To find the value of |a - b|, we need to determine the values of a and b. Given that the mean of the six observations is 10, we can find the sum of the observations by multiplying the mean by the number of observations:

Sum = Mean * Number of Observations
Sum = 10 * 6
Sum = 60

Next, we need to find the sum of the squares of the observations. This can be calculated using the formula for variance:

Variance = (Sum of Squares - (Sum)^2 / Number of Observations) / (Number of Observations - 1)

Substituting the given values into the equation:

20/3 = (Sum of Squares - 60^2 / 6) / 5

Simplifying the equation:

20/3 = (Sum of Squares - 3600) / 5
100/3 = Sum of Squares - 3600
Sum of Squares = 3600 + 100/3
Sum of Squares = 3800/3

We can now find the sum of the squares of the observations by subtracting the sum of the squares of the known numbers (7^2 + 10^2 + 11^2 + 15^2) from the total sum of squares:

Sum of Squares of a and b = 3800/3 - (7^2 + 10^2 + 11^2 + 15^2)
Sum of Squares of a and b = 3800/3 - (49 + 100 + 121 + 225)
Sum of Squares of a and b = 3800/3 - 495
Sum of Squares of a and b = 2305/3

Now, we can create two equations using the sum and the sum of squares of the six observations:

Equation 1: 7 + 10 + 11 + 15 + a + b = 60
Equation 2: 7^2 + 10^2 + 11^2 + 15^2 + a^2 + b^2 = 2305/3

Simplifying Equation 1, we get:
43 + a + b = 60
a + b = 17

Substituting this value in Equation 2, we get:
49 + 100 + 121 + 225 + a^2 + b^2 = 2305/3
495 + a^2 + b^2 = 2305/3
a^2 + b^2 = 2305/3 - 495
a^2 + b^2 = 2305/3 - 1485/3
a^2 + b^2 = 820/3

Now, using the identity (a - b)^2 = a^2 + b^2 - 2ab, we can substitute the values we have:

(a - b)^2 = 820/3 - 2ab
(a - b)^2 = (820 - 6ab) / 3

Since we want to find |a - b|, we take the square root of both sides:

|a - b| = sqrt((820 - 6ab) / 3)

To find the value of |a - b|, we need to determine the value