| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State exact binomial distribution |
| Difficulty | Moderate -0.8 This is a straightforward application of standard distribution approximations with clearly signposted steps. Part (i) requires simple identification of B(1200, 1/300), parts (ii-iii) apply the standard Poisson approximation with λ=4, and parts (iv-v) extend to Normal approximation using given formulas. All techniques are routine S2 content with no problem-solving or novel insight required—easier than average A-level. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.02i Poisson distribution: random events model5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Using trial and improvement with \(N(80,80)\) or \(N(80, 79.73)\) to find \(P(Y \leq k)\) for any \(k\) | M1 | Distribution being used needs to be made clear |
| \(P(Y \leq 66) = 0.0587\ldots\) \((0.0584\ldots\) from \(\sigma^2 = 79.73)\) or \(P(Y \leq 65) = 0.0467\ldots\) \((0.0464\ldots\) from \(\sigma^2 = 79.73)\) | A1 | |
| Both values correct | A1 | Final A1 not available if 66 and 65 used |
| Or \(P(Y \leq 65.5) = 0.0524\ldots\) \((0.0521\ldots\) from \(\sigma^2 = 79.73)\) or \(P(Y \leq 64.5) = 0.0415\ldots\) \((0.0412\ldots\) from \(\sigma^2 = 79.73)\) | A1 | |
| Both values correct | A1 | |
| Least value of \(k = 65\), dependent on previous two A marks earned | A1 |
## Question 2(v)B:
| Answer | Mark | Guidance |
|--------|------|----------|
| Using trial and improvement with $N(80,80)$ or $N(80, 79.73)$ to find $P(Y \leq k)$ for any $k$ | M1 | Distribution being used needs to be made clear |
| $P(Y \leq 66) = 0.0587\ldots$ $(0.0584\ldots$ from $\sigma^2 = 79.73)$ or $P(Y \leq 65) = 0.0467\ldots$ $(0.0464\ldots$ from $\sigma^2 = 79.73)$ | A1 | |
| Both values correct | A1 | Final A1 not available if 66 and 65 used |
| **Or** $P(Y \leq 65.5) = 0.0524\ldots$ $(0.0521\ldots$ from $\sigma^2 = 79.73)$ or $P(Y \leq 64.5) = 0.0415\ldots$ $(0.0412\ldots$ from $\sigma^2 = 79.73)$ | A1 | |
| Both values correct | A1 | |
| Least value of $k = 65$, dependent on previous two A marks earned | A1 | |
---2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.
\begin{enumerate}[label=(\roman*)]
\item State the exact distribution of $X$, the number of births in the sample which have the mutation.
\item Explain why $X$ has, approximately, a Poisson distribution.
\item Use a Poisson approximating distribution to find\\
(A) $\mathrm { P } ( X = 1 )$,\\
(B) $\mathrm { P } ( X > 4 )$.
\item Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
\item Use this Normal approximating distribution to\\
(A) find the probability that there are at least 90 births which have the mutation,\\
( $B$ ) find the least value of $k$ such that the probability that there are at most $k$ births with this mutation is greater than 5\%.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2012 Q2 [18]}}