| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution techniques from S2. Parts (a)-(b) involve basic probability calculations with scaling, (c) requires reverse lookup in tables (routine but slightly more involved), and (d)-(e) are standard hypothesis testing setup. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| 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) | \([\text{Po}(1)]\) | |
| \((P(X \geq 1))^2 = (1 - 0.3679)^2 = 0.39955041\ldots\) | M1 A1 | awrt 0.400 |
| Part (b) | \(\text{Po}(2)\) | B1 |
| \(P(X = 0) = 0.1353\) | B1 | awrt 0.135 |
| Part (c) | \(P(X = 4) = P(X \leq 4) - P(X \leq 3) = [0.0443]\) | M1 |
| Using tables: If \(\frac{w}{10} = 8\), \(P(X \leq 4) - P(X \leq 3) = 0.0996 - 0.0424 = 0.0572\) | ||
| If \(\frac{w}{10} = 8.5\), \(P(X \leq 4) - P(X \leq 3) = 0.0744 - 0.0301 = 0.0443\) | A1 | 1st A1 for 8.5 |
| \(\frac{w}{10} = 8.5\), so \(w = 85\) | A1 | 2nd A1 for 85 |
| Part (d) | \([\text{H}_0:] \mu = 10\) (\(\lambda = 1\)) | B1 |
| Part (e) | \(P(X \leq 14 \mid \mu = 10) = 0.9165\); \(P(X \leq 15 \mid \mu = 10) = 0.9513\); So critical region is \(X \geq 16\) | M1 A1 |
| Total (10) |
| **Part (a)** | $[\text{Po}(1)]$ | | |
| | $(P(X \geq 1))^2 = (1 - 0.3679)^2 = 0.39955041\ldots$ | M1 A1 | awrt **0.400** |
| **Part (b)** | $\text{Po}(2)$ | B1 | 1st B1 for writing or using Po(2) (may be implied by sight of $e^{-2}$) |
| | $P(X = 0) = 0.1353$ | B1 | awrt **0.135** |
| **Part (c)** | $P(X = 4) = P(X \leq 4) - P(X \leq 3) = [0.0443]$ | M1 | M1 for writing or using $P(X = 4) = P(X \leq 4) - P(X \leq 3)$ |
| | Using tables: If $\frac{w}{10} = 8$, $P(X \leq 4) - P(X \leq 3) = 0.0996 - 0.0424 = 0.0572$ | | |
| | If $\frac{w}{10} = 8.5$, $P(X \leq 4) - P(X \leq 3) = 0.0744 - 0.0301 = 0.0443$ | A1 | 1st A1 for 8.5 |
| | $\frac{w}{10} = 8.5$, so $w = 85$ | A1 | 2nd A1 for 85 |
| **Part (d)** | $[\text{H}_0:] \mu = 10$ ($\lambda = 1$) | B1 | |
| **Part (e)** | $P(X \leq 14 \mid \mu = 10) = 0.9165$; $P(X \leq 15 \mid \mu = 10) = 0.9513$; So critical region is $X \geq 16$ | M1 A1 | M1 for using Po(10); A1 for $X \geq 16$ or $X > 15$ (allow any letter for $X$) |
| | | | Total (10) |
---The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1
\begin{enumerate}[label=(\alph*)]
\item Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
\item Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]
\end{enumerate}
The probability that this volcano erupts exactly 4 times in a randomly selected $w$ year period is 0.0443 to 3 significant figures.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use the tables to find the value of $w$ [3]
\end{enumerate}
A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1
She selects a 100 year period at random in order to test her claim.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item State the null hypothesis for this test. [1]
\item Determine the critical region for the test at the 5\% level of significance. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q5 [10]}}