The mean of five observations is 4 and their variance is 5.2. If three of these observations are 2, 4 and 6, then the other two observations are :

Options

A. 3 and 5
B. 2 and 6
C. 4 and 4
D. 1 and 7

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have five observations with a mean of 4 and a variance of 5.2. Three of these observations are given as 2, 4, and 6.

### Step 2: Calculate the sum of the observations
The mean of the five observations is given by:
\[ \text{Mean} = \frac{\Sigma x_i}{n} \]
Where \( n = 5 \) and the mean is 4. Thus, we can write:
\[ \Sigma x_i = \text{Mean} \times n = 4 \times 5 = 20 \]

### Step 3: Calculate the sum of the squares of the observations
The variance is given by the formula:
\[ \text{Variance} = \frac{\Sigma x_i^2}{n} - \left(\frac{\Sigma x_i}{n}\right)^2 \]
Given that the variance is 5.2, we can rearrange this to find \(\Sigma x_i^2\):
\[ 5.2 = \frac{\Sigma x_i^2}{5} - 4^2 \]
This simplifies to:
\[ 5.2 = \frac{\Sigma x_i^2}{5} - 16 \]
Adding 16 to both sides gives:
\[ \frac{\Sigma x_i^2}{5} = 21.2 \]
Multiplying both sides by 5 results in:
\[ \Sigma x_i^2 = 106 \]

### Step 4: Set up equations with the known observations
Let the unknown observations be \( x_4 \) and \( x_5 \). We know:
\[ 2 + 4 + 6 + x_4 + x_5 = 20 \]
This simplifies to:
\[ 12 + x_4 + x_5 = 20 \implies x_4 + x_5 = 8 \quad \text{(Equation 1)} \]

For the sum of squares:
\[ 2^2 + 4^2 + 6^2 + x_4^2 + x_5^2 = 106 \]
Calculating the squares gives:
\[ 4 + 16 + 36 + x_4^2 + x_5^2 = 106 \]
This simplifies to:
\[ 56 + x_4^2 + x_5^2 = 106 \implies x_4^2 + x_5^2 = 50 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
From Equation 1, we have:
\[ x_5 = 8 - x_4 \]
Substituting \( x_5 \) into Equation 2:
\[ x_4^2 + (8 - x_4)^2 = 50 \]
Expanding this gives:
\[ x_4^2 + (64 - 16x_4 + x_4^2) = 50 \]
Combining like terms results in:
\[ 2x_4^2 - 16x_4 + 64 - 50 = 0 \]
This simplifies to:
\[ 2x_4^2 - 16x_4 + 14 = 0 \]
Dividing through by 2 gives:
\[ x_4^2 - 8x_4 + 7 = 0 \]

### Step 6: Factor the quadratic equation
Factoring the quadratic:
\[ (x_4 - 7)(x_4 - 1) = 0 \]
Thus, we have:
\[ x_4 = 7 \quad \text{or} \quad x_4 = 1 \]

### Step 7: Find the corresponding values for \( x_5 \)
If \( x_4 = 7 \), then:
\[ x_5 = 8 - 7 = 1 \]
If \( x_4 = 1 \), then:
\[ x_5 = 8 - 1 = 7 \]

### Conclusion
The other two observations are \( 1 \) and \( 7 \).