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 need to find two unknown observations given the mean and variance of five observations, along with three known observations. Let's break it down step by step.

### Step 1: Understand the Given Information
We have:
- Mean (\( \bar{x} \)) = 4
- Variance (\( \sigma^2 \)) = 5.2
- Three observations: \( x_1 = 2, x_2 = 4, x_3 = 6 \)

Let the unknown observations be \( a \) and \( b \).

### Step 2: Use the Mean to Set Up an Equation
The mean of the five observations can be calculated using the formula:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + a + b}{n} \]
where \( n \) is the total number of observations (which is 5).

Substituting the known values:
\[ 4 = \frac{2 + 4 + 6 + a + b}{5} \]

Calculating the sum of known observations:
\[ 2 + 4 + 6 = 12 \]

Now substituting this back into the equation:
\[ 4 = \frac{12 + a + b}{5} \]

Multiplying both sides by 5:
\[ 20 = 12 + a + b \]

Rearranging gives us:
\[ a + b = 20 - 12 = 8 \quad \text{(Equation 1)} \]

### Step 3: Use the Variance to Set Up Another Equation
The variance is given by the formula:
\[ \sigma^2 = \frac{\sum (x_i^2)}{n} - \bar{x}^2 \]

Substituting the known values:
\[ 5.2 = \frac{2^2 + 4^2 + 6^2 + a^2 + b^2}{5} - 4^2 \]

Calculating the squares of the known observations:
\[ 2^2 = 4, \quad 4^2 = 16, \quad 6^2 = 36 \]
So,
\[ 2^2 + 4^2 + 6^2 = 4 + 16 + 36 = 56 \]

Now substituting back into the variance equation:
\[ 5.2 = \frac{56 + a^2 + b^2}{5} - 16 \]

Adding 16 to both sides:
\[ 5.2 + 16 = \frac{56 + a^2 + b^2}{5} \]
\[ 21.2 = \frac{56 + a^2 + b^2}{5} \]

Multiplying both sides by 5:
\[ 106 = 56 + a^2 + b^2 \]

Rearranging gives us:
\[ a^2 + b^2 = 106 - 56 = 50 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( a + b = 8 \)
2. \( a^2 + b^2 = 50 \)

From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 8 - a \]

Substituting this into Equation 2:
\[ a^2 + (8 - a)^2 = 50 \]

Expanding the equation:
\[ a^2 + (64 - 16a + a^2) = 50 \]
\[ 2a^2 - 16a + 64 = 50 \]
\[ 2a^2 - 16a + 14 = 0 \]

Dividing the entire equation by 2:
\[ a^2 - 8a + 7 = 0 \]

### Step 5: Factor the Quadratic Equation
Factoring gives us:
\[ (a - 7)(a - 1) = 0 \]

Thus, we have:
\[ a = 7 \quad \text{or} \quad a = 1 \]

### Step 6: Find Corresponding Values of \( b \)
Using \( a + b = 8 \):
- If \( a = 7 \), then \( b = 1 \).
- If \( a = 1 \), then \( b = 7 \).

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