Question

The mean and variance of 'n' observations $x_1,x_2,x_3,......x_n$ are 5 and 0, respectively. If ${\sum }_{i=1}^n{x}_i^2=400$ then the value of n is

• A. 80
• B. 25
• C. 20
• D. 16

Answer 1

The correct option is D 16

Given, mean$=5$

$\Rightarrow \frac{\sum _{i=1}^{n}xi}{n}=5$

$\Rightarrow \sum _{i=1}^{n}xi=5n$ …………$\left(1\right)$

and variance$=0$

$\Rightarrow \frac{\sum _{i=1}^n(xi-5{)}^2}{n}=0$

$\Rightarrow \sum _{i=1}^n(xi-5{)}^2=0$

$\Rightarrow \sum _{i=1}^n(x{i}^2+25-10xi)=0$

$\Rightarrow \sum _{i=1}^{n}x{i}^2+25\sum _{i=1}^{n}1-10\sum _{i-1}^{n}xi=0$

$\Rightarrow 400+25n-10\left(5n\right)=0$

$\Rightarrow 400+25n-50n=0$

$\Rightarrow 400-25n=0$

$\Rightarrow 25n=400$

$\Rightarrow n=\frac{400}{25}$

$\Rightarrow n=16$.