The mean of five observations is 5 and their variance is 9.20.

Question:

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1,3 and 8, then a ratio of other two observations is :

1. $4: 9$

2. $6: 7$

3. $5: 8$

4. $10: 3$

Correct Option: 1

Solution:

Let two observations are $\mathrm{x}_{1}$ \& $\mathrm{x}_{2}$

mean $=\frac{\sum \mathrm{x}_{\mathrm{i}}}{5}=5 \Rightarrow 1+3+8+\mathrm{x}_{1}+\mathrm{x}_{2}=25$

$\Rightarrow x_{1}+x_{2}=13$ ..................(1)

variance $\left(\sigma^{2}\right)=\frac{\sum x_{i}^{2}}{5}-25=9.20$

$\Rightarrow \sum x_{\mathrm{i}}^{2}=171$

$\Rightarrow x_{1}^{2}+x_{2}^{2}=97$    .................(2)

by (1) \& (2)

$\left(\mathrm{x}_{1}+\mathrm{x}_{2}\right)^{2}-2 \mathrm{x}_{1} \mathrm{x}_{2}=97$

or $x_{1} x_{2}=36$

$\therefore \quad x_{1}: x_{2}=4: 9$