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 the observations.

### Step 1: Understand the given information
We are given:
- Mean of five observations = 4
- Variance of five observations = 5.2
- Three of the observations = 1, 2, and 6
- We need to find the other two observations, which we will denote as \( x \) and \( y \).

### Step 2: Set up the equation for the mean
The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
For our case, this can be written as:
\[ 4 = \frac{1 + 2 + 6 + x + y}{5} \]
Calculating the sum of the known observations:
\[ 1 + 2 + 6 = 9 \]
Thus, we can rewrite the equation:
\[ 4 = \frac{9 + x + y}{5} \]
Multiplying both sides by 5:
\[ 20 = 9 + x + y \]
Subtracting 9 from both sides gives:
\[ x + y = 11 \quad \text{(Equation 1)} \]

### Step 3: Set up the equation for the variance
The formula for variance is:
\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} \]
This can also be expressed as:
\[ \text{Variance} = \frac{\sum x_i^2}{n} - \text{Mean}^2 \]
For our case, we can write:
\[ 5.2 = \frac{1^2 + 2^2 + 6^2 + x^2 + y^2}{5} - 4^2 \]
Calculating the squares of the known observations:
\[ 1^2 + 2^2 + 6^2 = 1 + 4 + 36 = 41 \]
Substituting this into the variance equation:
\[ 5.2 = \frac{41 + x^2 + y^2}{5} - 16 \]
Adding 16 to both sides:
\[ 5.2 + 16 = \frac{41 + x^2 + y^2}{5} \]
\[ 21.2 = \frac{41 + x^2 + y^2}{5} \]
Multiplying both sides by 5:
\[ 106 = 41 + x^2 + y^2 \]
Subtracting 41 from both sides gives:
\[ x^2 + y^2 = 65 \quad \text{(Equation 2)} \]

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

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 11 - x \]
Substituting this into Equation 2:
\[ x^2 + (11 - x)^2 = 65 \]
Expanding the equation:
\[ x^2 + (121 - 22x + x^2) = 65 \]
Combining like terms:
\[ 2x^2 - 22x + 121 = 65 \]
Subtracting 65 from both sides:
\[ 2x^2 - 22x + 56 = 0 \]
Dividing the entire equation by 2:
\[ x^2 - 11x + 28 = 0 \]

### Step 5: Factor the quadratic equation
Factoring the quadratic:
\[ (x - 7)(x - 4) = 0 \]
Setting each factor to zero gives:
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

### Step 6: Find corresponding values of \( y \)
Using \( x + y = 11 \):
- If \( x = 7 \), then \( y = 11 - 7 = 4 \).
- If \( x = 4 \), then \( y = 11 - 4 = 7 \).

Thus, the other two observations are \( 4 \) and \( 7 \).

### Final Answer
The other two observations are \( 4 \) and \( 7 \).

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