Question

Two lines of regression intersect at

• A. $(\bar{X}, \bar{Y})$ where $\bar{X}, \bar{Y}$ are the mean of the two random variable $X$ and $Y$
• B. (0,0)
• C. (1,1)
• D. The two lines are always parallel. They never intersect with each other.

Hint:

If the two variables $X$ and $Y$ are perfectly correlated ($r = \pm 1$), the two regression lines coincide completely. If they are uncorrelated ($r = 0$), the lines are perpendicular, intersecting perpendicularly at $(\bar{X}, \bar{Y})$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In simple linear correlation, we define two distinct regression lines: the regression of $Y$ on $X$ and the regression of $X$ on $Y$.

Step 2: Detailed Explanation:
Let us write the equations of the two regression lines in their standard forms:
1. Regression line of $Y$ on $X$:
\[ Y - \bar{Y} = b_{yx} (X - \bar{X}) \]
2. Regression line of $X$ on $Y$:
\[ X - \bar{X} = b_{xy} (Y - \bar{Y}) \]
Let us substitute the point $(X = \bar{X}, Y = \bar{Y})$ into both equations:
- For line 1: $\bar{Y} - \bar{Y} = b_{yx} (\bar{X} - \bar{X}) \implies 0 = 0$, which is true.
- For line 2: $\bar{X} - \bar{X} = b_{xy} (\bar{Y} - \bar{Y}) \implies 0 = 0$, which is also true.
Because the point $(\bar{X}, \bar{Y})$ satisfies both regression equations, it lies on both lines.
Therefore, the two regression lines always intersect at the mean point $(\bar{X}, \bar{Y})$.

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