To solve the problem step by step, we will first find the values of \(a\) and \(b\) using the given mean and variance, and then calculate \(|a - b|\).
### Step 1: Calculate the Mean
The mean of the numbers \(5, 7, 12, 10, 15, 14, a, b\) is given as 10. The formula for the mean is:
\[
\text{Mean} = \frac{\text{Sum of all terms}}{\text{Total number of terms}}
\]
Here, the total number of terms is 8. Thus, we can write:
\[
10 = \frac{5 + 7 + 12 + 10 + 15 + 14 + a + b}{8}
\]
Calculating the sum of the known numbers:
\[
5 + 7 + 12 + 10 + 15 + 14 = 63
\]
Substituting this back into the equation gives:
\[
10 = \frac{63 + a + b}{8}
\]
Multiplying both sides by 8:
\[
80 = 63 + a + b
\]
Rearranging gives:
\[
a + b = 17 \quad \text{(Equation 1)}
\]
### Step 2: Calculate the Variance
The variance is given as 13.5. The formula for variance is:
\[
\sigma^2 = \frac{\sum x_i^2}{n} - \left(\text{Mean}\right)^2
\]
Substituting the known values:
\[
13.5 = \frac{5^2 + 7^2 + 12^2 + 10^2 + 15^2 + 14^2 + a^2 + b^2}{8} - 10^2
\]
Calculating the squares of the known numbers:
\[
5^2 = 25, \quad 7^2 = 49, \quad 12^2 = 144, \quad 10^2 = 100, \quad 15^2 = 225, \quad 14^2 = 196
\]
Now, summing these squares:
\[
25 + 49 + 144 + 100 + 225 + 196 = 839
\]
Substituting this into the variance equation gives:
\[
13.5 = \frac{839 + a^2 + b^2}{8} - 100
\]
Multiplying both sides by 8:
\[
108 = 839 + a^2 + b^2 - 800
\]
This simplifies to:
\[
a^2 + b^2 = 169 \quad \text{(Equation 2)}
\]
### Step 3: Solve for \(a\) and \(b\)
We have two equations now:
1. \(a + b = 17\) (Equation 1)
2. \(a^2 + b^2 = 169\) (Equation 2)
From Equation 1, we can express \(b\) in terms of \(a\):
\[
b = 17 - a
\]
Substituting this into Equation 2:
\[
a^2 + (17 - a)^2 = 169
\]
Expanding the equation:
\[
a^2 + (289 - 34a + a^2) = 169
\]
Combining like terms:
\[
2a^2 - 34a + 289 - 169 = 0
\]
This simplifies to:
\[
2a^2 - 34a + 120 = 0
\]
Dividing the entire equation by 2:
\[
a^2 - 17a + 60 = 0
\]
### Step 4: Factor the Quadratic
Now we factor the quadratic:
\[
(a - 12)(a - 5) = 0
\]
Thus, we have:
\[
a = 12 \quad \text{or} \quad a = 5
\]
Using \(a + b = 17\):
- If \(a = 12\), then \(b = 5\).
- If \(a = 5\), then \(b = 12\).
### Step 5: Calculate \(|a - b|\)
Now we calculate \(|a - b|\):
\[
|a - b| = |12 - 5| = 7
\]
### Final Answer
Thus, the value of \(|a - b|\) is:
\[
\boxed{7}
\]
