CAIE S2 2014 June — Question 1 4 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2014
SessionJune
Marks4
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Mark schemeDownload PDF ↗
TopicLinear combinations of normal random variables
TypeSingle sum threshold probability
DifficultyModerate -0.3 This is a straightforward application of the sum of independent normal random variables. Students need to recognize that the sum of 8 normal distributions is itself normal, calculate the mean (8×60.4=483.2) and standard deviation (√8×8.2≈23.2), then find P(X<436) using standardization. While it requires understanding of this key concept, the execution is mechanical with no problem-solving insight needed, making it slightly easier than average.
Spec2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions
1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(N(483.2, 537.92)\) or \(N(483.2, 23.2^2)\)B1 or \(\frac{8.2}{\sqrt{8}}\) or \(\frac{8.2^2}{8}\) seen or implied
\(\frac{436-483.2}{\sqrt{537.92}}\) or \(\frac{436-483.2}{23.2}\) \((= -2.035)\)M1 or \(\frac{\frac{436}{8}-60.4}{\frac{8.2}{\sqrt{8}}}\) standardising (no mixed methods)
\(\Phi("-2.035") = 1 - \Phi("2.035")\)M1 Correct area consistent with their working
\(= 0.021\) or \(2.1\%\)A1 [4] 
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $N(483.2, 537.92)$ or $N(483.2, 23.2^2)$ | B1 | or $\frac{8.2}{\sqrt{8}}$ or $\frac{8.2^2}{8}$ seen or implied |
| $\frac{436-483.2}{\sqrt{537.92}}$ or $\frac{436-483.2}{23.2}$ $(= -2.035)$ | M1 | or $\frac{\frac{436}{8}-60.4}{\frac{8.2}{\sqrt{8}}}$ standardising (no mixed methods) |
| $\Phi("-2.035") = 1 - \Phi("2.035")$ | M1 | Correct area consistent with their working |
| $= 0.021$ or $2.1\%$ | A1 [4] | |

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1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.

\hfill \mbox{\textit{CAIE S2 2014 Q1 [4]}}
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