| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - scaled period (normal approximation only) |
| Difficulty | Moderate -0.3 This is a straightforward Poisson distribution question requiring standard knowledge: stating model assumptions (routine recall), scaling the parameter for different time periods (mechanical calculation), and applying normal approximation (standard S2 technique). All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Planes arrive at constant mean rate | B1 | |
| Planes arrive at random | B1 | or Planes arrive independently. Must be in context |
| Answer | Marks | Guidance |
|---|---|---|
| \((\lambda =) 5.2 \div 4\) | M1 | |
| \(e^{-1.3}\left(\frac{1.3^2}{2} + \frac{1.3^3}{3!}\right)\) | M1 | Allow any \(\lambda\), allow one end error |
| \(= 0.330\) (3 sfs) | A1 | Accept 0.33 |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - e^{-3.467} \times \left(1 + 3.467 + \frac{3.467^2}{2!} + \frac{3.467^3}{3!}\right)\) | M1 | Allow any \(\lambda\) except 5.2 or 1.3, allow one end error |
| \(= 0.456\) (3 sfs) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(N(52, 52)\) stated or implied | B1 | |
| \(\frac{60.5 - 52}{\sqrt{52}} (= 1.179)\) | M1 | ft their mean and var. Allow wrong or no cc or no \(\sqrt{}\) |
| \(\Phi(``1.179")\) | M1 | |
| \(= 0.881\) (3 sf) | A1 |
## Question 7(i):
Planes arrive at constant mean rate | B1 |
Planes arrive at random | B1 | or Planes arrive independently. Must be in context
---
## Question 7(ii)(a):
$(\lambda =) 5.2 \div 4$ | M1 |
$e^{-1.3}\left(\frac{1.3^2}{2} + \frac{1.3^3}{3!}\right)$ | M1 | Allow any $\lambda$, allow one end error
$= 0.330$ (3 sfs) | A1 | Accept 0.33
---
## Question 7(ii)(b):
$1 - e^{-3.467} \times \left(1 + 3.467 + \frac{3.467^2}{2!} + \frac{3.467^3}{3!}\right)$ | M1 | Allow any $\lambda$ except 5.2 or 1.3, allow one end error
$= 0.456$ (3 sfs) | A1 |
---
## Question 7(iii):
$N(52, 52)$ stated or implied | B1 |
$\frac{60.5 - 52}{\sqrt{52}} (= 1.179)$ | M1 | ft their mean and var. Allow wrong or no cc or no $\sqrt{}$
$\Phi(``1.179")$ | M1 |
$= 0.881$ (3 sf) | A1 |7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable $X$ with distribution $\operatorname { Po } ( 5.2 )$.
\begin{enumerate}[label=(\roman*)]
\item State two assumptions required for the Poisson model to be valid in this context.
\item (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,\\
(b) Find the probability that more than 3 planes will arrive in a 40-minute period.
\item The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2017 Q7 [11]}}