CAIE S2 2024 November — Question 2 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2024
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear combinations of normal random variables
TypeSum versus sum comparison
DifficultyStandard +0.3 This is a standard linear combinations of normal random variables question requiring students to form the distribution of 3S - L, calculate its mean and variance using standard formulas, then find a single probability using normal tables. It's slightly easier than average because it's a direct application of well-practiced techniques with no conceptual surprises or multi-step reasoning.
Spec2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions
2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) 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. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\(E(S_1 + S_2 + S_3 - L) = 16 \times 3 - 51 \quad [= -3]\)B1 Oe, using \(L - (S_1 + S_2 + S_3)\)
\(\text{Var}(S_1 + S_2 + S_3 - L) = 3 \times 0.4 + 0.9 \quad [= 2.1]\)M1
\(\dfrac{0-(-3)}{\sqrt{2.1}} \quad [= 2.070]\)M1 For standardising with their values
\(1 - \Phi(\text{`}2.070\text{'})\)M1 For area consistent with their values
\(= 0.0192\) (3 sf)A1
5 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(S_1 + S_2 + S_3 - L) = 16 \times 3 - 51 \quad [= -3]$ | **B1** | Oe, using $L - (S_1 + S_2 + S_3)$ |
| $\text{Var}(S_1 + S_2 + S_3 - L) = 3 \times 0.4 + 0.9 \quad [= 2.1]$ | **M1** | |
| $\dfrac{0-(-3)}{\sqrt{2.1}} \quad [= 2.070]$ | **M1** | For standardising with their values |
| $1 - \Phi(\text{`}2.070\text{'})$ | **M1** | For area consistent with their values |
| $= 0.0192$ (3 sf) | **A1** | |
| | **5** | |
2 The masses, in kilograms, of small and large bags of wheat have the independent distributions $\mathrm { N } ( 16.0,0.4 )$ and $\mathrm { N } ( 51.0,0.9 )$ 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.\\

\includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}

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