If the angle between two lines regression is `90^(@)`, then it represents

Options

A. a perfect positive correlation
B. perfect negative correlation
C. no linear correlation i.e, r=0
D. None of these

Answer

To solve the question, we need to analyze the relationship between the angle between two regression lines and the correlation coefficient. Here’s a step-by-step solution:

### Step 1: Understand the relationship between angle and correlation
The angle \( \theta \) between two lines of regression is related to the correlation coefficient \( r \) between the two variables. The formula for the tangent of the angle \( \theta \) is given by:

\[ \tan \theta = \frac{1 - r^2}{r \cdot \frac{\sigma_x}{\sigma_y}} \]

Where \( \sigma_x \) and \( \sigma_y \) are the standard deviations of the variables \( x \) and \( y \).

### Step 2: Set the angle to 90 degrees
Given that the angle \( \theta \) is \( 90^\circ \), we know that:

\[ \tan 90^\circ = \infty \]

This implies that the expression on the right side must also approach infinity.

### Step 3: Analyze the implications for \( r \)
For \( \tan \theta \) to be infinite, the denominator of the right-hand side must equal zero. This occurs when:

\[ r = 0 \]

### Step 4: Interpret the result
A correlation coefficient \( r = 0 \) indicates that there is no linear correlation between the two variables. Therefore, if the angle between the two regression lines is \( 90^\circ \), it represents that there is no linear correlation between the variables.

### Conclusion
Thus, if the angle between two lines of regression is \( 90^\circ \), it represents that there is no linear correlation between the two variables.