Question 4 - A-Level Maths

CAIE S2 2019 June — Question 4 5 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2019
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear combinations of normal random variables
Type	Fixed container with random contents
Difficulty	Standard +0.3 This is a straightforward application of linear combinations of normal distributions with fixed coefficients (10 large + 20 small + box). Students need to find the mean (10×60 + 20×30 + 8 = 1208) and variance (10×1.2 + 20×0.7 = 26), then standardize and use tables. It's slightly above average difficulty due to the multi-component setup, but requires only direct application of standard formulas without problem-solving insight.
Spec	5.04b Linear combinations: of normal distributions

4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .

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Question 4:

Answer	Marks	Guidance
Answer	Marks	Guidance
Total \(\sim N(1208, \ldots)\)	B1
\(\text{Var(total)} = 10 \times 1.2 + 20 \times 0.7 \ (+ 0) = 26\)	B1	May be implied by next line
\(\pm \frac{1200 - \text{"1208"}}{\sqrt{\text{"26"}}} \ (= -1.569)\)	M1	FT their mean and var of total mass, e.g. allow 1200 and 11.24 (from \(10 \times 1.2^2 + 20 \times 0.7^2\))
\(1 - \Phi(\text{"1.569"})\)	M1	Correct area consistent with their working
\(= 0.0583\) (3 sf)	A1
Total: 5

## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Total $\sim N(1208, \ldots)$ | B1 | |
| $\text{Var(total)} = 10 \times 1.2 + 20 \times 0.7 \ (+ 0) = 26$ | B1 | May be implied by next line |
| $\pm \frac{1200 - \text{"1208"}}{\sqrt{\text{"26"}}} \ (= -1.569)$ | M1 | **FT** their mean and var of total mass, e.g. allow 1200 and 11.24 (from $10 \times 1.2^2 + 20 \times 0.7^2$) |
| $1 - \Phi(\text{"1.569"})$ | M1 | Correct area consistent with their working |
| $= 0.0583$ (3 sf) | A1 | |
| **Total: 5** | | |

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Show LaTeX source

4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions $\mathrm { N } ( 60,1.2 )$ and $\mathrm { N } ( 30,0.7 )$ respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .\\

\hfill \mbox{\textit{CAIE S2 2019 Q4 [5]}}

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