Question

The pair of values representing two correct regression coefficients in case of simple linear regression is

• A. $(-3, 1/2)$
• B. $(3, 1/2)$
• C. $(4/3, 1/2)$
• D. $(4/3, -1/2)$

Hint:

If one of the regression coefficients is greater than 1, the other coefficient must be less than 1 so that their product remains less than or equal to 1.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The two regression coefficients in simple linear regression, $b_{yx}$ and $b_{xy}$, must satisfy specific mathematical properties.
Key Formula or Approach:
The properties of regression coefficients are:
1. Both coefficients must have the same sign (either both positive or both negative).
2. The product of the two coefficients is equal to the square of the correlation coefficient:
\[ b_{yx} \cdot b_{xy} = r^2 \]
Since $0 \le r^2 \le 1$, the product of the coefficients must be bounded:
\[ 0 \le b_{yx} \cdot b_{xy} \le 1 \]

Step 2: Detailed Explanation:
Let us test each option against these properties:
- (A) $(-3, 1/2)$: The signs are different (one negative, one positive). This is impossible.
- (D) $(4/3, -1/2)$: The signs are different. This is impossible.
- (B) $(3, 1/2)$: Both are positive, but let us check their product:
\[ 3 \cdot \frac{1}{2} = 1.5 > 1 \]
Since $r^2$ cannot exceed 1, this pair is impossible.
- (C) $(4/3, 1/2)$: Both are positive, let us check their product:
\[ \frac{4}{3} \cdot \frac{1}{2} = \frac{4}{6} = \frac{2}{3} \approx 0.67 \]
Since $0 \le 0.67 \le 1$, this represents a valid pair of regression coefficients.
Therefore, the correct pair of values is $(4/3, 1/2)$.

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