| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Statistics (Further AS Paper 2 Statistics) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.3 This is a straightforward application of a one-tailed Poisson hypothesis test with standard structure: define hypotheses, calculate probability under H₀ (X ≤ 8 when λ = 14), compare to 5%, and conclude. Parts (b) and (c) test standard conceptual knowledge about Type II errors and significance levels. While it requires understanding of Poisson distribution and hypothesis testing framework, it follows a routine template with no novel problem-solving or geometric insight required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Part 1: \(H_0: \lambda = 14\); \(H_1: \lambda < 14\); 1 tail 5% (allow \(\lambda = 3.5\) used) | B1 | |
| Part 2: \(T \sim Po(14)\); \(P(\text{total sales} \leq 8)\) | M1 | |
| Part 3: \(= 0.062\) | A1 | |
| Part 4: \(p\text{-value} = 0.062 > 0.05\) | M1F | |
| Part 5: Accept \(H_0\) | E1 | |
| Part 6: No significant evidence to suggest Keith's mean number of properties sold per week in spring has reduced | E1 | Not definite |
| Answer | Marks |
|---|---|
| Type II error is to conclude there is no significant evidence to suggest Keith's mean number of properties sold per week in spring has reduced, when in fact the mean number sold has reduced | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Advantage clearly stated | E1 | Less likely to reject \(H_0\) when \(H_0\) is, in fact, true; or Less likely to make a Type I error |
| Disadvantage clearly stated | E1 | Less likely to accept \(H_1\) when \(H_0\) is, in fact, false; or More likely to make a Type II error |
## Question 8(a):
**Part 1:** $H_0: \lambda = 14$; $H_1: \lambda < 14$; 1 tail 5% (allow $\lambda = 3.5$ used) | B1 |
**Part 2:** $T \sim Po(14)$; $P(\text{total sales} \leq 8)$ | M1 |
**Part 3:** $= 0.062$ | A1 |
**Part 4:** $p\text{-value} = 0.062 > 0.05$ | M1F |
**Part 5:** Accept $H_0$ | E1 |
**Part 6:** No significant evidence to suggest Keith's mean number of properties sold per week in spring has reduced | E1 | Not definite
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## Question 8(b):
Type II error is to conclude there is no significant evidence to suggest Keith's mean number of properties sold per week in spring has reduced, when in fact the mean number sold has reduced | E1 |
## Question 8(c):
**Advantage clearly stated** | E1 | Less likely to reject $H_0$ when $H_0$ is, in fact, true; or Less likely to make a Type I error
**Disadvantage clearly stated** | E1 | Less likely to accept $H_1$ when $H_0$ is, in fact, false; or More likely to make a Type II error8 In a small town, the number of properties sold during a week in spring by a local estate agent, Keith, can be regarded as occurring independently and with constant mean $\mu$. Data from several years have shown the value of $\mu$ to be 3.5 .
A new housing development was built on the outskirts of the town and the properties on this development were offered for sale by the builder of the development, not by the local estate agents.
During the first four weeks in spring, when properties on the new development were offered for sale by the builder, Keith sold a total of 8 properties.
Keith claims that the sale of new properties by the builder reduced his mean number of properties sold during a week in spring.
8
\begin{enumerate}[label=(\alph*)]
\item Investigate Keith's claim, using the $5 \%$ level of significance.\\[0pt]
[6 marks]\\
8
\item For your test carried out in part (a) state, in context, the meaning of a Type II error.\\[0pt]
[1 mark]\\
8
\item State one advantage and one disadvantage of using a 1\% significance level rather than a 5\% level of significance in a hypothesis test.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics Q8 [9]}}