JEE Main 2021 — Statistics Question with Solution

We have, number of terms in the first group $n_1=100$ with their mean ${\overline{X}}_1=15$ and number of terms in whole group is $n_1+n_2=250$ and their mean $\overline{X}=15.6.$

Variance of first group$V_1\left(x\right)=9$ and whole group is $Var\left(x\right)=13.44.$

Now, ${\sigma }^2=\frac{n_1{\sigma }_1^2+n_2{\sigma }_2^2}{n_1+n_2}+\frac{n_1{n}_2}{{\left(n_1+n_2\right)}^2}{\left({\overline{x}}_1-{\overline{x}}_2\right)}^2$

and $n_2=150,{\overline{x}}_2=16,V_2\left(x\right)={\sigma }_2$

Therefore, $13.44=\frac{100\times 9+150\times {\sigma }_2^2}{250}+\frac{100\times 150}{{\left(250\right)}^2}\times 1$

$\Rightarrow {\sigma }_2^2=16$

$\Rightarrow {\sigma }_2=4$