If the mean and variance of six observations 7,10, 11, 15 a, b are 10 and `(20)/(3),` respectively, then value of `|a-b|` is equal to :

Options

A. 1
B. 11
C. 1
D. 9

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the mean and variance of the six observations: \( 7, 10, 11, 15, a, b \).

### Step 1: Calculate the Mean
The mean of the observations is given as \( 10 \). The formula for the mean is:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Substituting the values, we have:

\[ 10 = \frac{7 + 10 + 11 + 15 + a + b}{6} \]

Calculating the sum of the known observations:

\[ 7 + 10 + 11 + 15 = 43 \]

So, we can rewrite the equation:

\[ 10 = \frac{43 + a + b}{6} \]

Multiplying both sides by \( 6 \):

\[ 60 = 43 + a + b \]

Rearranging gives:

\[ a + b = 60 - 43 = 17 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance is given as \( \frac{20}{3} \). The formula for variance is:

\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2 \]

Substituting the known values, we have:

\[ \frac{20}{3} = \frac{49 + 100 + 121 + 225 + a^2 + b^2}{6} - 10^2 \]

Calculating \( 10^2 \):

\[ 10^2 = 100 \]

Now, substituting the known sums:

\[ \frac{20}{3} = \frac{49 + 100 + 121 + 225 + a^2 + b^2}{6} - 100 \]

Calculating the sum of the squares of the known observations:

\[ 49 + 100 + 121 + 225 = 495 \]

So we can rewrite the equation:

\[ \frac{20}{3} + 100 = \frac{495 + a^2 + b^2}{6} \]

Calculating \( 100 + \frac{20}{3} \):

\[ 100 = \frac{300}{3} \implies \frac{300 + 20}{3} = \frac{320}{3} \]

Now, we have:

\[ \frac{320}{3} = \frac{495 + a^2 + b^2}{6} \]

Multiplying both sides by \( 6 \):

\[ 640 = 495 + a^2 + b^2 \]

Rearranging gives:

\[ a^2 + b^2 = 640 - 495 = 145 \quad \text{(Equation 2)} \]

### Step 3: Solve for \( a \) and \( b \)
We have two equations now:

1. \( a + b = 17 \) (Equation 1)
2. \( a^2 + b^2 = 145 \) (Equation 2)

Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):

\[ 17^2 = 145 + 2ab \]

Calculating \( 17^2 \):

\[ 289 = 145 + 2ab \]

Rearranging gives:

\[ 2ab = 289 - 145 = 144 \implies ab = \frac{144}{2} = 72 \quad \text{(Equation 3)} \]

### Step 4: Find \( |a - b| \)
Using the identity \( (a - b)^2 = (a + b)^2 - 4ab \):

\[ (a - b)^2 = 17^2 - 4 \cdot 72 \]

Calculating:

\[ (a - b)^2 = 289 - 288 = 1 \]

Taking the square root gives:

\[ |a - b| = \sqrt{1} = 1 \]

Thus, the value of \( |a - b| \) is \( 1 \).

### Final Answer
\[ |a - b| = 1 \]