CAIE S2 2024 November — Question 2 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2024
SessionNovember
Marks5
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TopicLinear combinations of normal random variables
TypeSum versus sum comparison
DifficultyStandard +0.3 This is a standard linear combinations of normal random variables problem requiring students to form 3X - Y, find its mean and variance using independence, then calculate a single probability. While it involves multiple steps, it follows a routine template taught explicitly in S2 with no novel insight required, making it slightly easier than average.
Spec5.04b Linear combinations: of normal distributions
The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\text{N}(16.0, 0.4)\) and \(\text{N}(51.0, 0.9)\) respectively. Find the probability that the total mass of \(3\) randomly chosen small bags is greater than the mass of one randomly chosen large bag. [5]
Question 2:
AnswerMarks Guidance
2E(S S + S – L) = 16 x 3 – 51 [= −3 ]
1 + 2 3B1 Oe, using L – (S S + S ).
1 + 2 3
Var(S S + S – L) = 3 × 0.4 + 0.9 [= 2.1]
AnswerMarks
1 + 2 3M1
0−(−3)
[= 2.070]
AnswerMarks Guidance
'2.1'M1 For standardising with their values.
1 − Φ(‘2.070’)M1 For area consistent with their values.
= 0.0192 (3 sf)A1
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 2:
2 | E(S S + S – L) = 16 x 3 – 51 [= −3 ]
1 + 2 3 | B1 | Oe, using L – (S S + S ).
1 + 2 3
Var(S S + S – L) = 3 × 0.4 + 0.9 [= 2.1]
1 + 2 3 | M1
0−(−3)
[= 2.070]
'2.1' | M1 | For standardising with their values.
1 − Φ(‘2.070’) | M1 | For area consistent with their values.
= 0.0192 (3 sf) | A1
5
Question | Answer | Marks | Guidance
The masses, in kilograms, of small and large bags of wheat have the independent distributions $\text{N}(16.0, 0.4)$ and $\text{N}(51.0, 0.9)$ respectively.

Find the probability that the total mass of $3$ randomly chosen small bags is greater than the mass of one randomly chosen large bag. [5]

\hfill \mbox{\textit{CAIE S2 2024 Q2 [5]}}
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