Question

Given below are two statements
Statement I: If the variables are perfectly correlated the regression lines coincide.
Statement II: If the variables are not correlated, the regression lines are parallel to catch other.
In light of the above statements, choose the most appropriate answer from the options given below

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Hint:

- Perfect correlation (\(r = \pm 1\)) \(\rightarrow\) Lines coincide (identical).
- No correlation (\(r = 0\)) \(\rightarrow\) Lines are perpendicular (intersect at \(90^\circ\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Regression lines represent the average relationship between two variables.
The angle between the two regression lines (Y on X and X on Y) depends directly on the degree of correlation (\(r\)) between them.

Step 2: Detailed Explanation:
- Statement I: If two variables are perfectly correlated (\(r = \pm 1\)), the angle between the two regression lines becomes \(0^\circ\).
This means the two regression lines are identical and coincide with each other.
Thus, Statement I is correct.
- Statement II: If the variables are not correlated (\(r = 0\)), the angle between the two regression lines is \(90^\circ\).
They are perpendicular to each other, cutting at right angles at their mean intersection point \((\bar{X}, \bar{Y})\).
They are not parallel.
Thus, Statement II is incorrect.

Step 3: Final Answer:
Statement I is correct but Statement II is incorrect, which corresponds to option (C).