The mean of five observation is 4 and their variance is 2.8. If three of these observations are 2, 2 and 5, then the other two are

Options

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

Answer

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

### Step 1: Set Up the Problem
We know that the mean of five observations is 4 and their variance is 2.8. Three of these observations are 2, 2, and 5. We will denote the other two observations as \( x_1 \) and \( x_2 \).

### Step 2: Calculate the Total Sum of Observations
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 4 and there are 5 observations, we can set up the equation:
\[ \frac{2 + 2 + 5 + x_1 + x_2}{5} = 4 \]
Calculating the sum of the known observations:
\[ 2 + 2 + 5 = 9 \]
Thus, we can rewrite the equation:
\[ \frac{9 + x_1 + x_2}{5} = 4 \]
Multiplying both sides by 5:
\[ 9 + x_1 + x_2 = 20 \]
This simplifies to:
\[ x_1 + x_2 = 11 \quad \text{(Equation 1)} \]

### Step 3: Calculate the Variance
The variance is given by the formula:
\[ \text{Variance} = \frac{\sum (x_i^2)}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]
We know the variance is 2.8 and there are 5 observations:
\[ 2.8 = \frac{\sum (x_i^2)}{5} - \left(\frac{20}{5}\right)^2 \]
Calculating the square of the mean:
\[ \left(\frac{20}{5}\right)^2 = 4^2 = 16 \]
Substituting this back into the variance equation:
\[ 2.8 = \frac{\sum (x_i^2)}{5} - 16 \]
Multiplying both sides by 5:
\[ 14 = \sum (x_i^2) - 80 \]
Thus, we have:
\[ \sum (x_i^2) = 94 \quad \text{(Equation 2)} \]

### Step 4: Calculate the Sum of Squares of the Observations
Now, we can express the sum of squares of the observations:
\[ \sum (x_i^2) = 2^2 + 2^2 + 5^2 + x_1^2 + x_2^2 \]
Calculating the known squares:
\[ 2^2 = 4, \quad 2^2 = 4, \quad 5^2 = 25 \]
Thus:
\[ \sum (x_i^2) = 4 + 4 + 25 + x_1^2 + x_2^2 = 33 + x_1^2 + x_2^2 \]
Setting this equal to Equation 2:
\[ 33 + x_1^2 + x_2^2 = 94 \]
This simplifies to:
\[ x_1^2 + x_2^2 = 61 \quad \text{(Equation 3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( x_1 + x_2 = 11 \) (Equation 1)
2. \( x_1^2 + x_2^2 = 61 \) (Equation 3)

From Equation 1, we can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 11 - x_1 \]
Substituting this into Equation 3:
\[ x_1^2 + (11 - x_1)^2 = 61 \]
Expanding the equation:
\[ x_1^2 + (121 - 22x_1 + x_1^2) = 61 \]
Combining like terms:
\[ 2x_1^2 - 22x_1 + 121 - 61 = 0 \]
This simplifies to:
\[ 2x_1^2 - 22x_1 + 60 = 0 \]
Dividing the entire equation by 2:
\[ x_1^2 - 11x_1 + 30 = 0 \]

### Step 6: Factor the Quadratic Equation
Factoring the quadratic:
\[ (x_1 - 5)(x_1 - 6) = 0 \]
Thus, we have:
\[ x_1 = 5 \quad \text{or} \quad x_1 = 6 \]
Using Equation 1 to find \( x_2 \):
- If \( x_1 = 5 \), then \( x_2 = 11 - 5 = 6 \).
- If \( x_1 = 6 \), then \( x_2 = 11 - 6 = 5 \).

### Conclusion
The other two observations are \( x_1 = 5 \) and \( x_2 = 6 \).