OCR S3 2014 June — Question 4 7 marks

Exam BoardOCR
ModuleS3 (Statistics 3)
Year2014
SessionJune
Marks7
PaperDownload PDF ↗
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TopicLinear combinations of normal random variables
TypeComparison involving sums or multiples
DifficultyStandard +0.3 This is a straightforward application of linear combinations of normal distributions requiring students to set up 2B₁ + 2B₂ > 3C, find the mean and variance of the combined distribution using standard formulas, then calculate a single probability using normal tables. While it involves multiple random variables, the setup is mechanical and the calculation routine for S3 level.
Spec5.04b Linear combinations: of normal distributions
4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
Consider variable \(B_1 + B_2 - 3C\)M1 or use \(2\times10^2 + 3^2\times8^2\)
Mean \(= 10\) (or \(-10\))B1 Allow from \(2B - 3C\). Allow 1000–990
Variance \(= 776\)A1
\(\frac{0-10}{\sqrt{776}}\)M1 Allow reversed
\(= -0.359\)A1 Allow 0.359
\(\Phi(0.359)\)M1 Must be correct tail. Answer must be \(> 0.5\)
\(= 0.640\)A1 Allow 0.64
[7] 
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider variable $B_1 + B_2 - 3C$ | M1 | or use $2\times10^2 + 3^2\times8^2$ |
| Mean $= 10$ (or $-10$) | B1 | Allow from $2B - 3C$. Allow 1000–990 |
| Variance $= 776$ | A1 | |
| $\frac{0-10}{\sqrt{776}}$ | M1 | Allow reversed |
| $= -0.359$ | A1 | Allow 0.359 |
| $\Phi(0.359)$ | M1 | Must be correct tail. Answer must be $> 0.5$ |
| $= 0.640$ | A1 | Allow 0.64 |
| **[7]** | | |

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4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.

\hfill \mbox{\textit{OCR S3 2014 Q4 [7]}}
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