Given data for 7 observations

Mean $=8$, so the sum of all 7 observations is $S_7 = 7\times 8 = 56.$

Variance $=16$.

Using $\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2,$ we get $16 = \frac{\sum x_i^2}{7} - 8^2.$

So, $16 = \frac{\sum x_i^2}{7} - 64$ $\frac{\sum x_i^2}{7} = 80$ $\sum x_i^2 = 560.$

Omit the observation $14$

For the remaining 6 observations:

Sum becomes $S_6 = 56 - 14 = 42.$

Hence their mean is $a = \frac{42}{6} = 7.$

Find sum of squares for remaining 6 observations

$\sum x_i^2 = 560 - 14^2 = 560 - 196 = 364.$

Find variance $b$ of remaining 6 observations

Again, $b = \frac{364}{6} - 7^2.$

Compute: $\frac{364}{6} = \frac{182}{3}.$

Therefore, $b = \frac{182}{3} - 49 = \frac{182}{3} - \frac{147}{3} = \frac{35}{3}.$

Compute the required expression

$a+3b-5 = 7 + 3\left(\frac{35}{3}\right) - 5$ $= 7 + 35 - 5 = 37.$

Comparison with stored answer

Derived answer $=37$, which matches the stored correct answer.