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 solve the problem, we need to find the two remaining observations given the mean and variance of a set of observations. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
We have:
- Mean of 7 observations = 7
- Variance of 7 observations = 22
- 5 observations = 2, 4, 10, 12, 14

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

### Step 2: Set Up the Equation for Mean
The mean is calculated as follows:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Thus, we can write:
\[ 7 = \frac{2 + 4 + 10 + 12 + 14 + a + b}{7} \]
Calculating the sum of the known observations:
\[ 2 + 4 + 10 + 12 + 14 = 42 \]
Substituting this into the mean equation gives:
\[ 7 = \frac{42 + a + b}{7} \]
Multiplying both sides by 7:
\[ 49 = 42 + a + b \]
Rearranging gives us:
\[ a + b = 49 - 42 = 7 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Equation for Variance
The variance is given by:
\[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
For our case:
\[ 22 = \frac{1}{7} \left( (2 - 7)^2 + (4 - 7)^2 + (10 - 7)^2 + (12 - 7)^2 + (14 - 7)^2 + (a - 7)^2 + (b - 7)^2 \right) \]
Calculating the squared differences:
\[ (2 - 7)^2 = 25, \quad (4 - 7)^2 = 9, \quad (10 - 7)^2 = 9, \quad (12 - 7)^2 = 25, \quad (14 - 7)^2 = 49 \]
Summing these gives:
\[ 25 + 9 + 9 + 25 + 49 = 117 \]
Substituting into the variance equation:
\[ 22 = \frac{1}{7} (117 + (a - 7)^2 + (b - 7)^2) \]
Multiplying both sides by 7:
\[ 154 = 117 + (a - 7)^2 + (b - 7)^2 \]
Rearranging gives:
\[ (a - 7)^2 + (b - 7)^2 = 154 - 117 = 37 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
From Equation 1, we have:
\[ a + b = 7 \]
From Equation 2, we can expand:
\[ (a - 7)^2 + (b - 7)^2 = (a^2 - 14a + 49) + (b^2 - 14b + 49) = 37 \]
Substituting \( b = 7 - a \) into the equation:
\[ (a - 7)^2 + ((7 - a) - 7)^2 = 37 \]
This simplifies to:
\[ (a - 7)^2 + (-a)^2 = 37 \]
Expanding gives:
\[ (a - 7)^2 + a^2 = 37 \]
\[ a^2 - 14a + 49 + a^2 = 37 \]
Combining like terms:
\[ 2a^2 - 14a + 49 - 37 = 0 \]
\[ 2a^2 - 14a + 12 = 0 \]
Dividing by 2:
\[ a^2 - 7a + 6 = 0 \]
Factoring gives:
\[ (a - 1)(a - 6) = 0 \]
Thus, \( a = 1 \) or \( a = 6 \).

### Step 5: Find Corresponding Values of \( b \)
If \( a = 1 \), then \( b = 7 - 1 = 6 \).
If \( a = 6 \), then \( b = 7 - 6 = 1 \).

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