Question

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

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

Answer 1

The correct option is A

$18$

Explanation for the correct option :

We know that the arithmetic mean of $nth$ power $\left(\frac{\Sigma {x_i}^2}{n}\right)>$$nth$ power of arithmetic mean ${\left(\frac{\Sigma x_i}{n}\right)}^2$.

$$\begin{gathered} \Rightarrow \frac{400}{n}>{\left(\frac{80}{n}\right)}^2 \\ \Rightarrow \frac{400}{n}>\frac{6400}{n^2} \\ \Rightarrow n>16 \end{gathered}$$

In the given options, only $18>16$.

Hence, option A is correct.