CAIE S1 2017 June — Question 2 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2017
SessionJune
Marks5
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TopicApproximating Binomial to Normal Distribution
TypeSingle probability inequality
DifficultyStandard +0.3 This is a straightforward application of the normal approximation to the binomial distribution. Students need to identify n=270, p=1/3, calculate mean and variance, apply continuity correction (≥100 becomes >99.5), and use standard normal tables. While it requires multiple steps, each is routine and the question explicitly tells students to use an approximation, making it slightly easier than average.
Spec2.04d Normal approximation to binomial
2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.
Question 2:
AnswerMarks Guidance
AnswerMark Guidance
\(np = 270 \times \frac{1}{3} = 90\), \(npq = 270 \times \frac{1}{3} \times \frac{2}{3} = 60\)B1 Correct unsimplified \(np\) and \(npq\), SOI
\(P(x > 100) = P\left(z > \frac{99.5 - 90}{\sqrt{60}}\right) = P(z > 1.2264)\)M1 \(\pm\)Standardising using 100, need \(\sqrt{60}\)
M1Continuity correction, 99.5 or 100.5 used
\(= 1 - 0.8899\)M1 Correct area \(1 - \Phi\) implied by final prob. \(< 0.5\)
\(= 0.110\)A1
Total: 5 
# Question 2:

| Answer | Mark | Guidance |
|--------|------|----------|
| $np = 270 \times \frac{1}{3} = 90$, $npq = 270 \times \frac{1}{3} \times \frac{2}{3} = 60$ | B1 | Correct unsimplified $np$ and $npq$, SOI |
| $P(x > 100) = P\left(z > \frac{99.5 - 90}{\sqrt{60}}\right) = P(z > 1.2264)$ | M1 | $\pm$Standardising using 100, need $\sqrt{60}$ |
| | M1 | Continuity correction, 99.5 or 100.5 used |
| $= 1 - 0.8899$ | M1 | Correct area $1 - \Phi$ implied by final prob. $< 0.5$ |
| $= 0.110$ | A1 | |
| **Total: 5** | | |
2 The probability that George goes swimming on any day is $\frac { 1 } { 3 }$. Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.\\

\hfill \mbox{\textit{CAIE S1 2017 Q2 [5]}}
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