The arithmetic mean of the observation 10,8,5,a,b is 6 and their variance is 6.8. Then ab =

Options

A. 6
B. 4
C. 3
D. 12

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that the arithmetic mean of the observations \( 10, 8, 5, a, b \) is \( 6 \) and their variance is \( 6.8 \). Then we will calculate \( ab \).

### Step 1: Set up the equation for the arithmetic mean
The arithmetic mean (average) is calculated as follows:

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

Given that the mean is \( 6 \) and there are \( 5 \) observations, we can write:

\[ \frac{10 + 8 + 5 + a + b}{5} = 6 \]

### Step 2: Simplify the equation
Multiply both sides by \( 5 \):

\[ 10 + 8 + 5 + a + b = 30 \]

Now, combine the constants:

\[ 23 + a + b = 30 \]

### Step 3: Solve for \( a + b \)
Rearranging gives:

\[ a + b = 30 - 23 = 7 \]

### Step 4: Set up the equation for variance
The formula for variance is:

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

Given that the variance is \( 6.8 \), we can write:

\[ 6.8 = \frac{(10 - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (a - 6)^2 + (b - 6)^2}{5} \]

### Step 5: Calculate the squared differences
Calculating the squared differences:

\[ (10 - 6)^2 = 16, \quad (8 - 6)^2 = 4, \quad (5 - 6)^2 = 1 \]

So we have:

\[ 6.8 = \frac{16 + 4 + 1 + (a - 6)^2 + (b - 6)^2}{5} \]

### Step 6: Multiply both sides by 5
\[ 34 = 16 + 4 + 1 + (a - 6)^2 + (b - 6)^2 \]

### Step 7: Simplify
Combine the constants:

\[ 34 = 21 + (a - 6)^2 + (b - 6)^2 \]

### Step 8: Solve for the sum of squares
Rearranging gives:

\[ (a - 6)^2 + (b - 6)^2 = 34 - 21 = 13 \]

### Step 9: Substitute \( a \) and \( b \)
Let \( x = a - 6 \) and \( y = b - 6 \). Then:

\[ x^2 + y^2 = 13 \]

Also, since \( a + b = 7 \):

\[ (a - 6) + (b - 6) = 7 - 12 = -5 \quad \Rightarrow \quad x + y = -5 \]

### Step 10: Solve the system of equations
We have the equations:

1. \( x^2 + y^2 = 13 \)
2. \( x + y = -5 \)

Substituting \( y = -5 - x \) into the first equation:

\[ x^2 + (-5 - x)^2 = 13 \]

Expanding:

\[ x^2 + (25 + 10x + x^2) = 13 \]

Combining terms:

\[ 2x^2 + 10x + 25 - 13 = 0 \]

This simplifies to:

\[ 2x^2 + 10x + 12 = 0 \]

### Step 11: Solve the quadratic equation
Dividing through by 2:

\[ x^2 + 5x + 6 = 0 \]

Factoring gives:

\[ (x + 2)(x + 3) = 0 \]

Thus, \( x = -2 \) or \( x = -3 \).

### Step 12: Find corresponding \( y \) values
If \( x = -2 \):

\[ y = -5 - (-2) = -3 \]

If \( x = -3 \):

\[ y = -5 - (-3) = -2 \]

### Step 13: Find \( a \) and \( b \)
Recall \( a = x + 6 \) and \( b = y + 6 \):

1. If \( x = -2 \) and \( y = -3 \):
- \( a = 4 \), \( b = 3 \)

2. If \( x = -3 \) and \( y = -2 \):
- \( a = 3 \), \( b = 4 \)

### Step 14: Calculate \( ab \)
In both cases:

\[ ab = 4 \times 3 = 12 \]

Thus, the final answer is:

\[ \boxed{12} \]