1. Key Concepts and Formulas

To solve this problem efficiently, we rely on the fundamental definitions and computational formulas for mean and variance, along with a crucial algebraic identity.

• Arithmetic Mean ($\bar{x}$): The average of a dataset. For $N$ observations $x_1, x_2, \ldots, x_N$, the mean is given by: $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$ This formula helps establish a linear relationship involving the sum of the unknown variables.

• Variance ($\sigma^2$): A measure of the spread or dispersion of data points around the mean. The most convenient computational formula for variance, especially when dealing with unknown variables, is: $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - (\bar{x})^2$ This formula directly involves the sum of squares of the observations, which is key to finding the sum of squares of our unknown variables.

• Algebraic Identity: The square of a sum identity: $(a+b)^2 = a^2 + b^2 + 2ab$ This identity is vital for connecting the sum of variables ($a+b$) and their sum of squares ($a^2+b^2$) to their product ($ab$), which is required for the final expression.

2. Step-by-Step Solution

Let the given dataset be $X = \{6, 4, a, 8, b, 12, 10, 13\}$. The number of observations is $N=8$. The given mean is $\bar{x}=9$. The given variance is $\sigma^2=9.25$. We need to find the value of $a+b+ab$.

Step 1: Use the Mean to find the sum of the unknowns ($a+b$)

• What we are doing: We are using the definition of the arithmetic mean to form an equation involving $a$ and $b$.
• Why we are doing it: The mean formula provides a direct way to find the sum of all observations, from which we can isolate the sum of the unknown variables ($a+b$). This is usually the first step in such problems as it yields a linear equation.

The formula for the mean is $\bar{x} = \frac{\sum x_i}{N}$. Substitute the given values: $9 = \frac{6+4+a+8+b+12+10+13}{8}$ First, sum the known numerical values in the dataset: $6+4+8+12+10+13 = 53$ Substitute this sum back into the mean equation: $9 = \frac{53 + a + b}{8}$ Multiply both sides by $8$: $9 \times 8 = 53 + a + b$ $72 = 53 + a + b$ Isolate $a+b$: $a + b = 72 - 53$ $\mathbf{a+b = 19} \quad \text{(Equation 1)}$

Step 2: Use the Variance to find the sum of squares of the unknowns ($a^2+b^2$)

• What we are doing: We are applying the computational formula for variance to find the sum of squares of all observations.
• Why we are doing it: To find the product $ab$ later using the algebraic identity $(a+b)^2 = a^2+b^2+2ab$, we need the value of $a^2+b^2$. The variance formula is the most efficient way to obtain the sum of squares, $\sum x_i^2$, from which $a^2+b^2$ can be extracted.

The computational formula for variance is $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$. Substitute the given variance ($\sigma^2=9.25$), number of observations ($N=8$), and mean ($\bar{x}=9$): $9.25 = \frac{\sum x_i^2}{8} - (9)^2$ Calculate $9^2$: $9.25 = \frac{\sum x_i^2}{8} - 81$ Add $81$ to both sides to isolate the term $\frac{\sum x_i^2}{8}$: $9.25 + 81 = \frac{\sum x_i^2}{8}$ $90.25 = \frac{\sum x_i^2}{8}$ Multiply both sides by $8$ to find the total sum of squares $\sum x_i^2$: $\sum x_i^2 = 90.25 \times 8$ $\sum x_i^2 = 722$ The total sum of squares consists of the squares of known values and the squares of unknown values: $\sum x_i^2 = 6^2 + 4^2 + a^2 + 8^2 + b^2 + 12^2 + 10^2 + 13^2$ Calculate the sum of squares of the known numerical values: $6^2 = 36$ $4^2 = 16$ $8^2 = 64$ $12^2 = 144$ $10^2 = 100$ $13^2 = 169$ Sum of known squares: $36+16+64+144+100+169 = 529$. Substitute this sum back into the equation for $\sum x_i^2$: $722 = 529 + a^2 + b^2$ Isolate $a^2+b^2$: $a^2 + b^2 = 722 - 529$ $\mathbf{a^2 + b^2 = 193} \quad \text{(Equation 2)}$

Step 3: Use the Algebraic Identity to find the product of the unknowns ($ab$)

• What we are doing: We are applying the algebraic identity $(a+b)^2 = a^2+b^2+2ab$.
• Why we are doing it: We have found $a+b$ (from Step 1) and $a^2+b^2$ (from Step 2). This identity directly relates these two sums to the product $ab$, which is a component of the target expression $a+b+ab$.

Recall the identity: $(a+b)^2 = a^2 + b^2 + 2ab$ Substitute the values from Equation 1 ($a+b=19$) and Equation 2 ($a^2+b^2=193$): $(19)^2 = 193 + 2ab$ Calculate the square of $19$: $361 = 193 + 2ab$ Subtract $193$ from both sides: $2ab = 361 - 193$ $2ab = 168$ Divide by $2$ to find $ab$: $\mathbf{ab = 84} \quad \text{(Equation 3)}$

Step 4: Calculate the final desired expression ($a+b+ab$)

• What we are doing: We are substituting the values we found for $a+b$ and $ab$ into the target expression.
• Why we are doing it: This is the final step to arrive at the solution, as all necessary components have been determined.

We need to find $a+b+ab$. Substitute $a+b=19$ (from Equation 1) and $ab=84$ (from Equation 3): $a+b+ab = 19 + 84$ $a+b+ab = 103$

3. Common Mistakes & Tips

• Arithmetic Precision: Be extremely careful with calculations involving sums, squares, and decimals. Even a minor arithmetic error can lead to an incorrect final answer.
• Choice of Variance Formula: Always prefer the computational formula for variance, $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$, when dealing with unknown variables. It simplifies calculations by avoiding terms like $(a-\bar{x})^2$ and $(b-\bar{x})^2$.
• Algebraic Identity Recall: Ensure you correctly recall and apply the identity $(a+b)^2 = a^2+b^2+2ab$. Misremembering this can halt the solution process.

4. Summary

This problem effectively tests the application of fundamental statistical concepts alongside basic algebra. By first using the given mean to establish the sum of the unknown variables ($a+b=19$), then applying the computational variance formula to find the sum of their squares ($a^2+b^2=193$), we were able to use the algebraic identity $(a+b)^2 = a^2+b^2+2ab$ to determine their product ($ab=84$). Finally, combining these results yielded the required expression $a+b+ab = 103$.

The final answer is $\boxed{\text{103}}$, which corresponds to option (A).