CAIE S1 2013 November — Question 2 5 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2013
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNormal Distribution
TypeDirect binomial from normal probability
DifficultyStandard +0.3 This is a straightforward two-step problem: first find P(13.6 < X < 14.8) using normal distribution tables with standardization, then apply binomial probability formula for exactly 8 successes in 10 trials. Both steps are routine S1 techniques with no conceptual challenges, making it slightly easier than average.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .
AnswerMarks Guidance
\(P(13.6 < X < 14.8) = P\left(\frac{13.6 - 14}{0.52} < z < \frac{14.8 - 14}{0.52}\right) = P(-0.7692 < z < 1.538) = \Phi(1.538) - [1 - \Phi(0.7692)] = 0.9380 - [1 - 0.7791] = 0.7171\)M1 Standardising 1 expression, no cc, no sq rt, no sq, ±, mean on num.
M1\(\Phi1 + \Phi2 - 1\) (indep) oe or \((\Phi2 - \Phi1\) if cc used)
A1Correct probability rounding to 0.72 here
\(P(8) = (0.7171)^7(0.2829)^3 {}_10C_8 = 0.252\)M1 Binomial expression \(10C8 p^8q^2\), \(\Sigma p + q = 1\), any p
A1 5Correct answer (rounding to 0.252) 
$P(13.6 < X < 14.8) = P\left(\frac{13.6 - 14}{0.52} < z < \frac{14.8 - 14}{0.52}\right) = P(-0.7692 < z < 1.538) = \Phi(1.538) - [1 - \Phi(0.7692)] = 0.9380 - [1 - 0.7791] = 0.7171$ | M1 | Standardising 1 expression, no cc, no sq rt, no sq, ±, mean on num.

| M1 | $\Phi1 + \Phi2 - 1$ (indep) oe or $(\Phi2 - \Phi1$ if cc used)

| A1 | Correct probability rounding to 0.72 here

$P(8) = (0.7171)^7(0.2829)^3 {}_10C_8 = 0.252$ | M1 | Binomial expression $10C8 p^8q^2$, $\Sigma p + q = 1$, any p

| A1 5 | Correct answer (rounding to 0.252)
2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

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