Question

If the regression line is given by $X = 2 + 4Y$, then the correlation coefficient ($r$) is

• A. $1/4$
• B. $4$
• C. positive
• D. More than one

Hint:

The correlation coefficient \( r \), the slope of the regression of Y on X (\( b_{yx} \)), and the slope of the regression of X on Y (\( b_{xy} \)) always share the same sign.
Since the slope here is +4, \( r \) must be positive.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Linear regression and correlation are closely related.
The correlation coefficient (\( r \)) measures the strength and direction of the linear relationship between two variables, while the regression coefficients (\( b_{yx} \) and \( b_{xy} \)) describe the slopes of the regression lines.
Key Formula or Approach:
The relationship between the correlation coefficient and the regression coefficients is given by:
\[ r^2 = b_{yx} \cdot b_{xy} \]
Additionally, the correlation coefficient \( r \) must have the same algebraic sign (+ or -) as both of the regression slopes \( b_{yx} \) and \( b_{xy} \).

Step 2: Detailed Explanation:
Let us analyze the given regression line of $X$ on $Y$:
\[ X = 2 + 4Y \]
- The slope of this line is the regression coefficient of $X$ on $Y$, denoted as \( b_{xy} = +4 \).
- Because \( b_{xy} \) is positive, the regression coefficient of $Y$ on $X$ (\( b_{yx} \)) must also be positive, and therefore the correlation coefficient \( r \) must be positive.
- We can rule out the other options based on the properties of the correlation coefficient:
- The value of \( r \) must lie in the range \( [-1, 1] \). Therefore, option (B) (\( r = 4 \)) and option (D) (\( r > 1 \)) are mathematically impossible.
- Option (A) (\( r = 1/4 \)) cannot be confirmed because we do not know the value of \( b_{yx} \) (since \( r = \sqrt{b_{yx} \cdot 4} \)).
- Thus, the only certain statement is that the correlation coefficient is positive.

Step 3: Final Answer:
The correlation coefficient \( r \) is positive.