CAIE S2 2014 November — Question 1 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2014
SessionNovember
Marks5
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TopicLinear combinations of normal random variables
TypeDirect comparison with scalar multiple (different variables)
DifficultyChallenging +1.2 This question requires understanding that 'at least twice' translates to A ≥ 2B, reformulating as A - 2B ≥ 0, then finding the distribution of this linear combination (mean = 175 - 2(105) = -35, variance = 60² + 4(28²) = 6736). While the mechanics are standard S2 content, the conceptual leap from the word problem to the correct linear combination and the factor of 4 in the variance calculation elevate this above routine exercises. It's harder than average but not exceptionally difficult.
Spec2.04e Normal distribution: as model N(mu, sigma^2)5.04b Linear combinations: of normal distributions
1 The masses, in grams, of potatoes of types \(A\) and \(B\) have the distributions \(\mathrm { N } \left( 175,60 ^ { 2 } \right)\) and \(\mathrm { N } \left( 105,28 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen potato of type \(A\) has a mass that is at least twice the mass of a randomly chosen potato of type \(B\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(N(-35, 60^2 + 4 \times 28^2)\)B1 for \(\pm(175 - 2 \times 105)\) or \(\pm 35\)
\(N(35, 60^2 + 4 \times 28^2)\)B1 for \(60^2 + 4 \times 28^2\) or 6736
\(\frac{0-(-35)}{\sqrt{6736}}\) \((= 0.426)\)M1 For standardising with their mean and variance. Allow without \(\sqrt{}\); for use of tables and finding area consistent with working
\(1 - \Phi(``0.426")\)M1
\(= 0.335\) (3 sf)A1 Total: 5 
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $N(-35, 60^2 + 4 \times 28^2)$ | B1 | for $\pm(175 - 2 \times 105)$ or $\pm 35$ |
| $N(35, 60^2 + 4 \times 28^2)$ | B1 | for $60^2 + 4 \times 28^2$ or 6736 |
| $\frac{0-(-35)}{\sqrt{6736}}$ $(= 0.426)$ | M1 | For standardising with their mean and variance. Allow without $\sqrt{}$; for use of tables and finding area consistent with working |
| $1 - \Phi(``0.426")$ | M1 | |
| $= 0.335$ (3 sf) | A1 | **Total: 5** |

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1 The masses, in grams, of potatoes of types $A$ and $B$ have the distributions $\mathrm { N } \left( 175,60 ^ { 2 } \right)$ and $\mathrm { N } \left( 105,28 ^ { 2 } \right)$ respectively. Find the probability that a randomly chosen potato of type $A$ has a mass that is at least twice the mass of a randomly chosen potato of type $B$.

\hfill \mbox{\textit{CAIE S2 2014 Q1 [5]}}
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