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Step 1: Express the mean in terms of unknowns x and y
Since there are 7 observations in total, and five of them are given (2, 4, 10, 12, 14), let the remaining two be x and y. The mean is given as 8. Hence,
$$
\bar{x} = \frac{2 + 4 + 10 + 12 + 14 + x + y}{7} = 8.
$$
Multiplying both sides by 7,
$$
2 + 4 + 10 + 12 + 14 + x + y = 56.
$$
Simplifying,
$$
x + y = 56 - (2 + 4 + 10 + 12 + 14) = 14.
$$
Step 2: Express the variance in terms of x and y
The variance ($\sigma^2$) is given as 16. Recall the formula for variance of n observations:
$$
\sigma^2
= \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2.
$$
Here, $n = 7$. We know $\sum x_i = 2 + 4 + 10 + 12 + 14 + x + y = 56$. Therefore,
$$
\frac{\sum x_i^2}{7} - \left(\frac{56}{7}\right)^2 = 16.
$$
We need to compute $\sum x_i^2$ in terms of x and y:
$$
\sum x_i^2 = 2^2 + 4^2 + 10^2 + 12^2 + 14^2 + x^2 + y^2.
$$
That is
$$
4 + 16 + 100 + 144 + 196 + x^2 + y^2 = 460 + x^2 + y^2.
$$
So we substitute in the variance expression:
$$
\frac{460 + x^2 + y^2}{7} - (8)^2 = 16.
$$
Since $(8)^2 = 64$, we get
$$
\frac{460 + x^2 + y^2}{7} - 64 = 16.
$$
Rewriting:
$$
\frac{460 + x^2 + y^2}{7} = 16 + 64 = 80.
$$
Hence,
$$
460 + x^2 + y^2 = 7 \times 80 = 560.
$$
Thus,
$$
x^2 + y^2 = 560 - 460 = 100.
$$
Step 3: Use the sum and sum of squares to find |x - y|
From Step 1, we have $x + y = 14$. From Step 2, we have $x^2 + y^2 = 100$. To find $|x - y|$, recall the identity:
$$
(x - y)^2 = (x + y)^2 - 4xy.
$$
But first we need $xy$. Notice:
$$
(x + y)^2 = x^2 + y^2 + 2xy.
$$
Substitute $x + y = 14$ and $x^2 + y^2 = 100$:
$$
14^2 = 100 + 2xy.
$$
Hence,
$$
196 = 100 + 2xy \quad \Rightarrow \quad 2xy = 96 \quad \Rightarrow \quad xy = 48.
$$
Now,
$$
(x - y)^2 = (x + y)^2 - 4xy = 14^2 - 4 \times 48 = 196 - 192 = 4.
$$
Thus,
$$
|x - y| = \sqrt{4} = 2.
$$
Final Answer:
The absolute difference of the remaining two observations is 2.
