Key Concepts and Formulas

1. Mean ($\bar{x}$): The average of a set of $n$ observations $x_1, x_2, \ldots, x_n$. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ This formula helps relate the sum of all observations to the given mean.

2. Variance ($\sigma^2$): A measure of the spread of data points from the mean. The computational formula for variance is particularly useful: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ This formula allows us to directly calculate the sum of the squares of observations.

3. Algebraic Identities: These identities are crucial for manipulating expressions involving sums, products, and differences of variables:

   • $(a+b)^2 = a^2 + b^2 + 2ab$
   • $(a-b)^2 = (a+b)^2 - 4ab$

Step-by-Step Solution

We are given $n=7$ observations with a mean $\bar{x}=8$ and variance $\sigma^2=16$. Five of these observations are $2, 4, 10, 12, 14$. Let the two unknown observations be $x$ and $y$. Our goal is to find the absolute difference, $|x-y|$.

Step 1: Use the Mean Formula to Find the Sum of the Unknown Observations ($x+y$)

Why this step? The mean formula provides a direct relationship between the sum of all observations (including the unknowns) and the given mean. This will yield our first equation involving $x$ and $y$.

First, calculate the sum of the five known observations: $\text{Sum of known observations} = 2 + 4 + 10 + 12 + 14 = 42$

Now, apply the mean formula, including the unknown observations $x$ and $y$: $\bar{x} = \frac{\text{Sum of all observations}}{n}$ $8 = \frac{42 + x + y}{7}$

Multiply both sides by 7: $8 \times 7 = 42 + x + y$ $56 = 42 + x + y$

Subtract 42 from both sides to isolate $x+y$: $x + y = 56 - 42$ $\mathbf{x + y = 14 \quad \text{(Equation 1)}}$ We now have the sum of the two unknown observations.

Step 2: Use the Variance Formula to Find the Sum of Squares of the Unknown Observations ($x^2+y^2$)

Why this step? The computational variance formula involves the sum of squares of all observations ($\sum x_i^2$). Since we know the variance and mean, this formula allows us to set up a second equation to find $x^2+y^2$, which is essential for solving for $x$ and $y$.

First, calculate the sum of squares of the five known observations: $\text{Sum of squares of known observations} = 2^2 + 4^2 + 10^2 + 12^2 + 14^2$ $= 4 + 16 + 100 + 144 + 196$ $= 460$

Now, apply the variance formula, substituting the given values ($\sigma^2=16$, $\bar{x}=8$, $n=7$) and the sum of squares of known observations: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ $16 = \frac{460 + x^2 + y^2}{7} - (8)^2$ $16 = \frac{460 + x^2 + y^2}{7} - 64$

Add 64 to both sides to isolate the fraction term: $16 + 64 = \frac{460 + x^2 + y^2}{7}$ $80 = \frac{460 + x^2 + y^2}{7}$

Multiply both sides by 7: $80 \times 7 = 460 + x^2 + y^2$ $560 = 460 + x^2 + y^2$

Subtract 460 from both sides to isolate $x^2+y^2$: $x^2 + y^2 = 560 - 460$ $\mathbf{x^2 + y^2 = 100 \quad \text{(Equation 2)}}$ We now have the sum of squares of the two unknown observations.

Step 3: Combine Equations to Find the Product of the Unknown Observations ($xy$)

Why this step? We have $x+y$ (from Equation 1) and $x^2+y^2$ (from Equation 2). The algebraic identity $(x+y)^2 = x^2+y^2+2xy$ directly connects these two expressions to $xy$. Finding $xy$ is a necessary intermediate step to calculate $|x-y|$.

Recall the algebraic identity: $(x+y)^2 = x^2 + y^2 + 2xy$

Substitute the values from Equation 1 ($x+y=14$) and Equation 2 ($x^2+y^2=100$): $(14)^2 = 100 + 2xy$ $196 = 100 + 2xy$

Subtract 100 from both sides: $196 - 100 = 2xy$ $96 = 2xy$

Divide by 2 to find $xy$: $xy = \frac{96}{2}$ $\mathbf{xy = 48}$ We now have the product of the two unknown observations.

Step 4: Calculate the Absolute Difference ($|x-y|$)

Why this step? The ultimate objective is to find $|x-y|$. We have $x+y$ and $xy$, which can be directly used with the algebraic identity $(x-y)^2 = (x+y)^2 - 4xy$. This approach is more efficient than solving for $x$ and $y$ individually.

Recall the algebraic identity: $(x-y)^2 = (x+y)^2 - 4xy$

Substitute the values we found: $x+y=14$ and $xy=48$: $(x-y)^2 = (14)^2 - 4(48)$ $(x-y)^2 = 196 - 192$ $(x-y)^2 = 4$

To find the absolute difference, take the square root of both sides: $\sqrt{(x-y)^2} = \sqrt{4}$ $|x-y| = 2$ Thus, the absolute difference of the remaining two observations is 2.

Common Mistakes & Tips

• Variance Formula Error: A common mistake is using the formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ instead of the computational formula. While correct, the computational formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ is much more efficient when $\bar{x}$ and $\sum x_i^2$ are readily available or calculable.
• Arithmetic Errors: With multiple calculations involving squares and sums, it's easy to make small arithmetic mistakes. Always double-check your calculations, especially during summation and squaring.
• Ignoring Algebraic Identities: Attempting to solve for $x$ and $y$ individually by substituting $y=14-x$ into $x^2+y^2=100$ would lead to a quadratic equation ($x^2 + (14-x)^2 = 100$). While solvable, it's more time-consuming than directly using the identities to find $(x-y)^2$.

Summary

This problem effectively demonstrates the application of mean and variance definitions in conjunction with fundamental algebraic identities to solve for unknown data points. We systematically used the mean to establish the sum of the unknown observations, then employed the variance formula to find the sum of their squares. Finally, algebraic identities were leveraged to efficiently determine their product and, ultimately, their absolute difference. The absolute difference between the two remaining observations is 2.

The final answer is $\boxed{2}$, which corresponds to option (A).