| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate multiple probabilities using Poisson approximation |
| Difficulty | Moderate -0.3 This is a straightforward application of the Poisson approximation to the binomial distribution with clear parameters (n=240, p=0.01). The question explicitly tells students to use an approximation and requires only a standard probability calculation P(X>2) = 1-P(X≤2) plus a routine justification (n large, p small, np moderate). While it requires understanding when the approximation is valid, it involves no problem-solving insight or complex multi-step reasoning—just applying a standard technique covered in S2. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model |
| Answer | Marks |
|---|---|
| (ii) | Use of Poisson |
| Answer | Marks |
|---|---|
| 240 × 0.01 = 2.4 < 5 or np<5 or p<0.1 | B1 |
| Answer | Marks |
|---|---|
| B1 [2] | Allow any λ (Allow one end error) |
| Answer | Marks |
|---|---|
| (iii) | H0: Pop mean = 2.5 (or 7.5) |
| Answer | Marks |
|---|---|
| H0 was rejected oe | B1 |
| Answer | Marks |
|---|---|
| B1 [1] | or λ = 2.5(Not just “mean”) Allow µ |
Question 3:
--- 3 (i)
(ii) ---
3 (i)
(ii) | Use of Poisson
Mean = 2.4
-2.4 2.42
1 – e (1 + 2.4 + )
2
= 0.43(0) (3 sf)
240 > 50 or n>50
240 × 0.01 = 2.4 < 5 or np<5 or p<0.1 | B1
B1
M1
A1 [4]
B1
B1 [2] | Allow any λ (Allow one end error)
Final answer
SR Use of binomial: B1 for ans 0.431 (3 sf)
SR n large, p small: B1
4 (i)
(ii)
(iii) | H0: Pop mean = 2.5 (or 7.5)
H0: Pop mean < 2.5 (or 7.5)
λ = 7.5
P(X ⩽ 2) = e –7.5 (1+7.5+7.52 ) = 0.0203
2
P(X⩽3)=0.0203 + e –7.5 ×7.53 = 0.0591
3!
CR is X ⩽ 2
Reject H
0
Evidence that no of sightings fewer
P(Type I) = 0.0203 (3 sf)
H0 was rejected oe | B1
M1
A1
A1
A1 [5]
B1 [1]
B1 [1] | or λ = 2.5(Not just “mean”) Allow µ
or λ < 2.5
Either P(X⩽2) or P(X⩽3) , allow any λ
Both Correct
Clear statement
Follow through their CR/their P(X⩽2)
ft their P(X ⩽ 2)
or Type II is P(not reject H0)oe1\% of adults in a certain country own a yellow car.
\begin{enumerate}[label=(\roman*)]
\item Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car. [4]
\item Justify your approximation. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2016 Q3 [6]}}