| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | State conditions for Poisson approximation |
| Difficulty | Standard +0.3 This is a standard S2 question testing routine application of distribution approximations (Binomial→Poisson→Normal) with straightforward calculations. Part (iii) requires stating textbook conditions, and all parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04c Calculate binomial probabilities2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=1) = \binom{8}{1}(0.1)^1(0.9)^7\) | M1 | correct binomial calculation |
| \(= 0.3826\) | A1 | accept 0.383 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\lambda = 30 \times 0.1 = 3\) | B1 | Poisson mean stated |
| \(P(X=6) = e^{-3}\frac{3^6}{6!}\) | M1 | |
| \(= 0.0504\) | A1 |
| Answer | Marks |
|---|---|
| \(P(X \geq 8) = 1 - P(X \leq 7)\) | M1 |
| \(= 1 - 0.9881\) | A1 |
| \(= 0.0119\) | A1 |
| Answer | Marks |
|---|---|
| \(n\) is large | B1 |
| \(p\) is small | B1 |
| Answer | Marks |
|---|---|
| \(\mu = 120 \times 0.1 = 12\) | B1 |
| \(\sigma^2 = 120 \times 0.1 \times 0.9 = 10.8\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 15) = P\left(Z > \frac{15.5 - 12}{\sqrt{10.8}}\right)\) | M1 | continuity correction |
| \(= P(Z > 1.065)\) | A1 | |
| \(= 1 - 0.8566 = 0.1434\) | A1 | accept awrt 0.143 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq k) \geq 0.99\) | M1 | correct inequality |
| \(\frac{k + 0.5 - 12}{\sqrt{10.8}} = 2.326\) | M1 | z = 2.326 used |
| \(k = 12 - 0.5 + 2.326\sqrt{10.8}\) | M1 | |
| \(k = 19.14\), so minimum = 20 | A1 |
# Question 1:
## Part (i)
$P(X=1) = \binom{8}{1}(0.1)^1(0.9)^7$ | M1 | correct binomial calculation
$= 0.3826$ | A1 | accept 0.383
## Part (ii)(A)
$\lambda = 30 \times 0.1 = 3$ | B1 | Poisson mean stated
$P(X=6) = e^{-3}\frac{3^6}{6!}$ | M1 |
$= 0.0504$ | A1 |
## Part (ii)(B)
$P(X \geq 8) = 1 - P(X \leq 7)$ | M1 |
$= 1 - 0.9881$ | A1 |
$= 0.0119$ | A1 |
## Part (iii)
$n$ is large | B1 |
$p$ is small | B1 |
## Part (iv)
$\mu = 120 \times 0.1 = 12$ | B1 |
$\sigma^2 = 120 \times 0.1 \times 0.9 = 10.8$ | B1 |
## Part (v)
$P(X > 15) = P\left(Z > \frac{15.5 - 12}{\sqrt{10.8}}\right)$ | M1 | continuity correction
$= P(Z > 1.065)$ | A1 |
$= 1 - 0.8566 = 0.1434$ | A1 | accept awrt 0.143
## Part (vi)
$P(X \leq k) \geq 0.99$ | M1 | correct inequality
$\frac{k + 0.5 - 12}{\sqrt{10.8}} = 2.326$ | M1 | z = 2.326 used
$k = 12 - 0.5 + 2.326\sqrt{10.8}$ | M1 |
$k = 19.14$, so minimum = **20** | A1 |
---1 A low-cost airline charges for breakfasts on its early morning flights. On average, $10 \%$ of passengers order breakfast.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
\item Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is\\
(A) exactly 6,\\
(B) at least 8 .
\item State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
\item The aircraft carries 120 passengers and the flight is always full. Find the mean $\mu$ and variance $\sigma ^ { 2 }$ of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
\item Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
\item The airline wishes to be at least $99 \%$ certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2006 Q1 [18]}}