| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Find critical value for given significance |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question with routine Poisson calculations. Part (i) is bookwork recall, (ii) is straightforward Poisson probability with scaled parameter, (iii) requires finding critical region by cumulative probability (standard procedure), and (iv) tests understanding of Type II error. All parts follow textbook methods with no novel insight required, making it slightly easier than average A-level difficulty. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Coins occur at constant average rate and independently of one another | B1 | One contextualised condition, e.g. independent |
| B1 2 | A different one, e.g. constant average rate, or "not in boards" ["singly" not enough]. Treat "random" as equivalent to "independent". Allow "They..." | |
| (ii) \(R \sim Po(5.4)\) | B1 | Poisson (5.4) stated or implied |
| \(e^{-5.4} \times \frac{5.4^4}{4!} = 0.1185\) | M1 | Correct formula, any λ |
| A1 3 | Answer, in range [0.118, 0.119] | |
| (iii) \(R \sim Po(3)\) | B1 | Poisson (3) stated or implied |
| Tables, looking for 0.05 or 0.95 | M1 | Evidence of correct use of tables |
| \(P(R \geq 7) = 0.0335\) | A1V | One relevant correct probability seen |
| Therefore smallest number is 7 | A1 4 | \(r = 7\) only, ignore inequalities |
| (iv) \(R \sim Po(4.8)\) | B1 | Poisson (4.8) used |
| Type II error is \(R < 7\) when \(\mu = 4.8\) | M1 | Correct context for Type II error, √ on their r |
| \(P(< 7) = 0.7908\) | A1 3 | P(< 7), a.r.t. 0.791, c.w.o. [PC(≥ 7): M0] |
**(i)** Coins occur at constant average rate and independently of one another | B1 | One contextualised condition, e.g. independent
| B1 2 | A different one, e.g. constant average rate, or "not in boards" ["singly" not enough]. Treat "random" as equivalent to "independent". Allow "They..."
**(ii)** $R \sim Po(5.4)$ | B1 | Poisson (5.4) stated or implied
$e^{-5.4} \times \frac{5.4^4}{4!} = 0.1185$ | M1 | Correct formula, any λ
| A1 3 | Answer, in range [0.118, 0.119]
**(iii)** $R \sim Po(3)$ | B1 | Poisson (3) stated or implied
Tables, looking for 0.05 or 0.95 | M1 | Evidence of correct use of tables
$P(R \geq 7) = 0.0335$ | A1V | One relevant correct probability seen
Therefore smallest number is 7 | A1 4 | $r = 7$ only, ignore inequalities
**(iv)** $R \sim Po(4.8)$ | B1 | Poisson (4.8) used
Type II error is $R < 7$ when $\mu = 4.8$ | M1 | Correct context for Type II error, √ on their r
$P(< 7) = 0.7908$ | A1 3 | P(< 7), a.r.t. 0.791, c.w.o. [PC(≥ 7): M0]8 In excavating an archaeological site, Roman coins are found scattered throughout the site.\\
(i) State two assumptions needed to model the number of coins found per square metre of the site by a Poisson distribution.
Assume now that the number of coins found per square metre of the site can be modelled by a Poisson distribution with mean $\lambda$.\\
(ii) Given that $\lambda = 0.75$, calculate the probability that exactly 3 coins are found in a region of the site of area $7.20 \mathrm {~m} ^ { 2 }$.
A test is carried out, at the $5 \%$ significance level, of the null hypothesis $\lambda = 0.75$, against the alternative hypothesis $\lambda > 0.75$, in Region LVI which has area $4 \mathrm {~m} ^ { 2 }$.\\
(iii) Determine the smallest number of coins that, if found in Region LVI, would lead to rejection of the null hypothesis, stating also the values of any relevant probabilities.\\
(iv) Given that, in fact, $\lambda = 1.2$ in Region LVI, find the probability that the test results in a Type II error.
\hfill \mbox{\textit{OCR S2 2005 Q8 [12]}}