1. Key Concepts and Formulas

This problem requires a strong understanding of the fundamental statistical measures: Mean and Variance. For a set of $n$ observations $x_1, x_2, \ldots, x_n$:

• Mean ($\bar{x}$): The average of all observations. It is calculated as the sum of all observations divided by the total number of observations. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Variance ($\sigma^2$): A measure of the spread of data points around the mean. The most practical formula for calculations, especially when dealing with unknown observations and sums of squares, is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ This formula allows us to directly relate the variance to the sum of squares of the observations and the square of the mean, simplifying calculations significantly.
• Algebraic Identities: Essential for solving systems of equations involving sums and sums of squares of variables. Key identities include $(a+b)^2 = a^2+b^2+2ab$ and $(a-b)^2 = (a+b)^2 - 4ab$.

---

2. Step-by-Step Solution

Let the two unknown observations be $x$ and $y$. We are given:

• Total number of observations, $n = 8$.
• Mean of these 8 observations, $\bar{x} = 10$.
• Variance of these 8 observations, $\sigma^2 = 13.5$.
• Six of the observations are: $5, 7, 10, 12, 14, 15$. Our goal is to find the absolute difference, $|x-y|$.

Step 1: Use the Mean Formula to find the sum of the unknown observations ($x+y$).

• Reasoning: The mean formula directly links the total sum of observations to the number of observations and the mean. We can find the sum of all observations, subtract the sum of known observations, and thereby isolate the sum of the two unknowns.
• Calculation: We know $\bar{x} = \frac{\sum x_i}{n}$. Substituting the given values: $10 = \frac{\sum_{i=1}^{8} x_i}{8}$ Multiplying both sides by 8, we get the total sum of all eight observations: $\sum_{i=1}^{8} x_i = 10 \times 8 = 80$ Now, let's sum the six known observations: $5 + 7 + 10 + 12 + 14 + 15 = 63$ The sum of all observations is also the sum of known observations plus the sum of unknown observations: $(\text{Sum of known observations}) + (x+y) = \sum_{i=1}^{8} x_i$ $63 + (x+y) = 80$ Solving for $x+y$: $x+y = 80 - 63$ $\mathbf{x+y = 17 \quad \text{(Equation 1)}}$

Step 2: Use the Variance Formula to find the sum of squares of the unknown observations ($x^2+y^2$).

• Reasoning: The computational variance formula involves the sum of squares of all observations. By using this formula, we can find the total sum of squares, subtract the sum of squares of known observations, and then obtain the sum of squares of the two unknowns.
• Calculation: The variance formula is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. Substituting the given values: $13.5 = \frac{\sum_{i=1}^{8} x_i^2}{8} - (10)^2$ $13.5 = \frac{\sum_{i=1}^{8} x_i^2}{8} - 100$ Adding 100 to both sides: $113.5 = \frac{\sum_{i=1}^{8} x_i^2}{8}$ Multiplying by 8 to find the total sum of squares: $\sum_{i=1}^{8} x_i^2 = 113.5 \times 8 = 908$ Now, let's find the sum of squares for the six known observations: $5^2 + 7^2 + 10^2 + 12^2 + 14^2 + 15^2$ $= 25 + 49 + 100 + 144 + 196 + 225 = 739$ The total sum of squares is the sum of squares of known observations plus the sum of squares of unknown observations: $(\text{Sum of squares of known observations}) + (x^2+y^2) = \sum_{i=1}^{8} x_i^2$ $739 + (x^2+y^2) = 908$ Solving for $x^2+y^2$: $x^2+y^2 = 908 - 739$ $\mathbf{x^2+y^2 = 169 \quad \text{(Equation 2)}}$

Step 3: Solve the system of equations using algebraic identities to find $|x-y|$.

• Reasoning: We have two equations with $x+y$ and $x^2+y^2$. We need to find $|x-y|$. This can be done by first finding the product $xy$ using the identity $(x+y)^2 = x^2+y^2+2xy$, and then using the identity $(x-y)^2 = (x+y)^2 - 4xy$.

• Calculation: From Equation 1, $x+y = 17$. From Equation 2, $x^2+y^2 = 169$.

  First, let's find $2xy$: $(x+y)^2 = x^2+y^2+2xy$ $17^2 = 169 + 2xy$ $289 = 169 + 2xy$ $2xy = 289 - 169$ $2xy = 120$ $xy = 60$

  Now, let's find $(x-y)^2$: $(x-y)^2 = (x+y)^2 - 4xy$ Substitute the values of $x+y$ and $xy$: $(x-y)^2 = (17)^2 - 4(60)$ $(x-y)^2 = 289 - 240$ $(x-y)^2 = 49$ Taking the square root of both sides to find the absolute difference: $|x-y| = \sqrt{49}$ $\mathbf{|x-y| = 7}$

---

3. Common Mistakes & Tips

• Incorrect Variance Formula: Using the definitional formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ directly can be cumbersome. Always prefer the computational formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ for problems involving unknown observations or sums of squares.
• Algebraic Errors: Be careful with squaring numbers and performing arithmetic operations. A small calculation mistake can lead to an incorrect final answer. Ensure you correctly apply algebraic identities like $(a+b)^2$ and $(a-b)^2$.
• Forgetting Absolute Difference: The question asks for the "absolute difference," which means $|x-y|$, ensuring the result is non-negative.

---

4. Summary

This problem effectively tests the application of mean and variance definitions along with basic algebraic manipulations. We first used the mean to establish the sum of the two unknown observations. Then, we used the computational formula for variance to find the sum of squares of these unknowns. Finally, using key algebraic identities, we solved the system of equations to determine the absolute difference between the two observations. The derived absolute difference is 7.

---

5. Final Answer

The absolute difference of the remaining two observations is 7. The final answer is $\boxed{7}$, which corresponds to option (C).