Question

Let $x_{1},x_{2}\ldots ,x_{n}$ be $n$ observations such that $\displaystyle \sum x_{i}^{2} = 300$ and $\displaystyle \sum x_{i} = 90.$ Then a possible value of $n$ among the following is

• A. $25$
• B. $18$
• C. $29$
• D. $22$

Answer 1

Answer: (C)

Variance $=\frac{\Sigma x_{i}^{2}}{n}-\left(\frac{\Sigma x_{i}}{n}\right)^{2}$
$\Rightarrow \frac{\Sigma x_{i}^{2}}{n} \geq\left(\frac{\Sigma x_{i}}{n}\right)^{2}$
$\Rightarrow n \geq \frac{\left(\Sigma x_{i}\right)^{2}}{\Sigma x_{i}^{2}} \Rightarrow n \geq \frac{8100}{300} \Rightarrow n \geq 27$