If one `b_(yx) and b_(xy)` numerically greater than one, the other numerically is

Options

A. grater than 1
B. less than 1
C. equals to r
D. None of these

Answer

To solve the problem, we need to analyze the relationship between the regression coefficients \( b_{yx} \) (the regression coefficient of \( y \) on \( x \)) and \( b_{xy} \) (the regression coefficient of \( x \) on \( y \)).

### Step-by-Step Solution:

1. **Understanding the Regression Coefficients**:
- The regression coefficient \( b_{yx} \) represents the change in \( y \) for a unit change in \( x \).
- The regression coefficient \( b_{xy} \) represents the change in \( x \) for a unit change in \( y \).

2. **Sign of the Coefficients**:
- Both \( b_{yx} \) and \( b_{xy} \) share the same sign because they are derived from the same dataset. If one is positive, the other must also be positive; if one is negative, the other must also be negative.

3. **Correlation Coefficient**:
- Let \( r \) be the correlation coefficient between \( x \) and \( y \). The relationship between the regression coefficients and the correlation coefficient is given by:
\[ r^2 = b_{yx} \cdot b_{xy} \]
- The value of \( r \) lies between -1 and 1, which implies:
\[ 0 \leq r^2 \leq 1 \]

4. **Analyzing the Product of Coefficients**:
- Since \( r^2 \) is always less than or equal to 1, we have:
\[ b_{yx} \cdot b_{xy} \leq 1 \]

5. **Case Analysis**:
- If \( b_{yx} > 1 \), then to satisfy \( b_{yx} \cdot b_{xy} \leq 1 \), it must be that \( b_{xy} < 1 \).
- Conversely, if \( b_{xy} > 1 \), then \( b_{yx} < 1 \).

6. **Conclusion**:
- Therefore, if one of the coefficients \( b_{yx} \) or \( b_{xy} \) is numerically greater than 1, the other must be numerically less than 1.

### Final Answer:
If one of \( b_{yx} \) and \( b_{xy} \) is numerically greater than one, then the other is numerically less than one.

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