Question:

Step 1: Determine the sum of the remaining observations. Let the two unknown observations be $a$ and $b$. The total number of observations is $n=7$. The given observations are $2, 4, 10, 12, 14$. The mean of the seven observations is 8. The sum of all observations is given by $n \times \text{mean}$. $$ 2 + 4 + 10 + 12 + 14 + a + b = 7 \times 8 $$ $$ 42 + a + b = 56 $$ $$ a + b = 56 - 42 $$ $$ a + b = 14 \quad (1) $$ Step 2: Determine the sum of the squares of the remaining observations. The variance of the seven observations is 16. The formula for variance is $Var(X) = E(X^2) - [E(X)]^2$. First, calculate the sum of the squares of all observations, which is $n \times E(X^2)$: $$ \sum X^2 = 2^2 + 4^2 + 10^2 + 12^2 + 14^2 + a^2 + b^2 $$ $$ \sum X^2 = 4 + 16 + 100 + 144 + 196 + a^2 + b^2 $$ $$ \sum X^2 = 460 + a^2 + b^2 $$ Thus, $E(X^2) = \frac{460 + a^2 + b^2}{7}$. The mean $E(X)$ is 8, so $[E(X)]^2 = 8^2 = 64$. Substitute these values into the variance formula: $$ 16 = \frac{460 + a^2 + b^2}{7} - 64 $$ $$ 16 + 64 = \frac{460 + a^2 + b^2}{7} $$ $$ 80 = \frac{460 + a^2 + b^2}{7} $$ $$ 80 \times 7 = 460 + a^2 + b^2 $$ $$ 560 = 460 + a^2 + b^2 $$ $$ a^2 + b^2 = 560 - 460 $$ $$ a^2 + b^2 = 100 \quad (2) $$ Step 3: Calculate the product of the remaining observations. We use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$. Substitute the values from equations (1) and (2) into this identity: $$ (14)^2 = 100 + 2ab $$ $$ 196 = 100 + 2ab $$ $$ 196 - 100 = 2ab $$ $$ 96 = 2ab $$ $$ ab = \frac{96}{2} $$ $$ ab = 48 $$ The product of the remaining two observations is 48.