Mathematics - Statistics Question with Solution | TestHub

Let the two observations be $a$ and $b$.

Given:

Mean$=5$

$\Rightarrow \frac{1+3+5+a+b}{5}=5$

$\Rightarrow a+b=16 \dots \left(1\right)$

And

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

$\Rightarrow 8=\frac{1^2+3^2+5^2+a^2+b^2}{5}-25$

$\Rightarrow a^2+b^2=130 \dots \left(2\right)$

Solving $\left(1\right) \& \left(2\right)$, we get

${\left(16-b\right)}^2+b^2=130$

$\Rightarrow b=9$

So $a=7$

Hence, $a^3+b^3=343+729=1072$