For two variables x and y, the two regression lines are `x+2y-5=0, 2x+3y-8=0` and variance `(x)=12`. Then `sigma_(y)` is

Options

A. 4
B. 5
C. 2
D. none of these

Answer

To solve the problem step by step, we will find the variance of \( y \) given the regression lines and the variance of \( x \).

### Step 1: Identify the regression equations
The two regression lines given are:
1. \( x + 2y - 5 = 0 \)
2. \( 2x + 3y - 8 = 0 \)

### Step 2: Rearrange the regression equations to find the regression coefficients
From the first equation, we can express \( y \) in terms of \( x \):
\[ 2y = 5 - x \implies y = -\frac{1}{2}x + \frac{5}{2} \]
This gives us the regression coefficient \( b_{yx} = -\frac{1}{2} \).

From the second equation, we can express \( x \) in terms of \( y \):
\[ 2x = 8 - 3y \implies x = -\frac{3}{2}y + 4 \]
This gives us the regression coefficient \( b_{xy} = -\frac{3}{2} \).

### Step 3: Calculate the correlation coefficient \( r \)
The relationship between the regression coefficients and the correlation coefficient \( r \) is given by:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]
Substituting the values we found:
\[ r = \sqrt{\left(-\frac{1}{2}\right) \cdot \left(-\frac{3}{2}\right)} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 4: Use the formula for the relationship between variances and the correlation coefficient
We know that:
\[ b_{xy} = r \cdot \frac{\sigma_y}{\sigma_x} \]
We are given that \( \text{Var}(x) = 12 \), thus \( \sigma_x = \sqrt{12} = 2\sqrt{3} \).

Substituting the values into the equation:
\[ -\frac{3}{2} = \left(\frac{\sqrt{3}}{2}\right) \cdot \frac{\sigma_y}{2\sqrt{3}} \]

### Step 5: Solve for \( \sigma_y \)
Rearranging the equation:
\[ -\frac{3}{2} = \frac{\sqrt{3}}{4} \sigma_y \]
Multiplying both sides by \( 4 \):
\[ -6 = \sqrt{3} \sigma_y \]
Dividing both sides by \( \sqrt{3} \):
\[ \sigma_y = -\frac{6}{\sqrt{3}} = -2\sqrt{3} \]

Since standard deviation cannot be negative, we take the positive value:
\[ \sigma_y = 2 \]

### Final Answer
Thus, the value of \( \sigma_y \) is \( 2 \).

---