Question 3 - A-Level Maths

CAIE S2 2024 June — Question 3 5 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2024
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear combinations of normal random variables
Type	Sum versus sum comparison
Difficulty	Standard +0.8 This question requires understanding that sums of independent normal variables are normally distributed, forming the difference of two sums (5L - 10S), calculating means and variances correctly (including the multiplication by n and n² respectively), then standardizing and using tables. While systematic, it involves multiple conceptual steps beyond routine single-variable problems, making it moderately challenging for A-level Further Maths Statistics.
Spec	5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions

3 The masses in kilograms of large and small bags of cement have the independent distributions \(\mathrm { N } ( 50,2.4 )\) and \(\mathrm { N } ( 26,1.8 )\) respectively. Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-04_2714_34_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-05_2724_35_136_20}

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Question 3:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\text{Diff} \sim N(5\times50 - 10\times26,\ 5\times2.4 + 10\times1.8)\) \([= N(-10, 30)]\)	B1	For N and mean \(= \pm(5\times50 - 10\times26)\); SOI
B1	For var \(= 5\times2.4 + 10\times1.8\); SOI
\(\frac{0-(-10)}{\sqrt{30}}\) \([= 1.826]\)	M1	Standardising with their values
\(1 - \Phi(1.826)\)	M1	For area consistent with their values
\(= 0.0339\) or \(0.034[0]\) (3sf)	A1
Total: 5

## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Diff} \sim N(5\times50 - 10\times26,\ 5\times2.4 + 10\times1.8)$ $[= N(-10, 30)]$ | B1 | For N and mean $= \pm(5\times50 - 10\times26)$; SOI |
| | B1 | For var $= 5\times2.4 + 10\times1.8$; SOI |
| $\frac{0-(-10)}{\sqrt{30}}$ $[= 1.826]$ | M1 | Standardising with *their* values |
| $1 - \Phi(1.826)$ | M1 | For area consistent with *their* values |
| $= 0.0339$ or $0.034[0]$ (3sf) | A1 | |
| **Total: 5** | | |

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3 The masses in kilograms of large and small bags of cement have the independent distributions $\mathrm { N } ( 50,2.4 )$ and $\mathrm { N } ( 26,1.8 )$ respectively.

Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement.\\

\includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-04_2714_34_143_2012}\\
\includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-05_2724_35_136_20}

\hfill \mbox{\textit{CAIE S2 2024 Q3 [5]}}

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Q1 4 Q2 4 Q3 5 Q4 5 Q5 10 Q6 9 Q7 13