An online exam is attempted by 50 candidates out of which 20 are...

An online exam is attempted by

50

candidates out of which

20

are boys. The average marks obtained by boys is

12

with a variance

2 .

The variance of marks obtained by

30

girls is also

2 .

The average marks of all

50

candidates is

15 .

\mu

is the average marks of girls and

{\sigma}^{2}

is the variance of marks of

50

candidates, then

\mu + {\sigma}^{2}

is equal to

Correct answer

Step-by-step solution

$20$ Boys, $30$ Girls
${\sigma}_{B}^{2} = 2, \overset{¯}{X}B = 12{\sigma}_{G}^{2} = 2$

$\overset{¯}{X}G = \dfrac{50 \times 15 - 20 \times 12}{30} = \dfrac{510}{30} = 17 = \mu$
Variance of $50$ condidates
${\sigma}^{2} = \dfrac{20{\sigma}_{B}^{2} + 30{\sigma}_{G}^{2}}{20 + 30} + \dfrac{20 ⋅ 30}{( 20 + 30{)}^{2}}{\left(\overset{¯}{X}B - \overset{¯}{X}G\right)}^{2}$

$= \dfrac{20 \times 2 + 30 \times 2}{50} + \dfrac{600}{2500} ( 25 ) = 8$

$\mu + {\sigma}^{2} = 17 + 8 = 25$

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