Key Concepts and Formulas

This problem requires a solid understanding of the fundamental statistical measures: mean and variance. We'll use their definitions to establish relationships between the given observations and the unknown values.

1. Mean ($\bar{x}$): The mean is the average of a dataset. For $N$ observations $x_1, x_2, \dots, x_N$, the mean is calculated as: $\bar{x} = \frac{\sum_{i=1}^{N} x_i}{N}$ It provides a measure of the central tendency of the data.

2. Variance ($\sigma^2$): Variance quantifies the spread or dispersion of data points around the mean. A larger variance indicates data points are more spread out. The computational formula for variance is often more convenient for calculations, especially when dealing with unknown variables: $\sigma^2 = \frac{\sum_{i=1}^{N} x_i^2}{N} - (\bar{x})^2$ This formula relates the sum of the squares of the observations, the number of observations, and the mean.

Step-by-Step Solution

Step 1: Using the Mean Formula to Form an Equation for (a + b)

We are given six observations: $7, 10, 11, 15, a, b$. The total number of observations, $N$, is 6. The mean ($\bar{x}$) is given as 10.

• Why this step? The mean formula directly links the sum of all observations to the given mean and the total count. This allows us to form our first algebraic equation involving the unknown variables $a$ and $b$.

First, sum the known observations: $7 + 10 + 11 + 15 = 43$

Now, substitute these values into the mean formula: $\bar{x} = \frac{(7 + 10 + 11 + 15 + a + b)}{6}$ $10 = \frac{(43 + a + b)}{6}$

To solve for $(a+b)$, multiply both sides by 6: $10 \times 6 = 43 + a + b$ $60 = 43 + a + b$

Isolate $a + b$: $a + b = 60 - 43$ $\mathbf{a + b = 17 \quad \dots (1)}$ This is our first key equation.

Step 2: Using the Variance Formula to Form an Equation for ($a^2 + b^2$)

We are given that the variance ($\sigma^2$) is $\frac{20}{3}$, and the mean ($\bar{x}$) is 10.

• Why this step? The variance formula provides a second, independent relationship, this time involving the squares of the observations. By using the computational formula, we can efficiently calculate the sum of squares and derive an equation for $a^2 + b^2$.

First, calculate the sum of squares of the known observations: $7^2 = 49$ $10^2 = 100$ $11^2 = 121$ $15^2 = 225$ Sum of squares of known observations $= 49 + 100 + 121 + 225 = 495$

Now, substitute these values, along with the given variance and mean, into the computational variance formula: $\sigma^2 = \frac{(7^2 + 10^2 + 11^2 + 15^2 + a^2 + b^2)}{6} - (\bar{x})^2$ $\frac{20}{3} = \frac{(495 + a^2 + b^2)}{6} - (10)^2$ $\frac{20}{3} = \frac{(495 + a^2 + b^2)}{6} - 100$

To simplify, add 100 to both sides: $\frac{20}{3} + 100 = \frac{(495 + a^2 + b^2)}{6}$ Combine the terms on the left side by finding a common denominator: $\frac{20}{3} + \frac{300}{3} = \frac{(495 + a^2 + b^2)}{6}$ $\frac{320}{3} = \frac{(495 + a^2 + b^2)}{6}$

Now, multiply both sides by 6 to clear the denominators: $\frac{320}{3} \times 6 = 495 + a^2 + b^2$ $320 \times 2 = 495 + a^2 + b^2$ $640 = 495 + a^2 + b^2$

Isolate $a^2 + b^2$: $a^2 + b^2 = 640 - 495$ $\mathbf{a^2 + b^2 = 185 \quad \dots (2)}$ This is our second key equation.

Step 3: Solving the System of Equations to Find |a - b|

We now have a system of two equations with two unknowns, $a$ and $b$:

1. $a + b = 17$
2. $a^2 + b^2 = 185$

Our objective is to find the value of $|a - b|$. We can achieve this efficiently using algebraic identities.

• Why this step? Instead of solving for $a$ and $b$ individually (which would involve a quadratic equation), we can use identities that relate $(a+b)$, $(a^2+b^2)$, and $(a-b)^2$. This is a common and efficient strategy in such problems.

First, let's find the value of $2ab$ using the identity $(a+b)^2 = a^2 + b^2 + 2ab$: $2ab = (a+b)^2 - (a^2 + b^2)$ Substitute the values from equations (1) and (2): $2ab = (17)^2 - (185)$ $2ab = 289 - 185$ $2ab = 104$ $ab = 52$

Next, we use the identity for $(a-b)^2$: $(a-b)^2 = (a+b)^2 - 4ab$ Substitute the values for $(a+b)$ and $ab$: $(a-b)^2 = (17)^2 - 4(52)$ $(a-b)^2 = 289 - 208$ $(a-b)^2 = 81$

Finally, take the square root of both sides to find $a-b$: $a - b = \pm \sqrt{81}$ $a - b = \pm 9$

The question asks for the value of $|a - b|$. $|a - b| = | \pm 9 |$ $\mathbf{|a - b| = 9}$

Common Mistakes & Tips

• Arithmetic Precision: Double-check all calculations, especially sums of numbers and squares. A single arithmetic error can propagate through the entire solution.
• Choosing the Right Variance Formula: While the definitional formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}$ is correct, the computational formula $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$ is often more efficient for problems involving unknown variables $a$ and $b$, as it avoids calculating $(x_i - \bar{x})$ for each term.
• Algebraic Identities: Memorize and correctly apply algebraic identities like $(x+y)^2 = x^2+y^2+2xy$ and $(x-y)^2 = (x+y)^2 - 4xy$. These are crucial for simplifying the solution process.
• Absolute Value: Always ensure you provide the final answer in the requested format. If $|a-b|$ is asked, ensure the result is positive.

Summary

This problem effectively tests the application of fundamental statistical concepts (mean and variance) in conjunction with algebraic manipulation. We began by using the given mean to establish a linear equation for $(a+b)$. Then, we utilized the computational variance formula to form a second equation involving $(a^2+b^2)$. Finally, by applying algebraic identities that relate $(a+b)$, $(a^2+b^2)$, and $(a-b)^2$, we efficiently determined the value of $|a-b|$. The systematic approach ensures accuracy in solving for the unknowns.

The final answer is $\boxed{9}$ which corresponds to option (A).