| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State Poisson approximation with justification |
| Difficulty | Moderate -0.8 This is a straightforward application of the standard Poisson approximation to the binomial distribution. Students need to recognize n=3500, p=1/1000, identify B(3500, 1/1000) and Po(3.5), state the justification (large n, small p, np moderate), then perform a routine Poisson probability calculation. This requires only recall of conditions and basic calculator work, making it easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(B(3500, 0.001)\) | B1 | |
| Poisson with mean \(= 3.5\) | B1 | or \(Po(3.5)\) |
| \(n > 50\) and \(np < 5\) | B1 [3] | Both. Or \(n > 50\) and \(\lambda < 5\) or \(3.5 < 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(e^{-3.5}(1 + 3.5 + \frac{3.5^2}{2} + \frac{3.5^3}{3!})\) | M1 | Allow any \(\lambda\) |
| \(= 0.537\) (3 dp) | A1 [2] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $B(3500, 0.001)$ | B1 | |
| Poisson with mean $= 3.5$ | B1 | or $Po(3.5)$ |
| $n > 50$ and $np < 5$ | B1 [3] | Both. Or $n > 50$ and $\lambda < 5$ or $3.5 < 5$ |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e^{-3.5}(1 + 3.5 + \frac{3.5^2}{2} + \frac{3.5^3}{3!})$ | M1 | Allow any $\lambda$ |
| $= 0.537$ (3 dp) | A1 [2] | |
---4 The proportion of people who have a particular gene, on average, is 1 in 1000. A random sample of 3500 people in a certain country is chosen and the number of people, $X$, having the gene is found.\\
(i) State the distribution of $X$ and state also an appropriate approximating distribution. Give the values of any parameters in each case. Justify your choice of the approximating distribution.\\
(ii) Use the approximating distribution to find $\mathrm { P } ( X \leqslant 3 )$.
\hfill \mbox{\textit{CAIE S2 2014 Q4 [5]}}