CAIE S1 2013 June — Question 2 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2013
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApproximating Binomial to Normal Distribution
TypeSingle probability inequality
DifficultyModerate -0.3 This is a straightforward application of the normal approximation to the binomial distribution with clearly stated parameters (n=350, p=1/7). It requires identifying the distribution, applying continuity correction, standardizing to find a z-score, and reading from tables—all standard S1 techniques with no conceptual complications or multi-step reasoning beyond the routine procedure.
Spec2.04d Normal approximation to binomial
2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.
Question 2:
AnswerMarks Guidance
AnswerMark Guidance
\(np = 350 \times \frac{1}{7} = 50\)B1 Correct unsimplified \(np\) and \(npq\)
\(npq = 350 \times \frac{1}{7} \times \frac{6}{7} = 42.857\)M1 standardising, with or without cc, must have sq rt
\(P(x \leq 47) = P\left(z > \frac{46.5 - 50}{\sqrt{42.857}}\right)\)M1 continuity correction 46.5 or 47.5
M1correct area ie \(> 0.5\) must be a \(\Phi\)
\(P(z > -0.5346) = 0.704\)A1 [5] correct answer 
## Question 2:

| Answer | Mark | Guidance |
|--------|------|----------|
| $np = 350 \times \frac{1}{7} = 50$ | B1 | Correct unsimplified $np$ and $npq$ |
| $npq = 350 \times \frac{1}{7} \times \frac{6}{7} = 42.857$ | M1 | standardising, with or without cc, must have sq rt |
| $P(x \leq 47) = P\left(z > \frac{46.5 - 50}{\sqrt{42.857}}\right)$ | M1 | continuity correction 46.5 or 47.5 |
| | M1 | correct area ie $> 0.5$ must be a $\Phi$ |
| $P(z > -0.5346) = 0.704$ | A1 **[5]** | correct answer |

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2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.

\hfill \mbox{\textit{CAIE S1 2013 Q2 [5]}}
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