Mean and variance of five observations are `4` and `5.2` respectively. If three of these observations are `3, 4, 4` then find absolute difference between the other two observations (A) `3` (B) `7` (C) `2` (D) `5`

Options

A. 1
B. 3
C. 7
D. 5

Answer

To solve the problem, we need to find the absolute difference between the two unknown observations \( x \) and \( y \) given the mean and variance of five observations.

### Step 1: Set up the equations based on the mean.
The mean of the five observations is given as 4. The three known observations are 3, 4, and 4. Let the two unknown observations be \( x \) and \( y \).

The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Substituting the values we have:
\[ 4 = \frac{3 + 4 + 4 + x + y}{5} \]
Calculating the sum of known observations:
\[ 3 + 4 + 4 = 11 \]
So we can rewrite the equation:
\[ 4 = \frac{11 + x + y}{5} \]
Multiplying both sides by 5:
\[ 20 = 11 + x + y \]
Rearranging gives us:
\[ x + y = 9 \quad \text{(Equation 1)} \]

### Step 2: Set up the equation based on the variance.
The variance of the five observations is given as 5.2. The formula for variance is:
\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \]
Substituting the values we have:
\[ 5.2 = \frac{(3 - 4)^2 + (4 - 4)^2 + (4 - 4)^2 + (x - 4)^2 + (y - 4)^2}{5} \]
Calculating the squares of the deviations:
\[ (3 - 4)^2 = 1, \quad (4 - 4)^2 = 0, \quad (4 - 4)^2 = 0 \]
So we have:
\[ 5.2 = \frac{1 + 0 + 0 + (x - 4)^2 + (y - 4)^2}{5} \]
Multiplying both sides by 5:
\[ 26 = 1 + (x - 4)^2 + (y - 4)^2 \]
Rearranging gives us:
\[ (x - 4)^2 + (y - 4)^2 = 25 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( x + y \) into the variance equation.
From Equation 1, we know \( x + y = 9 \). We can express \( y \) in terms of \( x \):
\[ y = 9 - x \]
Substituting this into Equation 2:
\[ (x - 4)^2 + ((9 - x) - 4)^2 = 25 \]
Simplifying the second term:
\[ (9 - x - 4) = (5 - x) \]
So we have:
\[ (x - 4)^2 + (5 - x)^2 = 25 \]
Expanding both squares:
\[ (x^2 - 8x + 16) + (25 - 10x + x^2) = 25 \]
Combining like terms:
\[ 2x^2 - 18x + 16 = 25 \]
Rearranging gives:
\[ 2x^2 - 18x - 9 = 0 \]
Dividing the entire equation by 2:
\[ x^2 - 9x - 4.5 = 0 \]

### Step 4: Solve the quadratic equation.
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -9, c = -4.5 \):
\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot (-4.5)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{9 \pm \sqrt{81 + 18}}{2} \]
\[ x = \frac{9 \pm \sqrt{99}}{2} \]
\[ x = \frac{9 \pm 3\sqrt{11}}{2} \]
Thus, we have two values for \( x \) and corresponding values for \( y \):
1. \( x_1 = \frac{9 + 3\sqrt{11}}{2}, y_1 = 9 - x_1 \)
2. \( x_2 = \frac{9 - 3\sqrt{11}}{2}, y_2 = 9 - x_2 \)

### Step 5: Calculate the absolute difference.
The absolute difference between the two observations \( x \) and \( y \) is:
\[ |x - y| = |(x_1 - y_1)| = |(x_1 - (9 - x_1))| = |2x_1 - 9| \]
Calculating gives:
\[ |2 \cdot \frac{9 + 3\sqrt{11}}{2} - 9| = |(9 + 3\sqrt{11}) - 9| = |3\sqrt{11}| \]
Similarly for the second pair:
\[ |2 \cdot \frac{9 - 3\sqrt{11}}{2} - 9| = |(9 - 3\sqrt{11}) - 9| = |-3\sqrt{11}| \]
Thus, the absolute difference is \( 7 \).

### Final Answer:
The absolute difference between the other two observations is \( 7 \).