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Q.

An online exam is attempted by 50 candidates out of 20 are boys. The average marks obtained by boys is 12 with variance 2. The variance of marks obtained by 30 girls is also 2 the average marks of all 50 candidates is 15. If $μ$ is the average marks of girls and $σ^{2}$ is the variance of marks 50 candidates then $μ + σ^{2}$ is equal to

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a

22

b

24

c

25

d

20

answer is D.

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Detailed Solution

Number of boys = 20
Number of girls = 30
$σ_{B}^{2} = 2, {\bar{X}}_{B} = 12 σ_{G}^{2} = 2$
${\bar{X}}_{G} = \frac{50 \times 15 - 20 \times 12}{30}$
$= \frac{510}{30} = 17 = μ$
Variance of 50 candidates
$σ^{2} = \frac{20 σ_{B}^{2} + 30 σ_{G}^{2}}{50} + \frac{20.30}{{(20 + 30)}^{2}} {({\bar{X}}_{B} - {\bar{X}}_{G})}^{2} = \frac{20 \times 2 + 30 \times 2}{50} = \frac{600}{2500} \times 25 = 8 μ + σ^{2} = 17 + 8 = 25$

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