The mean and variance of seven observations are 8 and 16 respectively. If 5 of the observations are \(2,4,10,12,14\), then the square root of product of remaining two observations is

Let the unknown numbers be \(x\) and \(y\).
\(\begin{aligned}
& \text { Mean }=8 \\
& \Rightarrow \frac{2+4+10+12+14+x+y}{7}=8 \\
& \Rightarrow x+y=14...(i) \\
& \text { Variance }=16 \\
& \Rightarrow \frac{2^2+4^2+10^2+12^2+14^2+x^2+y^2}{7}
\end{aligned}\)
\(-(\text { mean })^2=16\)
\(\begin{aligned}
& \Rightarrow 460+x^2+y^2=7\left[16+(8)^2\right] \\
& \Rightarrow 460+x^2+y^2=560 \\
& \Rightarrow x^2+y^2=100
...(ii)\end{aligned}\)
Solving (i) and (ii), we get
\(\begin{array}{ll}
\quad & x=6, y=8 \text { or } x=8, y=6 \\
\therefore \quad & \text { Product }=48 \\
\therefore \quad & \sqrt{\text { Product }}=\sqrt{48}= 4 \sqrt{3}
\end{array}\)