CAIE S2 2017 June — Question 3 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2017
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicLinear combinations of normal random variables
TypeSingle sum threshold probability
DifficultyStandard +0.3 This is a straightforward application of the sum of independent normal random variables. Students need to recognize that the sum of 10 independent N(7.0, 0.46²) variables is N(70, 10×0.46²), then calculate a single probability P(X > 71) using standardization. It's slightly above average difficulty due to being Further Maths content and requiring understanding of variance scaling, but the calculation itself is routine.
Spec5.04b Linear combinations: of normal distributions
3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(10 \times 0.46^2 (= 2.116)\) or \(\frac{0.46}{\sqrt{10}}\)B1 SOI
Total mass of ore \(\sim N(70, 2.116)\) or \(\sim N\!\left(7, \left(\frac{0.46}{\sqrt{10}}\right)^{\!2}\right)\)B1
\(\pm\frac{71 - \text{"70"}}{\sqrt{\text{"2.116"}}}\) or \(\pm\frac{7.1 - \text{"7.0"}}{0.46/\sqrt{10}}\ (= 0.687)\)M1 Correct, using their sd or \(\sqrt{\text{(their var)}}\); e.g. allow \(\frac{71-\text{"70"}}{4.6}\) for M1
\(1 - \phi(\text{"0.687"})\)M1 For correct area consistent with their working
\(= 0.246\) (3 sf)A1 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $10 \times 0.46^2 (= 2.116)$ or $\frac{0.46}{\sqrt{10}}$ | B1 | SOI |
| Total mass of ore $\sim N(70, 2.116)$ or $\sim N\!\left(7, \left(\frac{0.46}{\sqrt{10}}\right)^{\!2}\right)$ | B1 | |
| $\pm\frac{71 - \text{"70"}}{\sqrt{\text{"2.116"}}}$ or $\pm\frac{7.1 - \text{"7.0"}}{0.46/\sqrt{10}}\ (= 0.687)$ | M1 | Correct, using their sd or $\sqrt{\text{(their var)}}$; e.g. allow $\frac{71-\text{"70"}}{4.6}$ for M1 |
| $1 - \phi(\text{"0.687"})$ | M1 | For correct area consistent with their working |
| $= 0.246$ (3 sf) | A1 | |

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3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.\\

\hfill \mbox{\textit{CAIE S2 2017 Q3 [5]}}
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