The mean and variance of 7 observations are 8 and 16 respectively....

Given,

Mean and variance of $7$ observation is given as $8 \text{ and } 16$ respectively,

So using the formula of mean we get, $\overset{¯}{x} = \dfrac{S}{7} ⇒ S = 8 \times 7 = 56$

Now if we remove $14$ sum will be $S = 56 - 14 = 42$ and new mean will be $a = \dfrac{42}{6} = 7$

Now using the formula of variance we get,

$\dfrac{\mathop{∑}_{i = 1}^{6}{x}_{i}^{2} + {14}^{2}}{7} - {\left(\overset{¯}{x}\right)}^{2} = 16$

$⇒ \dfrac{\mathop{∑}_{i = 1}^{6}{x}_{i}^{2} + {14}^{2}}{7} - {8}^{2} = 16$

$⇒ \dfrac{\mathop{∑}_{i = 1}^{6}{x}_{i}^{2} + {14}^{2}}{7} = 80$

$⇒ \mathop{∑}_{i = 1}^{6}{x}_{i}^{2} + {14}^{2} = 560$

$⇒ \mathop{∑}_{i = 1}^{6}{x}_{i} = 364$

Now, new variance $b = \dfrac{\mathop{∑}_{i = 1}^{6}{x}_{i}^{2}}{6} - {a}^{2} = \dfrac{364}{6} - {7}^{2} = \dfrac{35}{3}$

Hence, the value of $a + 3b - 5 = 7 + 35 - 5 = 37$