OCR S2 2012 January — Question 2 5 marks

Exam BoardOCR
ModuleS2 (Statistics 2)
Year2012
SessionJanuary
Marks5
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Mark schemeDownload PDF ↗
TopicApproximating the Binomial to the Poisson distribution
TypeCalculate single probability using Poisson approximation
DifficultyStandard +0.3 This is a straightforward application of the Poisson approximation to the binomial distribution with n large and p small (np = 4.2). Students need to recognize the conditions (n > 50, p < 0.1, np < 10), apply the approximation Y ~ Po(4.2), and calculate P(Y=5) using the Poisson formula. While it requires understanding when to use the approximation, the execution is routine for S2 students, making it slightly easier than average.
Spec2.04d Normal approximation to binomial5.02n Sum of Poisson variables: is Poisson
The random variable \(Y\) has the distribution B(140, 0.03). Use a suitable approximation to find P(\(Y = 5\)). Justify your approximation. [5]
AnswerMarks Guidance
\(\text{Po}(4.2)\)M1 \(\text{Po}(np)\) stated or implied
\(e^{-4.2} \frac{4.2^3}{3!} = 0.1633\)M1 Poisson formula or tables, allow for .1944, .1144, .16(0), .1663;
A1Answer, a.r.t. 0.163
B1One condition
B1The other condition Needs Poisson. If inequalities used, must be these, but allow \(p < 0.1\) if \(n > 50\) already stated
[5] 
$\text{Po}(4.2)$ | M1 | $\text{Po}(np)$ stated or implied
$e^{-4.2} \frac{4.2^3}{3!} = 0.1633$ | M1 | Poisson formula or tables, allow for .1944, .1144, .16(0), .1663;
| A1 | Answer, a.r.t. 0.163
| B1 | One condition
| B1 | The other condition | Needs Poisson. If inequalities used, must be these, but allow $p < 0.1$ if $n > 50$ already stated
| [5] |
The random variable $Y$ has the distribution B(140, 0.03). Use a suitable approximation to find P($Y = 5$). Justify your approximation. [5]

\hfill \mbox{\textit{OCR S2 2012 Q2 [5]}}
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