Question

The mean and variance of $7$ observations are $8$ and $16$, respectively. If five observations are $2,4,10,12,14,$ then the absolute difference of the remaining two observations is:

• A. $1$
• B. $4$
• C. $3$
• D. $2$

Answer 1

The correct option is D $2$

$Var\left(x\right)=\sum \frac{x_i^2}{n}-{\left(\overline{x}\right)}^2$
$\Rightarrow 16=\frac{x_1^2+x_2^2+x_4^2+x_5^2+x_6^2+x_7^2}{7}-64$
$\Rightarrow 80\times 7=x_1^2+x_2^2+x_3^2+\cdots +x_7^2$

Now, $x_6^2+x_7^2=560-(x_1^2+\cdots +x_5^2)$
$\Rightarrow x_6^2+x_7^2=560-(4+16+100+144+196)$
$\Rightarrow x_6^2+x_7^2=100\cdots \left(1\right)$

Now, $\frac{x_1+x_2+\cdots +x_7}{7}=8$
$\Rightarrow x_6+x_7=14\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$
${(x_6+x_7)}^2-2x_6{x}_7=100$
$\Rightarrow 2x_6{x}_7=96\Rightarrow x_6{x}_7=48\cdots \left(3\right)$

Now, $|x_6-x_7|=\sqrt{{(x_6+x_7)}^2-4x_6{x}_7}$
$=\sqrt{196-192}=2$