The mean and variance of 5 observations are 5 and 8 respectively....

Let the two observations be $a and$b$.</p><p>Given:</p><p>Mean$= 5$</p><p>$⇒ \dfrac{1 + 3 + 5 + a + b}{5} = 5$</p><p>$⇒ a + b = 16 \ldots \left(1\right)$</p><p>And</p><p>${\sigma}^{2} = \dfrac{∑ {x}_{i}^{2}}{5} - {\left(\dfrac{∑ {x}_{i}}{5}\right)}^{2}$</p><p>$⇒ 8 = \dfrac{{1}^{2} + {3}^{2} + {5}^{2} + {a}^{2} + {b}^{2}}{5} - 25$</p><p>$⇒ {a}^{2} + {b}^{2} = 130 \ldots \left(2\right)$</p><p>Solving \left(1\right) \text{ and } \left(2\right)$, we get

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

$⇒ b = 9$

So $a = 7$

Hence, ${a}^{3} + {b}^{3} = 343 + 729 = 1072$