| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2003 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find Type I error probability |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question on Poisson distribution with standard definitions and calculations. Part (a) requires recall of error type definitions, part (b) is a routine one-tailed Poisson test, and parts (c)-(d) involve direct probability calculations from the Poisson distribution. While it requires understanding of hypothesis testing concepts, all steps are mechanical applications of standard S4 techniques with no novel problem-solving required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Type I - \(H_0\) rejected when it is true | B1 | |
| Type II - \(H_0\) is accepted when it is false | B1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0 : \lambda = 5\), \(H_1 : \lambda > 5\) | both | B1 |
| \(P(X \geq 7 | \lambda = 5) = 1 - 0.7622 = 0.2378 > 0.05\) | |
| (OR \(P(X \geq 9) = 0.0681\), \(P(X \geq 10) = 0.0318\), CV=10, 7 not in CR. No evidence of an increase in the number of chicks reared per year.) | probabs, 10 context | M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq c | \lambda = 5) < 0.05\) | |
| \(P(X \geq 9) = 0.0681\), \(P(X \geq 10) = 0.0318\), c=10 | may be seen in (b) | M1 |
| \(P(\text{Type I Error}) = 0.0318\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\lambda = 8\) | M1A1 | |
| \(P(X \leq 9 | \lambda = 8) = 0.7166\) | |
| (OR if c=9 in (d), \(P(X \leq 8 | \lambda = 8) = 0.5925\)) |
## Part (a)(i)-(ii)
Type I - $H_0$ rejected when it is true | B1 |
Type II - $H_0$ is accepted when it is false | B1 | (2)
## Part (b)
$H_0 : \lambda = 5$, $H_1 : \lambda > 5$ | both | B1 |
$P(X \geq 7 | \lambda = 5) = 1 - 0.7622 = 0.2378 > 0.05$ | | M1A1 |
(OR $P(X \geq 9) = 0.0681$, $P(X \geq 10) = 0.0318$, CV=10, 7 not in CR. No evidence of an increase in the number of chicks reared per year.) | probabs, 10 context | M1A1 | (4)
## Part (c)
$P(X \geq c | \lambda = 5) < 0.05$ | | M1 |
$P(X \geq 9) = 0.0681$, $P(X \geq 10) = 0.0318$, c=10 | may be seen in (b) | M1 |
$P(\text{Type I Error}) = 0.0318$ | | A1 | (3)
## Part (d)
$\lambda = 8$ | | M1A1 |
$P(X \leq 9 | \lambda = 8) = 0.7166$ | | M1A1 |
(OR if c=9 in (d), $P(X \leq 8 | \lambda = 8) = 0.5925$) | | M1A1) | (2)
---\begin{enumerate}[label=(\alph*)]
\item Define
\begin{enumerate}[label=(\roman*)]
\item a Type I error,
\item a Type II error. [2]
\end{enumerate}
\end{enumerate}
A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
\item Calculate the probability of the Type I error for this test. [3]
\item Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2003 Q5 [11]}}