| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Multiple approximations in one question |
| Difficulty | Standard +0.3 This question tests standard approximation techniques (Binomial→Poisson and Poisson→Normal) with straightforward application of continuity correction. While it requires knowledge of when approximations are valid and careful execution, these are routine S2 procedures with no conceptual challenges or multi-step reasoning beyond looking up values in tables. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Po(1.2) | B1 | Po(1.2) stated or implied |
| Tables or correct formula used; 0.8795 | M1 | Correct method for Poisson probability, allow "1 –" |
| A1 3 | Answer, 0.8795 or 0.879(0.880) | |
| (b) \(N(30, 30)\) | B1 | Normal, mean 30 stated or implied |
| \(38.5 - 30 = [1.55]\) | B1 | Variance 30 stated or implied, allow √30 or 30² |
| \(\sqrt{30}\) | M1 | Standardise using \(\sigma = \sqrt{30}\), allow √ or cc errors |
| \([\Phi(1.55) = ] \quad 0.9396\) | A1 | √μ and 38.5 both correct |
| A1 5 | Answer in range [0.939, 0.94(0)] |
**(a)** Po(1.2) | B1 | Po(1.2) stated or implied
Tables or correct formula used; 0.8795 | M1 | Correct method for Poisson probability, allow "1 –"
| A1 3 | Answer, 0.8795 or 0.879(0.880)
**(b)** $N(30, 30)$ | B1 | Normal, mean 30 stated or implied
$38.5 - 30 = [1.55]$ | B1 | Variance 30 stated or implied, allow √30 or 30²
$\sqrt{30}$ | M1 | Standardise using $\sigma = \sqrt{30}$, allow √ or cc errors
$[\Phi(1.55) = ] \quad 0.9396$ | A1 | √μ and 38.5 both correct
| A1 5 | Answer in range [0.939, 0.94(0)]3
\begin{enumerate}[label=(\alph*)]
\item The random variable $X$ has a $\mathrm { B } ( 60,0.02 )$ distribution. Use an appropriate approximation to find $\mathrm { P } ( X \leqslant 2 )$.
\item The random variable $Y$ has a $\operatorname { Po } ( 30 )$ distribution. Use an appropriate approximation to find $\mathrm { P } ( Y \leqslant 38 )$.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2005 Q3 [8]}}