Question

IF $x_1,x_2,....x_n$ are $n$ observations such that ${\sum }_{i=1}^n{x}_i^2=400$ and ${\sum }_{i=1}^n{x}_i=80$ then the least value of $n$ is

• A. $12$
• B. $15$
• C. $16$
• D. $18$

Answer 1

The correct option is C $16$

$\frac{x_1+x_2+......+x_n}{n}\le \left(\frac{x_1^2+x_2^p+....+x_n^2}{n}\right)$

$\frac{80}{n}\le {\left(\frac{400}{n}\right)}^{1/2}$

$\frac{6400}{n^2}\le \frac{400}{n}$

$\frac{64}{4}\le \frac{n^2}{n}$

$n\ge 16$

least value of $n=16$