If `b_(yx) lt0` and `b_(xy) lt0` then `r` is _____

Options

A. `a lt0`
B. `b gt0`
C. `c =0`
D. `d gt1`

Answer

To solve the problem, we need to analyze the implications of the given regression coefficients \( b_{yx} < 0 \) and \( b_{xy} < 0 \).

### Step-by-Step Solution:

1. **Understanding Regression Coefficients**:
- The regression coefficient \( b_{yx} \) represents the relationship between the dependent variable \( y \) and the independent variable \( x \).
- The regression coefficient \( b_{xy} \) represents the relationship between the dependent variable \( x \) and the independent variable \( y \).

2. **Given Conditions**:
- We are given that \( b_{yx} < 0 \) and \( b_{xy} < 0 \). This indicates that both regression coefficients are negative.

3. **Correlation Coefficient**:
- The correlation coefficient \( r \) can be expressed in terms of the regression coefficients:
\[ r^2 = b_{xy} \cdot b_{yx} \]
- Since both \( b_{xy} \) and \( b_{yx} \) are negative, their product \( b_{xy} \cdot b_{yx} \) will be positive.

4. **Determining the Sign of \( r \)**:
- The correlation coefficient \( r \) can be either positive or negative. However, since both regression coefficients are negative, we can conclude that:
\[ r = -\sqrt{b_{xy} \cdot b_{yx}} \]
- This indicates that \( r \) must be negative because it is the negative square root of a positive value.

5. **Conclusion**:
- Therefore, since \( r \) is derived from the negative square root of a positive product, we conclude that:
\[ r < 0 \]

### Final Answer:
Thus, \( r \) is **less than zero**.

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