This problem requires a solid understanding of the definitions and formulas for the mean and variance of a data set. We will use the given information to form a system of equations involving the unknown numbers $a$ and $b$, and then solve this system.

1. Key Concepts and Formulas

• Mean ($\bar{x}$): The average of a set of numbers. For a data set $x_1, x_2, \ldots, x_n$, the mean is given by: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ where $\sum x_i$ is the sum of all observations and $n$ is the number of observations.

• Variance ($\sigma^2$): A measure of the spread of numbers in a data set from their mean. For a data set $x_1, x_2, \ldots, x_n$ with mean $\bar{x}$, the variance is defined as: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$ An alternative and often more convenient formula for calculation, especially when the mean is an integer, is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ This formula helps avoid calculating individual deviations and is generally more efficient.

2. Step-by-Step Solution

We are given the numbers $a, b, 8, 5, 10$. The total number of observations, $n = 5$. The mean of these numbers, $\bar{x} = 6$. The variance of these numbers, $\sigma^2 = 6.80$.

• Step 1: Use the mean to form the first equation. What we are doing: We are using the definition of the mean to establish a relationship between $a$ and $b$. Why this step: The problem provides the mean, so applying its formula is the direct way to use this information.

  First, sum all the observations: $\sum x_i = a + b + 8 + 5 + 10 = a + b + 23$ Now, apply the mean formula: $\bar{x} = \frac{\sum x_i}{n}$ $6 = \frac{a + b + 23}{5}$ Multiply both sides by 5: $30 = a + b + 23$ Rearrange to find the first equation: $a + b = 30 - 23$ $a + b = 7 \quad \text{(Equation 1)}$

• Step 2: Use the variance to form the second equation. What we are doing: We are using the given variance and mean to establish another relationship between $a$ and $b$. Why this step: With two unknown variables ($a$ and $b$), we need two independent equations to solve for them. The variance provides this second piece of information.

  We will use the computationally efficient variance formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. First, calculate the sum of the squares of all observations: $\sum x_i^2 = a^2 + b^2 + 8^2 + 5^2 + 10^2$ $\sum x_i^2 = a^2 + b^2 + 64 + 25 + 100$ $\sum x_i^2 = a^2 + b^2 + 189$ Now, substitute the values into the variance formula: $6.80 = \frac{a^2 + b^2 + 189}{5} - (6)^2$ Calculate the squared mean: $6.80 = \frac{a^2 + b^2 + 189}{5} - 36$ Add 36 to both sides: $6.80 + 36 = \frac{a^2 + b^2 + 189}{5}$ $42.80 = \frac{a^2 + b^2 + 189}{5}$ Multiply both sides by 5: $42.80 \times 5 = a^2 + b^2 + 189$ $214 = a^2 + b^2 + 189$ Rearrange to find the second equation: $a^2 + b^2 = 214 - 189$ $a^2 + b^2 = 25 \quad \text{(Equation 2)}$

• Step 3: Solve the system of equations for $a$ and $b$. What we are doing: We now have two equations with two unknowns, which we can solve simultaneously. Why this step: Solving these equations will give us the specific values of $a$ and $b$ that satisfy both the mean and variance conditions.

  Our system of equations is:

  1. $a + b = 7$
  2. $a^2 + b^2 = 25$

  From Equation 1, express $b$ in terms of $a$: $b = 7 - a$ Substitute this expression for $b$ into Equation 2: $a^2 + (7 - a)^2 = 25$ Expand the squared term: $(7-a)^2 = 7^2 - 2(7)(a) + a^2 = 49 - 14a + a^2$. $a^2 + (49 - 14a + a^2) = 25$ Combine like terms: $2a^2 - 14a + 49 = 25$ Move all terms to one side to form a standard quadratic equation: $2a^2 - 14a + 49 - 25 = 0$ $2a^2 - 14a + 24 = 0$ Divide the entire equation by 2 to simplify: $a^2 - 7a + 12 = 0$ Factorize the quadratic equation. We need two numbers that multiply to 12 and add up to -7. These are -3 and -4. $(a - 3)(a - 4) = 0$ This gives two possible values for $a$: $a = 3 \quad \text{or} \quad a = 4$ Now, find the corresponding values of $b$ using $b = 7 - a$: If $a = 3$, then $b = 7 - 3 = 4$. If $a = 4$, then $b = 7 - 4 = 3$. So, the possible pairs for $(a, b)$ are $(3, 4)$ or $(4, 3)$.

• Step 4: Check the given options. What we are doing: We compare our derived possible values for $(a, b)$ with the given options to find the correct match. Why this step: This is the final verification to ensure our solution aligns with one of the provided choices.

  Our derived pairs are $(3, 4)$ and $(4, 3)$. Let's examine the options: (A) $a = 0, b = 7$ (B) $a = 5, b = 2$ (C) $a = 1, b = 6$ (D) $a = 3, b = 4$

  Option (D) directly matches one of our derived pairs. Let's verify option (D) against the original problem conditions for completeness: For $a=3, b=4$:

  1. Mean: $(3 + 4 + 8 + 5 + 10) / 5 = 30 / 5 = 6$. (Matches the given mean)
  2. Variance: The numbers are $3, 4, 8, 5, 10$. The mean is 6. Sum of squares: $\sum x_i^2 = 3^2 + 4^2 + 8^2 + 5^2 + 10^2 = 9 + 16 + 64 + 25 + 100 = 214$. Variance $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 = \frac{214}{5} - 6^2 = 42.8 - 36 = 6.8$. (Matches the given variance) Since both conditions are satisfied, $(3,4)$ is a possible pair.

  Let's also quickly check option (A) as it was designated as the correct answer in the prompt, to highlight any discrepancy: For $a=0, b=7$:

  1. Mean: $(0 + 7 + 8 + 5 + 10) / 5 = 30 / 5 = 6$. (Matches the given mean)
  2. Variance: The numbers are $0, 7, 8, 5, 10$. The mean is 6. Sum of squares: $\sum x_i^2 = 0^2 + 7^2 + 8^2 + 5^2 + 10^2 = 0 + 49 + 64 + 25 + 100 = 238$. Variance $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 = \frac{238}{5} - 6^2 = 47.6 - 36 = 11.6$. This variance ($11.6$) does not match the given variance of $6.80$. Therefore, option (A) is not a possible value.

  Based on the problem statement and standard statistical formulas, the values $(3,4)$ or $(4,3)$ are the correct solutions.

3. Common Mistakes & Tips

• Formula Recall: Ensure you correctly recall the formulas for mean and variance. A common mistake is to forget to divide by $n$ in the variance formula or to use the sum of absolute deviations instead of squared deviations.
• Algebraic Errors: Be meticulous with arithmetic and algebraic manipulations, especially when expanding $(7-a)^2$ or combining terms. Small calculation errors can lead to incorrect quadratic equations.
• Checking Options: Always substitute your calculated values back into the original conditions (mean and variance) or check the options against the derived equations to verify your answer. This is a crucial step for catching errors.

4. Summary

We systematically used the given mean and variance to construct two algebraic equations involving the unknown numbers $a$ and $b$. The mean condition led to $a+b=7$, and the variance condition led to $a^2+b^2=25$. Solving this system of equations revealed that the possible pairs for $(a,b)$ are $(3,4)$ or $(4,3)$. Comparing these with the given options, we found that option (D) matches our derived values.

The final answer is $\boxed{\text{a = 3, b = 4}}$ which corresponds to option (D).