The mean and variance of the 4 observations `3,7,x,y ("where" x gt y)` is 5 and 10 respectively . the mean of the observation `4+x+y , 7+x , x+y , x-y` will be

Options

A. 10
B. 12
C. 8
D. 14

Answer

To solve the problem step by step, we need to find the values of \( x \) and \( y \) based on the given mean and variance, and then calculate the mean of the new set of observations.

### Step 1: Set up the equations for mean and variance

Given the observations: \( 3, 7, x, y \)

1. **Mean Calculation**:
\[ \text{Mean} = \frac{3 + 7 + x + y}{4} = 5 \]
Simplifying this:
\[ 10 + x + y = 20 \implies x + y = 10 \quad \text{(Equation 1)} \]

2. **Variance Calculation**:
The variance is given as 10. The formula for variance is:
\[ \sigma^2 = \frac{(3^2 + 7^2 + x^2 + y^2)}{4} - \left(\frac{3 + 7 + x + y}{4}\right)^2 \]
Plugging in the known values:
\[ 10 = \frac{9 + 49 + x^2 + y^2}{4} - 5^2 \]
Simplifying this:
\[ 10 = \frac{58 + x^2 + y^2}{4} - 25 \]
\[ 10 + 25 = \frac{58 + x^2 + y^2}{4} \]
\[ 35 = \frac{58 + x^2 + y^2}{4} \]
Multiplying through by 4:
\[ 140 = 58 + x^2 + y^2 \]
\[ x^2 + y^2 = 82 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations

Now we have two equations:
1. \( x + y = 10 \)
2. \( x^2 + y^2 = 82 \)

Using Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 10 - x \]

Substituting into Equation 2:
\[ x^2 + (10 - x)^2 = 82 \]
Expanding this:
\[ x^2 + (100 - 20x + x^2) = 82 \]
\[ 2x^2 - 20x + 100 - 82 = 0 \]
\[ 2x^2 - 20x + 18 = 0 \]
Dividing through by 2:
\[ x^2 - 10x + 9 = 0 \]
Factoring:
\[ (x - 1)(x - 9) = 0 \]
Thus, \( x = 1 \) or \( x = 9 \). Since \( x > y \), we take \( x = 9 \) and substituting back:
\[ y = 10 - 9 = 1 \]

### Step 3: Calculate the mean of the new observations

Now we need to find the mean of the observations \( 4 + x + y, 7 + x, x + y, x - y \):
1. \( 4 + x + y = 4 + 9 + 1 = 14 \)
2. \( 7 + x = 7 + 9 = 16 \)
3. \( x + y = 9 + 1 = 10 \)
4. \( x - y = 9 - 1 = 8 \)

Now, we calculate the mean:
\[ \text{Mean} = \frac{14 + 16 + 10 + 8}{4} = \frac{48}{4} = 12 \]

### Final Answer
The mean of the observations \( 4 + x + y, 7 + x, x + y, x - y \) is **12**.