The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1,2 and 6, then the other two are

Options

A. 2 and 9
B. 3 and 8
C. 4 and 7
D. 5 and 6

Explanation

To solve the problem step by step, we will follow the given information about the mean and variance of five observations, and we will find the two unknown observations.

### Step 1: Understand the given data
We know:
- Mean of five observations = 4
- Variance of five observations = 5.2
- Three observations are 1, 2, and 6.
- Let the other two observations be \( x \) and \( y \).

### Step 2: Use the mean to find the sum of all observations
The mean is calculated as:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 4 for 5 observations:
\[ 4 = \frac{1 + 2 + 6 + x + y}{5} \]
Multiplying both sides by 5:
\[ 20 = 1 + 2 + 6 + x + y \]
Calculating the sum of the known observations:
\[ 20 = 9 + x + y \]
Thus, we can express \( x + y \):
\[ x + y = 20 - 9 = 11 \]

### Step 3: Use the variance to find another equation
The formula for variance is:
\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} \]
We can also express it as:
\[ \text{Variance} = \frac{\sum x_i^2}{n} - \text{Mean}^2 \]
Given that the variance is 5.2:
\[ 5.2 = \frac{1^2 + 2^2 + 6^2 + x^2 + y^2}{5} - 4^2 \]
Calculating \( 4^2 \):
\[ 5.2 = \frac{1 + 4 + 36 + x^2 + y^2}{5} - 16 \]
Calculating the sum of squares of known observations:
\[ 5.2 = \frac{41 + x^2 + y^2}{5} - 16 \]
Multiplying both sides by 5:
\[ 26 = 41 + x^2 + y^2 - 80 \]
Rearranging gives:
\[ x^2 + y^2 = 26 + 80 - 41 = 65 \]

### Step 4: Set up a system of equations
Now we have two equations:
1. \( x + y = 11 \)
2. \( x^2 + y^2 = 65 \)

### Step 5: Substitute \( y \) in terms of \( x \)
From the first equation, we can express \( y \):
\[ y = 11 - x \]
Substituting into the second equation:
\[ x^2 + (11 - x)^2 = 65 \]
Expanding the equation:
\[ x^2 + (121 - 22x + x^2) = 65 \]
Combining like terms:
\[ 2x^2 - 22x + 121 - 65 = 0 \]
This simplifies to:
\[ 2x^2 - 22x + 56 = 0 \]
Dividing by 2:
\[ x^2 - 11x + 28 = 0 \]

### Step 6: Factor the quadratic equation
Factoring gives:
\[ (x - 7)(x - 4) = 0 \]
Thus, we have:
\[ x = 7 \quad \text{or} \quad x = 4 \]

### Step 7: Find corresponding \( y \) values
If \( x = 7 \):
\[ y = 11 - 7 = 4 \]
If \( x = 4 \):
\[ y = 11 - 4 = 7 \]

### Conclusion
The two unknown observations are \( 4 \) and \( 7 \).