This problem requires a systematic application of the definitions of mean and variance to determine unknown data points within a set of observations. We are given the overall statistical summaries (mean and variance) and a partial list of observations. Our goal is to find the product of the two missing observations.

1. Key Concepts and Formulas

For a set of $N$ observations, $x_1, x_2, \ldots, x_N$:

• Arithmetic Mean ($\mu$): The average value of all observations. It is calculated as the sum of all observations divided by the total number of observations. $\mu = \frac{\sum_{i=1}^{N} x_i}{N}$

• Variance ($\sigma^2$): A measure of the spread or dispersion of the data points around the mean. It quantifies how much the observations deviate from the average. A widely used computational formula for variance is: $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - \mu^2$ This formula is often preferred for calculations as it avoids computing individual deviations from the mean.

• Algebraic Identity: For two numbers $x$ and $y$, the square of their sum is related to the sum of their squares and their product by the identity: $(x+y)^2 = x^2 + y^2 + 2xy$ This identity is crucial for connecting the information derived from the mean and variance formulas to find the product of the unknown observations.

2. Step-by-Step Solution

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

• Total number of observations ($N$) = 7
• Mean ($\mu$) = 8
• Variance ($\sigma^2$) = 16
• Known observations: 2, 4, 10, 12, 14

Step 1: Using the Mean Formula to Find the Sum of Unknowns ($x+y$)

The mean formula directly relates the sum of all observations to the mean and the number of observations. This is our first step to establish a relationship between $x$ and $y$.

1. Calculate the sum of the known observations: Sum of known observations $= 2 + 4 + 10 + 12 + 14 = 42$.

2. Express the total sum of all seven observations: The sum of all observations, $\sum x_i$, is the sum of the known observations plus the sum of the unknown observations: $\sum x_i = 42 + x + y$

3. Apply the Mean Formula: We know $\mu = 8$ and $N = 7$. $\mu = \frac{\sum x_i}{N} \implies 8 = \frac{42 + x + y}{7}$

4. Solve for $x+y$: Multiply both sides by 7: $8 \times 7 = 42 + x + y$ $56 = 42 + x + y$ Subtract 42 from both sides: $x + y = 56 - 42$ $x + y = 14 \quad \ldots(1)$ Reasoning: The mean provides a direct link to the total sum of all observations. By using the given mean and the sum of the known observations, we can isolate the sum of the unknown observations, giving us our first equation.

Step 2: Using the Variance Formula to Find the Sum of Squares of Unknowns ($x^2+y^2$)

The variance formula involves the sum of the squares of the observations. This will provide a second, independent equation involving $x$ and $y$.

1. Calculate the sum of the squares of the known observations: $2^2 = 4$ $4^2 = 16$ $10^2 = 100$ $12^2 = 144$ $14^2 = 196$ Sum of squares of known observations $= 4 + 16 + 100 + 144 + 196 = 460$.

2. Express the total sum of squares of all seven observations: The sum of squares of all observations, $\sum x_i^2$, is the sum of squares of the known observations plus the sum of squares of the unknown observations: $\sum x_i^2 = 460 + x^2 + y^2$

3. Apply the Variance Formula: We know $\sigma^2 = 16$, $N = 7$, and $\mu = 8$. $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2 \implies 16 = \frac{460 + x^2 + y^2}{7} - (8)^2$

4. Solve for $x^2+y^2$: First, calculate $8^2 = 64$: $16 = \frac{460 + x^2 + y^2}{7} - 64$ Add 64 to both sides: $16 + 64 = \frac{460 + x^2 + y^2}{7}$ $80 = \frac{460 + x^2 + y^2}{7}$ Multiply both sides by 7: $80 \times 7 = 460 + x^2 + y^2$ $560 = 460 + x^2 + y^2$ Subtract 460 from both sides: $x^2 + y^2 = 560 - 460$ $x^2 + y^2 = 100 \quad \ldots(2)$ Reasoning: The variance formula provides a second, independent relationship between the unknown observations, specifically their squares. By substituting the given variance and mean, we can determine the sum of the squares of the unknown observations.

Step 3: Combining the Equations to Find the Product ($xy$)

We now have a system of two algebraic equations:

1. $x + y = 14$
2. $x^2 + y^2 = 100$

We can use the algebraic identity $(x+y)^2 = x^2 + y^2 + 2xy$ to find the product $xy$.

1. Substitute the values from equations (1) and (2) into the identity: $(14)^2 = 100 + 2xy$

2. Calculate $(14)^2$: $196 = 100 + 2xy$

3. Solve for $xy$: Subtract 100 from both sides: $196 - 100 = 2xy$ $96 = 2xy$ Divide both sides by 2: $xy = \frac{96}{2}$ $xy = 48$ Reasoning: This step uses a fundamental algebraic identity to relate the sum and sum of squares of two variables to their product. This is a standard technique to find the product when the sum and sum of squares are known.

3. Common Mistakes & Tips

• Arithmetic Errors: Be meticulous with calculations, especially squaring numbers and sums. A small error early on can propagate and lead to an incorrect final answer.
• Incorrect Variance Formula: Ensure you use the correct formula for variance. The computational formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$ is generally easier to use than the definition involving deviations from the mean.
• Forgetting Algebraic Identities: Remember key algebraic identities like $(x+y)^2 = x^2+y^2+2xy$. These are frequently used in problems involving sums and products of variables.

4. Summary

This problem demonstrates a structured approach to finding missing data points using statistical measures. We first used the mean formula to establish the sum of the two unknown observations. Next, we utilized the variance formula to find the sum of the squares of these observations. Finally, a common algebraic identity was employed to combine these two pieces of information and determine the product of the unknown observations. The product of the remaining two observations is 48.

5. Final Answer

The product of the remaining two observations is $\boxed{48}$, which corresponds to option (B).