| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.3 This is a straightforward application of Poisson approximation to binomial with standard conditions (n large, p small, np moderate). Part (i) requires recognizing n=40, p=0.225 gives np=9 and using normal approximation instead (or recognizing Poisson isn't suitable here), while part (ii) is basic sampling methodology. The question tests understanding of when approximations apply rather than complex calculation, making it slightly easier than average A-level statistics questions. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(B(40, 0.225)\) | M1 | \(B(40, 0.225)\) stated or implied |
| \(= N(9, 6.975)\) | M1 | Normal, mean 9 |
| \(\frac{5.5 - 9}{\sqrt{6.975}} = -1.325\) | A1 | Variance 6.975 or SD 2.641 or 6.975 |
| M1 | Standardise with \(np\) and \(\sqrt{npq}\), allow \(npq\), no or wrong cc | |
| \(0.9074\) | A1 | CC and \(\sqrt{npq}\) correct, allow from N(3600, 0.225) |
| A1 | Answer, in range [0.907, 0.908] | |
| B2 | Full conditions B2; partial, B1 (assertions OK). Allow \(npq\), allow from e.g. \(n = 3600\) | |
| 8 | ||
| (ii) Number list sequentially and select using random numbers | B1 | Number list, don't need "sequentially" |
| If \(\# > 3600\), ignore (etc) | B1 | Mention random numbers (not "select numbers randomly") |
| B1 | Deal with issues like \(\# > 3600\) or "ignore repeats" | |
| 3 | α "Randomly pick numbers from 0 to 3599": (B1) B0 B1 |
**(i)** $B(40, 0.225)$ | M1 | $B(40, 0.225)$ stated or implied
$= N(9, 6.975)$ | M1 | Normal, mean 9
$\frac{5.5 - 9}{\sqrt{6.975}} = -1.325$ | A1 | Variance 6.975 or SD 2.641 or 6.975
| M1 | Standardise with $np$ and $\sqrt{npq}$, allow $npq$, no or wrong cc
$0.9074$ | A1 | CC and $\sqrt{npq}$ correct, allow from N(3600, 0.225)
| A1 | Answer, in range [0.907, 0.908]
| B2 | Full conditions B2; partial, B1 (assertions OK). Allow $npq$, allow from e.g. $n = 3600$
| | 8
**(ii)** Number list sequentially and select using random numbers | B1 | Number list, don't need "sequentially"
If $\# > 3600$, ignore (etc) | B1 | Mention random numbers (not "select numbers randomly")
| B1 | Deal with issues like $\# > 3600$ or "ignore repeats"
| | 3 | α "Randomly pick numbers from 0 to 3599": (B1) B0 B18 A company has 3600 employees, of whom $22.5 \%$ live more than 30 miles from their workplace. A random sample of 40 employees is obtained.\\
(i) Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.\\
(ii) Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.
\hfill \mbox{\textit{OCR S2 2011 Q8 [11]}}