Question

If regression line of X on Y is linear, then the regression line of Y on X is:

• A. Linear
• B. Curvilinear
• C. Polynomial
• D. Linear or curvilinear depending on nature of data

Hint:

Exam Tip:
In simple linear regression, both regression lines are linear: \[ X = a + bY \quad \text{and} \quad Y = c + dX \] The coefficients are related by \(b \cdot d = r^2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
If the regression of X on Y is linear, it means the conditional expectation \(E(X \mid Y)\) is a linear function of Y.
For bivariate normal distributions, both regressions are linear.
In general, if one regression is linear, the other need not be linear.
However, in standard regression theory, when we talk about regression lines, we are usually referring to the least squares linear regression lines.
Both regression lines are linear by definition.
The regression line of X on Y is linear if the data are such that the linear fit is appropriate.
The regression line of Y on X is also linear.
So, if the regression of X on Y is linear, the regression of Y on X is also linear.

Step 2: Analyzing the Options:

• (A) Linear: True.
• (B) Curvilinear: False.
• (C) Polynomial: False.
• (D) Linear or curvilinear depending on nature of data: False.

Step 3: Final Answer:
Therefore, option (A) is correct.