Question

if bxy = +0.80, the value of byx can be

• A. +1.25
• B. -1.25
• C. +1.26
• D. -1.24

Hint:

Regression Coefficient Rules: 1. $b_{xy}$ and $b_{yx}$ MUST have the SAME sign. 2. $b_{xy} \times b_{yx} = r^2 \le 1$. If $b_{xy} = 0.80$, then $b_{yx} \le 1/0.80 = +1.25$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Mathematical properties of linear regression coefficients: both regression coefficients ($b_{xy}$ and $b_{yx}$) must share the same mathematical sign, and their geometric product cannot exceed unity ($b_{xy} \cdot b_{yx} = r^2 \le 1$).
Key Formula or Approach:
\[ r^2 = b_{xy} \cdot b_{yx} \le 1 \implies b_{yx} \le \frac{1}{b_{xy}} \quad \text{and} \quad \text{Sign}(b_{yx}) = \text{Sign}(b_{xy}) = \text{Sign}(r) \]

Step 2: Detailed Explanation:
Evaluating the mathematical properties of regression coefficients:
1. Same Sign Rule: Given \( b_{xy} = +0.80 \) (positive), \( b_{yx} \) must also be positive (ruling out negative options $-1.25$ and $-1.24$).
2. Coefficient Bound Rule: The product of regression coefficients equals the coefficient of determination ($r^2$), which must satisfy $r^2 \le 1$:
\[ b_{xy} \cdot b_{yx} \le 1 \implies 0.80 \cdot b_{yx} \le 1 \implies b_{yx} \le \frac{1}{0.80} = 1.25 \]
3. Testing candidates:
- For \( b_{yx} = +1.25 \): \( r^2 = 0.80 \times 1.25 = 1.00 \le 1 \) (Valid, representing perfect linear correlation $r = +1.0$).
- For \( b_{yx} = +1.26 \): \( r^2 = 0.80 \times 1.26 = 1.008 > 1 \) (Mathematically impossible).

Step 3: Final Answer:
Therefore, the value of byx can be +1.25, corresponding to option (A).