| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Binomial of Poisson approximations |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard Poisson distribution calculations and the normal approximation to Poisson. Part (i) requires basic Poisson probability calculation, part (ii) applies binomial distribution using the result from (i), and part (iii) uses normal approximation with continuity correction—all routine S2 techniques with clear signposting and no novel problem-solving required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\lambda = 1.2\) | B1 | Mean 1.2 stated or implied |
| Tables or formula used | M1 | Tables or formula [allow ± 1 term, or "1 −"] correctly used |
| 0.6626 | A1 | 3 |
| \([.3012, .6990, .6268 \text{ or } .8795: B1M1A0]\) | ||
| (ii) \(B(20, 0.6626v)\) | M1 | \(B(20, p), p\) from (i), stated or implied |
| \(^{20}C_{13} 0.6626^{13} \times 0.3374^7\) | M1 | Correct formula for their \(p\) |
| 0.183 | A1 | 3 |
| (iii) Let \(S\) be the number of stars | B1 | \(\text{Po}(24)\) stated or implied |
| \(S \sim \text{Po}(24)\) | B1 | Normal, mean 24 |
| B1V | Variance 24 or \(24^2\) or \(\sqrt{24}\), \(\sqrt{\text{if 24 wrong}}\) | |
| \(\frac{29.5-24}{\sqrt{24}}\) | M1 | Standardise with \(\lambda, x\), allow errors in cc or √ or both √/a, and cc both correct |
| \(= [1.1227]\) | A1 | Answer, in range [0.868, 0.8694] |
| 0.8692 | A1 | 6 |
(i) $\lambda = 1.2$ | B1 | Mean 1.2 stated or implied
Tables or formula used | M1 | Tables or formula [allow ± 1 term, or "1 −"] correctly used
**0.6626** | A1 | 3 | Answer in range [0.662, 0.663]
| | $[.3012, .6990, .6268 \text{ or } .8795: B1M1A0]$ | |
(ii) $B(20, 0.6626v)$ | M1 | $B(20, p), p$ from (i), stated or implied
$^{20}C_{13} 0.6626^{13} \times 0.3374^7$ | M1 | Correct formula for their $p$
**0.183** | A1 | 3 | Answer, a.r.t. 0.183
(iii) Let $S$ be the number of stars | B1 | $\text{Po}(24)$ stated or implied
$S \sim \text{Po}(24)$ | B1 | Normal, mean 24
| | | B1V | Variance 24 or $24^2$ or $\sqrt{24}$, $\sqrt{\text{if 24 wrong}}$
$\frac{29.5-24}{\sqrt{24}}$ | M1 | Standardise with $\lambda, x$, allow errors in cc or √ or both √/a, and cc both correct
$= [1.1227]$ | A1 | Answer, in range [0.868, 0.8694]
**0.8692** | A1 | 6 |5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution $\operatorname { Po(0.2). }$\\
(i) Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.\\
(ii) Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .\\
(iii) Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.
\hfill \mbox{\textit{OCR S2 2007 Q5 [12]}}