Q111 / 160 · TS EAMCET 2014 · Mathematics
If $x_1, x_2, \ldots, x_n$ are $n$ observations such that $\sum_{i=1}^n x_1^2=400$ and $\sum_{i=1}^n x_1=80$, then the least value of $n$ is
- A$18$
- B$12$
- C$15$
- Correct answerD$16$
Solution
Given, $\quad \sum_{i=1}^n x_i^2=400$ and
$$
\sum_{i=1}^n x_i=80
$$
We know, $\frac{\Sigma\left(x_i\right)^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2 \geq 0$
$$
\begin{aligned}
\Rightarrow & & \frac{400}{n}-\left(\frac{80}{n}\right)^2 & \geq 0 \\
\Rightarrow & & 400 n & \geq 6400 \\
\Rightarrow & & n & \geq 16 \\
\therefore & & n & =16
\end{aligned}
$$
Hence, the correct answer is option: 'D'.