CAIE S1 2005 June — Question 1 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2005
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicApproximating Binomial to Normal Distribution
TypeSingle probability inequality
DifficultyStandard +0.3 This is a straightforward application of the normal approximation to the binomial distribution with continuity correction. It requires identifying n=400, p=2/5, calculating mean and variance, applying continuity correction (X<165 becomes Z<164.5), and using standard normal tables—all standard S1 techniques with no conceptual challenges.
Spec2.04d Normal approximation to binomial
1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.
\(\mu = 160, \sigma^2 = 96\)
\(P(\leq 165) = \Phi\left(\frac{164.5 - 160}{\sqrt{96}}\right) = \Phi(0.4593) = 0.677\)
AnswerMarks Guidance
AnswerMarks Guidance
For 160 and 96 seen or implied by 9.798B1
For standardising, must have square rootM1
For continuity correction, either 165.5 or 164.5M1
For using tables and finding correct area (i.e. > 0.5)M1
For correct answerA1 [5] 
$\mu = 160, \sigma^2 = 96$

$P(\leq 165) = \Phi\left(\frac{164.5 - 160}{\sqrt{96}}\right) = \Phi(0.4593) = 0.677$

| Answer | Marks | Guidance |
|--------|-------|----------|
| For 160 and 96 seen or implied by 9.798 | B1 | |
| For standardising, must have square root | M1 | |
| For continuity correction, either 165.5 or 164.5 | M1 | |
| For using tables and finding correct area (i.e. > 0.5) | M1 | |
| For correct answer | A1 [5] | |

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1 It is known that, on average, 2 people in 5 in a certain country are overweight. A random sample of 400 people is chosen. Using a suitable approximation, find the probability that fewer than 165 people in the sample are overweight.

\hfill \mbox{\textit{CAIE S1 2005 Q1 [5]}}
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