CAIE S2 2018 November — Question 3 5 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2018
SessionNovember
Marks5
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TopicLinear combinations of normal random variables
TypeMixed sum threshold probability
DifficultyStandard +0.3 This is a straightforward application of linear combinations of independent normal distributions requiring students to form 2S + 5F, find its mean (2×250 + 5×160 = 1300) and variance (4×100 + 25×81 = 2425), then calculate P(X > 1310) using standardization. While it involves multiple steps, each is routine and the question clearly signposts what to do, making it slightly easier than average.
Spec5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions
3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.
Question 3:
AnswerMarks Guidance
\(E(T) = 2 \times 250 + 5 \times 160 = 1300\)B1
\(\text{Var}(T) = 2 \times 10 + 5 \times 9 = 65\)B1
\(\frac{1310 - 1300}{\sqrt{65}} = 1.240\)M1 Standardise using their values (must come from a combination attempt). Ignore cc
\(1 - \phi(1.240)\)M1 Correct area consistent with their working
\(= 0.1075\)A1 Allow 0.107 to 0.108 (no errors seen) 
## Question 3:

| $E(T) = 2 \times 250 + 5 \times 160 = 1300$ | B1 | |
|---|---|---|
| $\text{Var}(T) = 2 \times 10 + 5 \times 9 = 65$ | B1 | |
| $\frac{1310 - 1300}{\sqrt{65}} = 1.240$ | M1 | Standardise using their values (must come from a combination attempt). Ignore cc |
| $1 - \phi(1.240)$ | M1 | Correct area consistent with their working |
| $= 0.1075$ | A1 | Allow 0.107 to 0.108 (no errors seen) |

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3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution $\mathrm { N } ( 250,10 )$. The mass, in grams, of one cup of flour has the independent distribution $\mathrm { N } ( 160,9 )$. Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.\\

\hfill \mbox{\textit{CAIE S2 2018 Q3 [5]}}
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