CAIE S2 2016 March — Question 4 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2016
SessionMarch
Marks5
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Mark schemeDownload PDF ↗
TopicLinear combinations of normal random variables
TypeDirect comparison with scalar multiple (different variables)
DifficultyStandard +0.8 This requires forming the linear combination X - 2Y, finding its distribution (mean and variance using independence), then calculating a single probability. It's a standard S2 application requiring careful algebraic manipulation of the variance term (4×0.1²) but no novel insight beyond the core technique.
Spec2.04e Normal distribution: as model N(mu, sigma^2)5.04a Linear combinations: E(aX+bY), Var(aX+bY)
4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)\). Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.
Question 4:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(X - 2Y \sim N(0.1,\ 0.2^2 + 4 \times 0.1^2)\) soi \((= N(0.1, 0.08))\)B1 B1 B1 for \(\pm 0.1\); B1 for \(0.2^2 + 4 \times 0.1^2\)
\(\frac{0 - 0.1}{\sqrt{0.08}}\) \((= -0.354)\)M1 For standardising. Allow without \(\sqrt{\phantom{x}}\) sign
\(\Phi(\text{"}{-0.354}\text{"}) = 1 - \Phi(\text{"0.354"}) = 0.362\) (3 sf)M1 A1 [5] For correct area consistent with their working 
## Question 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $X - 2Y \sim N(0.1,\ 0.2^2 + 4 \times 0.1^2)$ soi $(= N(0.1, 0.08))$ | **B1 B1** | **B1** for $\pm 0.1$; **B1** for $0.2^2 + 4 \times 0.1^2$ |
| $\frac{0 - 0.1}{\sqrt{0.08}}$ $(= -0.354)$ | **M1** | For standardising. Allow without $\sqrt{\phantom{x}}$ sign |
| $\Phi(\text{"}{-0.354}\text{"}) = 1 - \Phi(\text{"0.354"}) = 0.362$ (3 sf) | **M1 A1** [5] | For correct area consistent with their working |

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4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by $X$ and $Y$ respectively, where $X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)$ and $Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)$. Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.

\hfill \mbox{\textit{CAIE S2 2016 Q4 [5]}}
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