CAIE S2 2016 November — Question 3 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2016
SessionNovember
Marks5
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TopicLinear combinations of normal random variables
TypeSingle normal population sample mean
DifficultyStandard +0.3 This is a straightforward application of sampling distribution theory with normal variables. Students must sum three independent normals (giving mean 178.0, SD 6.5), then apply the Central Limit Theorem for n=15 to find the distribution of the sample mean, and finally calculate a single probability using standardization. While it requires multiple steps, each is routine and the question clearly signposts the approach—slightly above average due to the multi-stage setup but well within standard S2 material.
Spec5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem
3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.
Question 3:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(N(178, \ldots)\)B1 stated or implied
\(\text{Var} = 3.2^2 + 4.1^2 + 3.8^2\) or \(41.49\)B1 or sd \(= 6.44\) stated or implied
\(\frac{175 - 178}{\sqrt{41.49} \div \sqrt{15}} \quad (= {-1.804})\)M1 need \(\sqrt{15}\) but allow var/sd mix for "41.49"; allow cc for method marks
\(\Phi({-1.804}) = 1 - \Phi(1.804)\)M1 independent M1 for area/prob consistent with working
\(= 0.0356\) (3 sf)A1 [5] 
## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $N(178, \ldots)$ | B1 | stated or implied |
| $\text{Var} = 3.2^2 + 4.1^2 + 3.8^2$ or $41.49$ | B1 | or sd $= 6.44$ stated or implied |
| $\frac{175 - 178}{\sqrt{41.49} \div \sqrt{15}} \quad (= {-1.804})$ | M1 | need $\sqrt{15}$ but allow var/sd mix for "41.49"; allow cc for method marks |
| $\Phi({-1.804}) = 1 - \Phi(1.804)$ | M1 | independent M1 for area/prob consistent with working |
| $= 0.0356$ (3 sf) | A1 | [5] |

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3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.

\hfill \mbox{\textit{CAIE S2 2016 Q3 [5]}}
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