The arithmetic mean of the observations $10,8,5, a, b$ is 6 and their variance is 6.8 , then $a b$ is equal to

Question

Aicharya · Accepted Answer

Given observations are $10,8,5, a, b$. Arithmetic Mean, $\begin{aligned} & \text { A.M. }=\frac{10+8+5+a+b}{5}=6 \\ & 23+a+b=30 \end{aligned}$

Here,

$\therefore \quad \sum\left(x_i-\bar{x}\right)^2=a^2+b^2+93-12(a+b)$ Variance, $\begin{aligned} & \frac{1}{n} \sum\left(x_i-\bar{x}\right)^2=6.8 \\ & \frac{a^2+b^2+93-12(a+b)}{5}=6.8 \\ & a^2+b^2+93-12 \times 7=6.8 \times 5 \\ & a^2+b^2+93-84=34 \quad[\because a+b=7] \end{aligned}$

$\begin{aligned} & {\left[\because(a+b)^2=a^2+b^2+2 a b\right]} \\ & (7)^2=25+2 a b \\ & \text { [using Eqs. (i) and (ii)] } \\ & 49=25+2 a b \\ & a b=12\end{aligned}$

The arithmetic mean of the observations $10,8,5, a, b$ is 6 and their variance is 6.8 , then $a b$ is equal to

Solution