CAIE S1 2017 March — Question 3 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2017
SessionMarch
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 continuity correction. Students need to identify n=160, p=0.1, check np≥5 and nq≥5, apply the continuity correction (P(X>17) = P(X≥18) → P(Y≥17.5)), then standardize and use tables. It's slightly easier than average because it's a single probability calculation with clear parameters and standard method, though the continuity correction adds a small layer of complexity.
Spec2.04d Normal approximation to binomial

3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.

Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(np = 160\times0.1\ (16)\), \(npq = 160\times0.1\times0.9\ (14.4)\)B1 Correct unsimplified \(np\) and \(npq\)
\(P(>17) = P\!\left(z > \dfrac{17.5-16}{\sqrt{14.4}}\right) = P(z > 0.3953)\)M1 Standardising, need \(\sqrt{\phantom{x}}\)
M116.5 or 17.5 seen in standardised eqn for continuity correction
\(= 1 - 0.6536\)M1 Correct area from their mean \((1-\Phi)\), final solution
\(= 0.346\)A1
# Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $np = 160\times0.1\ (16)$, $npq = 160\times0.1\times0.9\ (14.4)$ | B1 | Correct unsimplified $np$ and $npq$ |
| $P(>17) = P\!\left(z > \dfrac{17.5-16}{\sqrt{14.4}}\right) = P(z > 0.3953)$ | M1 | Standardising, need $\sqrt{\phantom{x}}$ |
| | M1 | 16.5 or 17.5 seen in standardised eqn for continuity correction |
| $= 1 - 0.6536$ | M1 | Correct area from their mean $(1-\Phi)$, final solution |
| $= 0.346$ | A1 | |

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3 It is found that $10 \%$ of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.\\

\hfill \mbox{\textit{CAIE S1 2017 Q3 [5]}}