Question:
The mean and the variance of five observation are 4 and $5.20$, respectively. If three of the observations are 3,4 and 4 ; then then absolute value of the difference of the other two observations, is:
Correct Option: , 3
Solution:
mean $\overline{\mathrm{x}}=4, \sigma^{2}=5.2, \mathrm{n}=5, . \mathrm{x}_{1}=3 \mathrm{x}_{2}=4=\mathrm{x}_{3}$
$\sum \mathrm{x}_{\mathrm{i}}=20$
$x_{4}+x_{5}=9$ ...........(i)
$\frac{\sum x_{i}^{2}}{x}-(\bar{x})^{2}=\sigma^{2} \Rightarrow \sum x_{i}^{2}=106$
$x_{4}^{2}+x_{5}^{2}=65$ ..............(ii)
Using (i) and (ii) $\left(\mathrm{x}_{4}-\mathrm{x}_{5}\right)^{2}=49$
$\left|x_{4}-x_{5}\right|=7$