CAIE S1 2015 June — Question 3 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2015
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicApproximating Binomial to Normal Distribution
TypeSingle probability inequality
DifficultyModerate -0.8 This is a straightforward application of the normal approximation to the binomial distribution with clearly stated parameters (n=300, p=0.072). It requires only calculating mean and variance, applying continuity correction, and looking up a single z-value—a routine textbook exercise with no conceptual challenges or multi-step reasoning.
Spec2.04d Normal approximation to binomial
3 On a production line making cameras, the probability of a randomly chosen camera being substandard is 0.072 . A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\mu = 300 \times 0.072 = 21.6\), \(\sigma^2 = 20.0448\)B1 \(300 \times 0.072\) seen and \(300 \times 0.072 \times 0.928\) seen or implied (\(\sigma = 4.4771\), \(\sigma^2 = 20(.0)\)) oe
\(P(x < 18) = P\!\left(z < \frac{17.5 - 21.6}{\sqrt{20.0448}}\right)\)M1 \(\pm\)Standardising, their mean/var, with sq root
M1Cont corr 17.5 or 18.5
\(= P(z < -0.9157)\)M1 Correct area \(1 - \Phi\)
\(= 1 - 0.8201 = 0.180\)A1 [5] Answer wrt 0.180, nfww 
## Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mu = 300 \times 0.072 = 21.6$, $\sigma^2 = 20.0448$ | B1 | $300 \times 0.072$ seen and $300 \times 0.072 \times 0.928$ seen or implied ($\sigma = 4.4771$, $\sigma^2 = 20(.0)$) oe |
| $P(x < 18) = P\!\left(z < \frac{17.5 - 21.6}{\sqrt{20.0448}}\right)$ | M1 | $\pm$Standardising, their mean/var, with sq root |
| | M1 | Cont corr 17.5 or 18.5 |
| $= P(z < -0.9157)$ | M1 | Correct area $1 - \Phi$ |
| $= 1 - 0.8201 = 0.180$ | A1 **[5]** | Answer wrt 0.180, nfww |

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3 On a production line making cameras, the probability of a randomly chosen camera being substandard is 0.072 . A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.

\hfill \mbox{\textit{CAIE S1 2015 Q3 [5]}}
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