Question

The mean of five observations is $5$ and their variance is $9.20$. If three of the given five observations are $1,3$ and $8$, then a ratio of other two observations is :

• A. $10:3$
• B. $4:9$
• C. $5:8$
• D. $6:7$

Answer 1

The correct option is B $4:9$

$\frac{\sum x_i}{5}=5$
$\sum x_i=25$
Let other two observations are $a$ and $b$
$\therefore 1+3+8+a+b=25$
$a+b=13\cdots \left(1\right)$
variance ${\sigma }^2=\frac{\sum x_i^2}{n}-{\left(\frac{\sum x_i}{n}\right)}^2$
$\Rightarrow 9.20=\frac{1^2+3^2+8^2+a^2+b^2}{5}-5^2$
$\Rightarrow a^2+b^2=97$
$\Rightarrow (a+b{)}^2-2ab=97$
$\Rightarrow 169-2ab=97$
$\Rightarrow ab=36\cdots \left(2\right)$
From eqn $\left(1\right)$ and $\left(2\right)$, we get
$\frac{36}{a}+a=13\Rightarrow a^2-13a+36=0$
$\Rightarrow a=9,4\Rightarrow b=4,9$
Hence, ratio $=9:4$ or $4:9$