Question

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 product of the remaining two observations is :

• A. 40
• B. 49
• C. 48
• D. 45

Answer 1

Answer: (C)

Let $7$ observations be $x_1,x_2,x_3,x_4,x_5,x_6,x_7$
$\bar{x}=8\Rightarrow {\sum }_{i=1}^7{x}_i=56$ .......(1)
Also ${\sigma }^2=16$
$\Rightarrow 16=\frac{1}{7}\left({\sum }_{i=1}^7{x}_i^2\right)-{\left(\bar{x}\right)}^2$
$\Rightarrow 16=\frac{1}{7}\left({\sum }_{i=1}^7{x}_i^2\right)-64$
$\Rightarrow \left({\sum }_{i=1}^7{x}_i^2\right)=560$
$x_1=2,x_2=4,x_3=10,x_4=12,x_5=14$
$\Rightarrow x_6+x_7=14$ (from (1))
& $x_6^2+x_7^2=100$ (from (2))
$\therefore x_6^2+x_7^2={\left(x_6+x_7\right)}^2-2x_6.x_7\Rightarrow x_6.x_7=48$