If the lines of regreasion of y on x and x on y makes angle `30 ^(@) and 60^(@) ` respectively with positive direction of x - axis the correlation coefficient between x on y is

Options

A. ` +-(1)/(3)`
B. ` sqrt3 `
C. ` +-(1)/(sqrt3) `
D. ` ( 1)/(sqrt3)`

Answer

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

### Step 1: Understand the relationships
The lines of regression of \( y \) on \( x \) and \( x \) on \( y \) are represented by their slopes. The slope of the regression line of \( y \) on \( x \) is given by:
\[ b_{xy} = \tan(\theta_{xy}) \]
where \( \theta_{xy} \) is the angle made by the regression line of \( y \) on \( x \) with the positive direction of the x-axis.

Similarly, the slope of the regression line of \( x \) on \( y \) is given by:
\[ b_{yx} = \tan(\theta_{yx}) \]
where \( \theta_{yx} \) is the angle made by the regression line of \( x \) on \( y \) with the positive direction of the x-axis.

### Step 2: Calculate the slopes
Given:
- \( \theta_{xy} = 30^\circ \)
- \( \theta_{yx} = 60^\circ \)

Now, we calculate the slopes:
\[ b_{xy} = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
\[ b_{yx} = \tan(60^\circ) = \sqrt{3} \]

### Step 3: Relationship between slopes and correlation coefficient
The relationship between the slopes and the correlation coefficient \( r \) is given by:
\[ b_{xy} \cdot b_{yx} = r^2 \]

### Step 4: Substitute the values
Substituting the values we found:
\[ \left(\frac{1}{\sqrt{3}}\right) \cdot \left(\sqrt{3}\right) = r^2 \]
\[ 1 = r^2 \]

### Step 5: Solve for \( r \)
Taking the square root of both sides gives:
\[ r = \pm 1 \]

### Conclusion
The correlation coefficient between \( x \) and \( y \) is \( \pm 1 \).