The mean and variance of 7 observations are 7 and 22 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the remaining 2 observations are

Options

A. 4, 3
B. 2, 5
C. 6, 1
D. 4, 2

Answer

To find the remaining two observations given that the mean and variance of 7 observations are 7 and 22 respectively, and that 5 of the observations are 2, 4, 10, 12, and 14, we can follow these steps:

### Step 1: Calculate the sum of the observations
The mean of the observations is given by:

\[ \text{Mean} = \frac{\text{Sum of observations}}{n} \]

Where \( n \) is the number of observations. Here, \( n = 7 \) and the mean is 7.

Thus, we can calculate the sum of the observations:

\[ \text{Sum of observations} = \text{Mean} \times n = 7 \times 7 = 49 \]

### Step 2: Sum of the known observations
Next, we need to find the sum of the known observations:

\[ 2 + 4 + 10 + 12 + 14 = 42 \]

### Step 3: Set up the equation for the remaining observations
Let the remaining two observations be \( A \) and \( B \). We can set up the equation:

\[ 42 + A + B = 49 \]

This simplifies to:

\[ A + B = 49 - 42 = 7 \quad \text{(Equation 1)} \]

### Step 4: Calculate the variance
The variance is given by the formula:

\[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \text{Mean})^2 \]

Given that the variance is 22, we can write:

\[ 22 = \frac{1}{7} \left( (2 - 7)^2 + (4 - 7)^2 + (10 - 7)^2 + (12 - 7)^2 + (14 - 7)^2 + (A - 7)^2 + (B - 7)^2 \right) \]

### Step 5: Calculate the squared differences
Calculating the squared differences for the known observations:

\[ (2 - 7)^2 = 25, \quad (4 - 7)^2 = 9, \quad (10 - 7)^2 = 9, \quad (12 - 7)^2 = 25, \quad (14 - 7)^2 = 49 \]

Adding these:

\[ 25 + 9 + 9 + 25 + 49 = 117 \]

### Step 6: Set up the equation for variance
Now substituting back into the variance equation:

\[ 22 = \frac{1}{7} \left( 117 + (A - 7)^2 + (B - 7)^2 \right) \]

Multiplying both sides by 7:

\[ 154 = 117 + (A - 7)^2 + (B - 7)^2 \]

### Step 7: Solve for the squared differences
Rearranging gives:

\[ (A - 7)^2 + (B - 7)^2 = 154 - 117 = 37 \quad \text{(Equation 2)} \]

### Step 8: Substitute \( A + B \) into the squared differences
From Equation 1, we know \( A + B = 7 \). We can express \( B \) in terms of \( A \):

\[ B = 7 - A \]

Substituting this into Equation 2:

\[ (A - 7)^2 + ((7 - A) - 7)^2 = 37 \]

This simplifies to:

\[ (A - 7)^2 + (-A)^2 = 37 \]

Expanding gives:

\[ (A - 7)^2 + A^2 = 37 \]

### Step 9: Expand and simplify
Expanding \( (A - 7)^2 \):

\[ A^2 - 14A + 49 + A^2 = 37 \]

Combining like terms:

\[ 2A^2 - 14A + 49 - 37 = 0 \]

This simplifies to:

\[ 2A^2 - 14A + 12 = 0 \]

### Step 10: Solve the quadratic equation
Dividing through by 2:

\[ A^2 - 7A + 6 = 0 \]

Factoring gives:

\[ (A - 6)(A - 1) = 0 \]

Thus, \( A = 6 \) or \( A = 1 \).

### Step 11: Find corresponding \( B \)
If \( A = 6 \), then \( B = 1 \). If \( A = 1 \), then \( B = 6 \).

### Conclusion
The remaining two observations are \( 6 \) and \( 1 \).