CAIE S2 2013 November — Question 1 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2013
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicPoisson distribution
TypeStandard Poisson approximation to binomial
DifficultyModerate -0.3 This is a straightforward application of Poisson approximation to binomial with n=1000, p=1/30000, giving λ=1/30. Requires recognizing the approximation is valid (large n, small p), calculating λ, and finding P(X≥1)=1-P(X=0) using standard Poisson formula. Slightly easier than average due to being a direct textbook-style application with minimal steps.
Spec2.04d Normal approximation to binomial5.02i Poisson distribution: random events model
1 Each computer made in a factory contains 1000 components. On average, 1 in 30000 of these components is defective. Use a suitable approximate distribution to find the probability that a randomly chosen computer contains at least 1 faulty component.
AnswerMarks Guidance
\(\lambda = \frac{1}{30}\)B1 o.e
\(1 - e^{-\frac{1}{30}}\)M1 \(1 - P(X = 0)\) by Poisson, any \(\lambda\) allow 1 end error
\(= 0.0328\) (3 s.f.)M1 \(1 - P(X = 0)\) by Poisson, correct \(\lambda\) no end errors
A1S.R. Binomial with final answer 0.0328 B2 Correct answer, no working scores B2 
$\lambda = \frac{1}{30}$ | B1 | o.e

$1 - e^{-\frac{1}{30}}$ | M1 | $1 - P(X = 0)$ by Poisson, any $\lambda$ allow 1 end error

$= 0.0328$ (3 s.f.) | M1 | $1 - P(X = 0)$ by Poisson, correct $\lambda$ no end errors
| A1 | S.R. Binomial with final answer 0.0328 B2 Correct answer, no working scores B2 |
1 Each computer made in a factory contains 1000 components. On average, 1 in 30000 of these components is defective. Use a suitable approximate distribution to find the probability that a randomly chosen computer contains at least 1 faulty component.

\hfill \mbox{\textit{CAIE S2 2013 Q1 [4]}}
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