| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| 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 Poisson approximation to the binomial with standard conditions (n large, p small, np moderate). Part (i) requires only calculating λ = np = 0.2 and finding P(X > 2) from tables. Part (ii) is routine recall of the justification criteria. No problem-solving or insight needed beyond recognizing the standard setup. |
| Spec | 2.04d Normal approximation to binomial5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Poisson with \(\lambda = 0.2\) | B1 | |
| \(1 - e^{-0.2}(1 + 0.2 + \frac{0.2^2}{2})\) | M1 | \(1 -\) Poisson P(0,1,2,3) attempted, any \(\lambda\), allow one end error |
| \(= 0.00115\) (3 sf) | A1 | SR: using Bin, ans 0.00115: B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n\) large (\(n > 50\)) | B1 | |
| \(np = 0.2 < 5\) or \(p\) small | B1 |
## Question 1(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Poisson with $\lambda = 0.2$ | B1 | |
| $1 - e^{-0.2}(1 + 0.2 + \frac{0.2^2}{2})$ | M1 | $1 -$ Poisson P(0,1,2,3) attempted, any $\lambda$, allow one end error |
| $= 0.00115$ (3 sf) | A1 | SR: using Bin, ans 0.00115: **B1** |
## Question 1(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n$ large ($n > 50$) | B1 | |
| $np = 0.2 < 5$ or $p$ small | B1 | |1 On average, 1 clover plant in 10000 has four leaves instead of three.\\
(i) Use an approximating distribution to calculate the probability that, in a random sample of 2000 clover plants, more than 2 will have four leaves.\\
(ii) Justify your approximating distribution.\\
\hfill \mbox{\textit{CAIE S2 2017 Q1 [5]}}