Let $${x_1},{x_2},......\,{x_n}$$   be $$n$$ observations such that $$\sum {x_i^2 = 400\,\,{\text{and }}\sum {{x_i} = 80.} } $$      Then the possible value of $$n$$ among the following is

Solution :
We know that for positive real numbers $${x_1},{x_2},......\,{x_n}$$    A.M. of $${k^{th}}$$ power of $$x{'_i}s \geqslant {k^{th}}$$   the power of A.M. of $$x{'_i}s$$
$$\eqalign{ & \Rightarrow \,\,\frac{{\sum {x_1^2} }}{n} \geqslant {\left( {\frac{{\sum {{x_1}} }}{n}} \right)^2} \cr & \Rightarrow \,\,\frac{{400}}{n} \geqslant {\left( {\frac{{80}}{n}} \right)^2} \cr} $$
$$ \Rightarrow \,\,n \geqslant 16.$$   So only possible value for $$n$$ = 18