| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Two-tailed test setup or execution |
| Difficulty | Challenging +1.2 This is a multi-part hypothesis testing question requiring setup of a two-tailed Poisson test with critical regions, understanding of Type I/II errors, and binomial probability calculations. While it involves several concepts and calculations (critical regions for Poisson with λ=7.5, probability calculations, Type II error with new parameter), these are standard Further Statistics 1 techniques applied systematically rather than requiring novel insight. The computational steps are straightforward once the framework is understood, placing it moderately above average difficulty. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: \lambda = 2.5\) (or \(\mu = 7.5\)), \(H_1: \lambda \neq 2.5\) (or \(\mu \neq 7.5\)) | B1 | For both hypotheses in terms of \(\lambda\) or \(\mu\) (either way around) |
| \([X = \text{no. of accidents in a 3-month period}]\) \(X \sim Po(7.5)\) | M1 | For selecting correct Po model; sight or use of \(Po(7.5)\) may be implied by 2nd M1 |
| \(P(X \leq 2) = 0.0203\) (calc: 0.020256...) \(\{\)or \(P(X \leq 3) = 0.0591\}\) | M1 | For using correct model to find one of these probs with correct label (2sf or better) |
| \(P(X \leq 13) = 0.9784\) so \(P(X \geq 14) = 0.0216\) (calc: 0.0215646...) \(\{\)or \(P(X \geq 15) = 0.0103\}\) | ||
| Critical region: \(X \leq 2\) | A1 | 1st A1 for one end correct; allow any letter, even CR \(\leq 2\) or set notation but not \(P(X \leq 2)\) |
| \(X \geq 14\) | A1 | 2nd A1 for fully correct CR; can have \(X < 3\) and \(X > 13\) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([0.0203 + 0.0216] = \text{awrt } \mathbf{0.0419}\) or (calc: 0.041821366... awrt 0.0418) | B1ft | For awrt 0.0419 or awrt 0.0418; or ft addition of their two probs provided both are \(0 < \text{prob} < 0.025\) (awrt 3sf) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([M = \text{no of 3-month periods with a significant result}]\) \(M \sim B(8, \text{"0.0419"})\) | M1 | For selecting correct binomial model, ft their answer to part (b) |
| \([P(M \geq 2)] = 1 - P(M \leq 1)\) | M1 | For correct probability statement of \(1 - P(M \leq 1)\) dep on a binomial selected |
| \(= 1 - 0.9584...\) | ||
| \(= 0.04153...\) (calc: 0.041394...) \([\mathbf{0.04139 \sim 0.04154}]\) | A1cso | For answer in range \([0.04139, 0.04154]\) dep on use of \(B(8, \text{"0.0419"})\) or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Y \sim Po(6.3)\) | M1 | For selecting a \(Po(6.3)\) model |
| \(P(\text{Type II error}) = P(3 \leq Y \leq 13)\) or \(P(Y \leq 13) - P(Y \leq 2)\) | M1 | For correct probability statement using their Poisson model and their CR in (a) which may have just one tail |
| \([= 0.9945147... - 0.049846...]\) | ||
| \(= 0.9446...\) awrt \(\mathbf{0.945}\) | A1 | For awrt 0.945 |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda = 2.5$ (or $\mu = 7.5$), $H_1: \lambda \neq 2.5$ (or $\mu \neq 7.5$) | B1 | For both hypotheses in terms of $\lambda$ or $\mu$ (either way around) |
| $[X = \text{no. of accidents in a 3-month period}]$ $X \sim Po(7.5)$ | M1 | For selecting correct Po model; sight or use of $Po(7.5)$ may be implied by 2nd M1 |
| $P(X \leq 2) = 0.0203$ (calc: 0.020256...) $\{$or $P(X \leq 3) = 0.0591\}$ | M1 | For using correct model to find one of these probs with correct label (2sf or better) |
| $P(X \leq 13) = 0.9784$ so $P(X \geq 14) = 0.0216$ (calc: 0.0215646...) $\{$or $P(X \geq 15) = 0.0103\}$ | | |
| Critical region: $X \leq 2$ | A1 | 1st A1 for one end correct; allow any letter, even CR $\leq 2$ or set notation but **not** $P(X \leq 2)$ |
| $X \geq 14$ | A1 | 2nd A1 for fully correct CR; can have $X < 3$ and $X > 13$ etc |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[0.0203 + 0.0216] = \text{awrt } \mathbf{0.0419}$ or (calc: 0.041821366... awrt **0.0418**) | B1ft | For awrt 0.0419 or awrt 0.0418; or ft addition of their two probs provided both are $0 < \text{prob} < 0.025$ (awrt 3sf) |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[M = \text{no of 3-month periods with a significant result}]$ $M \sim B(8, \text{"0.0419"})$ | M1 | For selecting correct binomial model, ft their answer to part (b) |
| $[P(M \geq 2)] = 1 - P(M \leq 1)$ | M1 | For correct probability statement of $1 - P(M \leq 1)$ dep on a binomial selected |
| $= 1 - 0.9584...$ | | |
| $= 0.04153...$ (calc: 0.041394...) $[\mathbf{0.04139 \sim 0.04154}]$ | A1cso | For answer in range $[0.04139, 0.04154]$ dep on use of $B(8, \text{"0.0419"})$ or better |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $Y \sim Po(6.3)$ | M1 | For selecting a $Po(6.3)$ model |
| $P(\text{Type II error}) = P(3 \leq Y \leq 13)$ or $P(Y \leq 13) - P(Y \leq 2)$ | M1 | For correct probability statement using their Poisson model and their CR in (a) which may have just one tail |
| $[= 0.9945147... - 0.049846...]$ | | |
| $= 0.9446...$ awrt $\mathbf{0.945}$ | A1 | For awrt 0.945 |
---\begin{enumerate}
\item Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5 Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.\\
(a) Stating your hypotheses clearly and using a $5 \%$ level of significance, find the critical region for this test. You should state the probability in each tail.\\
(b) State P(Type I error) using this test.
\end{enumerate}
Data from the series of 3-month periods are recorded for 2 years.\\
(c) Find the probability that at least 2 of these 3-month periods give a significant result.
Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,\\
(d) find P (Type II error) for the test in part (a)
\hfill \mbox{\textit{Edexcel FS1 2019 Q5 [12]}}