Let x_1,x_2,,x_n be n observations such that x_i^2=400 and x_i=80.

Question

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

Accepted Answer

By the power-mean (QM $\ge$ AM) inequality: $\dfrac{\sum x_i^2}{n} \ge \left(\dfrac{\sum x_i}{n}\right)^2$

$$\frac{400}{n} \ge \left(\frac{80}{n}\right)^2 \implies n \ge 16$$

Among the options, only $n=18$ satisfies this.

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