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 use the information given about the mean and variance of the observations.

### Step 1: Understand the given information
We have 5 observations, and we know:
- Mean (x̄) = 4
- Variance (σ²) = 2.8
- Three of the observations are: 2, 2, and 5.

Let the other two observations be \( a \) and \( b \).

### Step 2: Calculate the total sum of the observations
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{N} \]
where \( N \) is the number of observations.

Given that the mean is 4 and there are 5 observations:
\[ 4 = \frac{2 + 2 + 5 + a + b}{5} \]
Multiplying both sides by 5:
\[ 20 = 2 + 2 + 5 + a + b \]
\[ 20 = 9 + a + b \]
Subtracting 9 from both sides:
\[ a + b = 11 \quad \text{(Equation 1)} \]

### Step 3: Use the variance to find another equation
The variance is given by the formula:
\[ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N} \]
Substituting the values we have:
\[ 2.8 = \frac{(2 - 4)^2 + (2 - 4)^2 + (5 - 4)^2 + (a - 4)^2 + (b - 4)^2}{5} \]
Calculating the squared differences:
\[ (2 - 4)^2 = 4, \quad (2 - 4)^2 = 4, \quad (5 - 4)^2 = 1 \]
So we have:
\[ 2.8 = \frac{4 + 4 + 1 + (a - 4)^2 + (b - 4)^2}{5} \]
Simplifying:
\[ 2.8 = \frac{9 + (a - 4)^2 + (b - 4)^2}{5} \]
Multiplying both sides by 5:
\[ 14 = 9 + (a - 4)^2 + (b - 4)^2 \]
Subtracting 9 from both sides:
\[ 5 = (a - 4)^2 + (b - 4)^2 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( a + b = 11 \)
2. \( (a - 4)^2 + (b - 4)^2 = 5 \)

From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 11 - a \]

Substituting \( b \) into Equation 2:
\[ (a - 4)^2 + ((11 - a) - 4)^2 = 5 \]
Simplifying:
\[ (a - 4)^2 + (7 - a)^2 = 5 \]
Expanding both squares:
\[ (a^2 - 8a + 16) + (49 - 14a + a^2) = 5 \]
Combining like terms:
\[ 2a^2 - 22a + 65 = 5 \]
Subtracting 5 from both sides:
\[ 2a^2 - 22a + 60 = 0 \]
Dividing the entire equation by 2:
\[ a^2 - 11a + 30 = 0 \]

### Step 5: Factor the quadratic equation
Factoring:
\[ (a - 6)(a - 5) = 0 \]
Thus, \( a = 6 \) or \( a = 5 \).

### Step 6: Find corresponding values of \( b \)
If \( a = 6 \):
\[ b = 11 - 6 = 5 \]
If \( a = 5 \):
\[ b = 11 - 5 = 6 \]

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