Question

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 $15.$ If $\mu$ is the average marks of girls and ${\sigma }^2$ is the variance of marks of $50$ candidates, then $\mu +{\sigma }^2$ is equal to ________

Answer 1

Answer: 25

${\sigma }_b^2=2$ (variance of boys)
$n_1=$ no. of boys
${\overline{x}}_b=12$
$n_2=$ no. of girls
${\sigma }_g^2=2$ ${\overline{x}}_g=\frac{50\times 15-12\times {\sigma }_b}{30}$
$=\frac{750-12\times 20}{30}=17=\mu$
variance of combined series
${\sigma }^2=\frac{n_1{\sigma }_b^2+n_2{\sigma }_g^2}{n_1+n_2}+\frac{n_1\cdot n_2}{{\left(n_1+n_2\right)}^2}{\left({\bar{x}}_b-{\bar{x}}_g\right)}^2$
${\sigma }^2=\frac{20\times 2+30\times 2}{20+30}+\frac{20\times 30}{(20+30{)}^2}(12-17{)}^2$
${\sigma }^2=8$
$\Rightarrow \mu +{\sigma }^2=17+8=25$