fish caught. | | |
| | | | Total (12) |A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.
The fisherman takes 5 fishing trips each lasting 1 hour.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]
\end{enumerate}
The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour.
Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q7 [12]}}