OCR S2 2011 January — Question 3 6 marks

Exam BoardOCR
ModuleS2 (Statistics 2)
Year2011
SessionJanuary
Marks6
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TopicApproximating the Binomial to the Poisson distribution
TypeState Poisson approximation with justification
DifficultyModerate -0.8 This is a straightforward application of the Poisson approximation to the binomial distribution. Students need to verify n is large and p is small (np = 4.56 < 5, n = 200 large, p = 0.0228 small), then calculate P(X ≤ 2) using Poisson tables. The justification is standard bookwork and the calculation is routine with no problem-solving required.
Spec2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial
3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.
AnswerMarks Guidance
\(B(200, 0.0228)\)M1 \(B(200, 0.0228)\) stated or implied
\(Po(4.56)\)A1 \(Po(4.56)\) stated or implied
\(e^{-4.56}(1 + 4.56 + \frac{4.56^2}{2})\)M1 Correct formula for \(P(\le 2) \geq 1\) term, any \(\lambda\) (tables: M0)
\(= 0.167\)A1 Correct formula, 4.56 needed
A1Answer, a.r.t. 0.167 [0.16694]
B1Both, can be merely asserted. If numbers, must be these SR interpolation: clear method M1, answer A2
6MR: typically \(B(200, 0.228) = N(45.6, 3.52)\): M1A1; standardise correctly, M1; state \(np > 5\), B1 
$B(200, 0.0228)$ | M1 | $B(200, 0.0228)$ stated or implied
$Po(4.56)$ | A1 | $Po(4.56)$ stated or implied
$e^{-4.56}(1 + 4.56 + \frac{4.56^2}{2})$ | M1 | Correct formula for $P(\le 2) \geq 1$ term, any $\lambda$ (tables: M0)
$= 0.167$ | A1 | Correct formula, 4.56 needed
| A1 | Answer, a.r.t. 0.167 [0.16694]
| B1 | Both, can be merely asserted. If numbers, must be these SR interpolation: clear method M1, answer A2
| | 6 | MR: typically $B(200, 0.228) = N(45.6, 3.52)$: M1A1; standardise correctly, M1; state $np > 5$, B1
3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.

\hfill \mbox{\textit{OCR S2 2011 Q3 [6]}}
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