The two lines of of regressions are `x+2y-5=0` and `2x+3y-8=0` and the variance of x is 12. Find the variance of y and the coefficient of correlation.

Answer

To solve the problem, we need to find the variance of \( y \) and the coefficient of correlation \( r \) given the two lines of regression and the variance of \( x \).

### Step-by-Step Solution:

1. **Identify the Regression Lines:**
The two lines of regression are given as:
\[ x + 2y - 5 = 0 \quad \text{(1)} \]
\[ 2x + 3y - 8 = 0 \quad \text{(2)} \]

2. **Convert the Regression Lines to Slope-Intercept Form:**
Rearranging equation (1):
\[ 2y = -x + 5 \implies y = -\frac{1}{2}x + \frac{5}{2} \]
Thus, the slope \( b_{yx} = -\frac{1}{2} \).

Rearranging equation (2):
\[ 3y = -2x + 8 \implies y = -\frac{2}{3}x + \frac{8}{3} \]
Thus, the slope \( b_{xy} = -\frac{2}{3} \).

3. **Calculate the Coefficient of Correlation \( r \):**
The relationship between the slopes of the regression lines and the coefficient of correlation is given by:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]
Substituting the values:
\[ r = \sqrt{\left(-\frac{1}{2}\right) \cdot \left(-\frac{2}{3}\right)} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]
Since both slopes are negative, the coefficient of correlation will also be negative:
\[ r = -\frac{1}{\sqrt{3}} \approx -0.577 \]

4. **Given Variance of \( x \):**
We know that the variance of \( x \) is given as:
\[ \sigma^2_x = 12 \]

5. **Use the Relationship Between Variance and Regression Coefficients:**
The relationship between the regression coefficients and the variances is given by:
\[ b_{yx} = r \frac{\sigma_y}{\sigma_x} \]
Rearranging gives:
\[ \sigma_y = \frac{b_{yx} \cdot \sigma_x}{r} \]

6. **Substituting the Known Values:**
We substitute \( b_{yx} = -\frac{1}{2} \), \( \sigma_x = \sqrt{12} = 2\sqrt{3} \), and \( r = -\frac{1}{\sqrt{3}} \):
\[ \sigma_y = \frac{-\frac{1}{2} \cdot 2\sqrt{3}}{-\frac{1}{\sqrt{3}}} \]
Simplifying:
\[ \sigma_y = \frac{\sqrt{3}}{1} = \sqrt{3} \]

7. **Calculate the Variance of \( y \):**
The variance of \( y \) is given by:
\[ \sigma^2_y = \sigma_y^2 = (\sqrt{3})^2 = 3 \]

### Final Results:
- Variance of \( y \): \( \sigma^2_y = 3 \)
- Coefficient of correlation: \( r = -\frac{1}{\sqrt{3}} \)