Mathematics - Statistics Question with Solution | TestHub

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

$n_1=10,n_2=n,{\sigma }_1^2=2,{\sigma }_2^2=1$

${\overline{x}}_1=2,{\overline{x}}_2=3,{\sigma }^2=\frac{17}{9}$

$\frac{17}{9}=\frac{10\times 2+n}{n+10}+\frac{10n}{(n+10{)}^2}{\left(3-2\right)}^2$

$\Rightarrow \frac{17}{9}=\frac{(n+20)(n+10)+10n}{(n+10{)}^2}$

$\Rightarrow 17n^2+1700+340n=90n+9\left(n^2+30n+200\right)$

$\Rightarrow 8n^2-20n-100=0$

$2n^2-5n-25=0$

$\Rightarrow \left(2n+5\right)\left(n-5\right)=0\Rightarrow n=\frac{-5}{{\displaystyle \underset{\underset{\left(\mathrm{Rejected}\right)}{\downarrow }}{2}}},5$

Hence $n=5$