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Step-by-Step Solution
Step 1: Express the Sum of the Observations
The mean of the 8 observations is given as 10. Let the remaining two observations be $x$ and $y$. The sum of the 8 observations is therefore $8 \times 10 = 80.$
We already have six observations: 5, 7, 10, 12, 14, 15. Their sum is $5 + 7 + 10 + 12 + 14 + 15 = 63.$
Hence, $x + y = 80 - 63 = 17 \quad ...(1)
Step 2: Express the Variance and Form an Equation
The variance of the 8 observations is 13.5. Variance is given by:
$\text{Var} = \frac{\sum \text{(each observation)}^2}{8} - (\text{mean})^2 = 13.5.$
The sum of squares of the 6 given observations is:
$25 + 49 + 100 + 144 + 196 + 225 = 739.$
(These are the squares of 5, 7, 10, 12, 14, 15 respectively.)
Hence:
$
\frac{(739 + x^2 + y^2)}{8} - (10)^2 = 13.5.
$
Since $(10)^2 = 100,$ rearranging gives:
$
\frac{739 + x^2 + y^2}{8} = 113.5 \quad \Rightarrow \quad 739 + x^2 + y^2 = 908.
$
Thus:
$
x^2 + y^2 = 908 - 739 = 169 \quad ...(2)
$
Step 3: Solve the System of Equations
From (1) we have $x + y = 17,$ and from (2) we have $x^2 + y^2 = 169.$
Using $(x + y)^2 = x^2 + 2xy + y^2,$ we get:
$
(17)^2 = x^2 + y^2 + 2xy.
$
Substituting $x^2 + y^2 = 169,$ we get:
$
289 = 169 + 2xy \quad \Rightarrow \quad 2xy = 120 \quad \Rightarrow \quad xy = 60.
$
The pair of solutions satisfying $x + y = 17$ and $xy = 60$ is $(12, 5)$ or $(5, 12).$
Step 4: Find the Absolute Difference
The absolute difference is $\left| x - y \right| = \left|12 - 5\right| = 7.$
Hence, the absolute difference of the remaining two observations is 7.
