Question

The mean of $6$ distinct observations is $6.5$ and their variance is $\mathrm{10.25.}$ If $4$ out of $6$ observations are $2,4,5$ and $7,$ then the remaining two observations are

• A. $1,20$
• B. $10,11$
• C. $3,18$
• D. $8,13$

Answer 1

The correct option is B $10,11$

Let the observations be $2,4,5,7,x$ and $y$
$\overline{x}=\frac{18+x+y}{6}=6.5\Rightarrow x+y=21\dots \left(1\right)$
and
${\sigma }^2=\frac{2^2+4^2+5^2+7^2+x^2+y^2}{6}-(6.5{)}^2=10.25$
$\Rightarrow x^2+y^2=221\dots \left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we get
$(x,y)=(10,11)$ or $(11,10)$