| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring understanding of hypothesis testing with Poisson distributions, including Type I and Type II errors. Part (a) requires setting up hypotheses and finding the critical region (non-trivial with discrete distributions), part (b) is straightforward application, and part (c) requires calculating Type II error probability. The conceptual demand of error types and discrete distribution hypothesis testing places this above average difficulty, though it follows standard S4 procedures. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \lambda = 6\), \(H_1: \lambda > 6\) | B1 | Both hypotheses; allow use of \(\mu\) |
| \(P(X \geq 10) = 0.0839\) | M1 | For seeing \(P(X \geq 10) = 0.0839\) or \(P(X \geq 11) = 0.0426\) or \(P(X \leq 9) = 0.9161\) or \(P(X \leq 10) = 0.9574\); allow sideways slip of 1 |
| \(P(X \geq 11) = 0.0426\) | ||
| \(CR: X \geq 11\) | A1 | For seeing \(P(X \leq 10) = 0.9574\) or \(P(X \geq 11) = 0.0426\) or \(CR: X \geq 11\) |
| \(P(\text{Type I Error}) = 0.0426\) | A1 | An answer of 0.0426 implies M1A1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| 9 is not in the critical region therefore there is no evidence of an increase in the number of accidents per year, or there is no evidence to support Jonty's claim | M1 | Must have 9/value not in CR; allow \(0.153 > 0.05\) |
| A1ft | Correct statement in context – need accidents or Jonty |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lambda = 8\) | ||
| \(P(X \leq 10 \mid \lambda = 8) = 0.8159\) | M1A1 | M1: \(P(X \leq c-1 \mid \lambda = 8)\) with \(c-1\) correct or using their \(c\); allow if CR stated as \(X \leq c\); for \(1 - P(X \leq c \mid \lambda = 8)\); A1 awrt 0.816 |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 6$, $H_1: \lambda > 6$ | B1 | Both hypotheses; allow use of $\mu$ |
| $P(X \geq 10) = 0.0839$ | M1 | For seeing $P(X \geq 10) = 0.0839$ or $P(X \geq 11) = 0.0426$ or $P(X \leq 9) = 0.9161$ or $P(X \leq 10) = 0.9574$; allow sideways slip of 1 |
| $P(X \geq 11) = 0.0426$ | | |
| $CR: X \geq 11$ | A1 | For seeing $P(X \leq 10) = 0.9574$ or $P(X \geq 11) = 0.0426$ or $CR: X \geq 11$ |
| $P(\text{Type I Error}) = 0.0426$ | A1 | An answer of 0.0426 implies M1A1A1 |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| 9 is not in the critical region therefore there is no evidence of an increase in the number of accidents per year, or there is no evidence to support Jonty's claim | M1 | Must have 9/value not in CR; allow $0.153 > 0.05$ |
| | A1ft | Correct statement in context – need accidents or Jonty |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lambda = 8$ | | |
| $P(X \leq 10 \mid \lambda = 8) = 0.8159$ | M1A1 | M1: $P(X \leq c-1 \mid \lambda = 8)$ with $c-1$ correct or using their $c$; allow if CR stated as $X \leq c$; for $1 - P(X \leq c \mid \lambda = 8)$; A1 awrt 0.816 |
---\begin{enumerate}
\item The number of accidents per year in Daftstown follows a Poisson distribution with mean $\lambda$. The value of $\lambda$ has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of $\lambda$ has increased.
\end{enumerate}
Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the $5 \%$ significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.\\
(a) Stating your hypotheses clearly, calculate the probability of a Type I error for this test.
Given that there were 9 accidents in the first year after the speed limit was increased,\\
(b) state, giving a reason, whether or not there is evidence to support Jonty's claim.\\
(c) Given that the value of $\lambda$ has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased.\\
\hfill \mbox{\textit{Edexcel S4 2017 Q2 [8]}}