Mathematics - Statistics Question with Solution | TestHub

Mean $\overline{x}=\frac{\sum x_i}{5}=\frac{24}{5}\Rightarrow \sum x_i=24 ...\left(i\right)$

Variance ${\sigma }^2=\frac{\sum x_i^2}{5}-{\left(\frac{24}{5}\right)}^2=\frac{194}{25}$

$\Rightarrow \sum {\mathrm{x}}_{\mathrm{i}}^2=154 ...\left(ii\right)$

Also given mean$\frac{x_1+x_2+x_3+x_4}{4}=\frac{7}{2}\Rightarrow x_1+x_2+x_3+x_4=14$

$\Rightarrow x_5=10$ (from $\left(i\right)$)

and variance ${\sigma }^2=\frac{x_1^2+x_2^2+x_3^2+x_4^2}{4}-\frac{49}{4}=a$

$\Rightarrow x_1^2+x_2^2+x_3^2+x_4^2=4a+49$

$\Rightarrow x_5^2=154-4a-49$ (from $\left(ii\right)$)

$\Rightarrow 100=105-4a\Rightarrow 4a=5$

Hence $4a+x_5=15$