| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| 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 multi-part hypothesis testing question requiring: (a) one-tailed Poisson test with continuity considerations, (b) critical region determination from Poisson tables, and (c) Type II error calculation under a specific alternative hypothesis. Part (c) is particularly demanding as it requires understanding of power/Type II errors and computing P(accept H₀|H₁ true) using a different parameter value, which goes beyond routine hypothesis testing and requires deeper statistical reasoning. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(H_0: \lambda = 13\) | B1 | CAO; OE |
| \(H_1: \lambda < 13\) | B1 | CAO; OE |
| \(P(R \leq 10 \mid \text{Po}(13))\) | M1 | Used or implied |
| \(= 0.2517\) | A1 | AWFW 0.251 to 0.252 |
| Prob of \(0.2517 > 0.10\) (10%), \(z = -0.83\) to \(-0.70 > -1.28\) | M1 | Comparison of prob with 0.10 / Comparison of \(z\) with \(-1.28\) |
| Thus no evidence, at 10% level, of a reduction in the mean value of \(R\) | A1\(\checkmark\) | \(\checkmark\) on probability or \(z\); in 'context' and qualified |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Require \(P(R \leq r \mid \text{Po}(13)) \approx 0.10\) | M1 | Stated or implied |
| Critical Region is \(R \leq 8\) or \(R < 9\) | A1 | Accept \(R=8\); may be scored in (a) |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Require \(P(\text{accept } H_0 \mid H_0 \text{ false})\) | B1 | OE; PI |
| \(= P(R > 8 \mid \text{Po}(6.5))\) | M1 | Use of Po(6.5) |
| \(= 1 - P(R \leq 8 \mid \text{Po}(6.5))\) | m1 | |
| \(= 1 - 0.7916\) | ||
| \(= 0.208\) to \(0.209\) | A1 | AWFW (0.2084) |
| Total | 4 |
# Question 7(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 13$ | B1 | CAO; OE |
| $H_1: \lambda < 13$ | B1 | CAO; OE |
| $P(R \leq 10 \mid \text{Po}(13))$ | M1 | Used or implied |
| $= 0.2517$ | A1 | AWFW 0.251 to 0.252 |
| Prob of $0.2517 > 0.10$ (10%), $z = -0.83$ to $-0.70 > -1.28$ | M1 | Comparison of prob with 0.10 / Comparison of $z$ with $-1.28$ |
| Thus no evidence, at 10% level, of a reduction in the mean value of $R$ | A1$\checkmark$ | $\checkmark$ on probability or $z$; in 'context' and qualified |
| **Total** | **6** | |
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# Question 7(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Require $P(R \leq r \mid \text{Po}(13)) \approx 0.10$ | M1 | Stated or implied |
| Critical Region is $R \leq 8$ or $R < 9$ | A1 | Accept $R=8$; may be scored in (a) |
| **Total** | **2** | |
---
# Question 7(c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Require $P(\text{accept } H_0 \mid H_0 \text{ false})$ | B1 | OE; PI |
| $= P(R > 8 \mid \text{Po}(6.5))$ | M1 | Use of Po(6.5) |
| $= 1 - P(R \leq 8 \mid \text{Po}(6.5))$ | m1 | |
| $= 1 - 0.7916$ | | |
| $= 0.208$ to $0.209$ | A1 | AWFW (0.2084) |
| **Total** | **4** | |7 In a town, the total number, $R$, of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 .
A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.
\begin{enumerate}[label=(\alph*)]
\item Using the $10 \%$ level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of $R$.
\item Determine, for your test in part (a), the critical region for $R$.
\item Assuming that the offer for sale of houses on the new housing development has reduced the mean value of $R$ to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA S3 2007 Q7 [12]}}