If the sign of the correlation coefficient is negative, then the sign of the slope of the respective regression line is _____

Answer

To solve the problem, we need to analyze the relationship between the correlation coefficient and the slope of the regression line.

### Step-by-Step Solution:

1. **Understanding Correlation Coefficient (r)**:
The correlation coefficient (denoted as r) measures the strength and direction of a linear relationship between two variables. If r is negative, it indicates an inverse relationship between the variables.

**Hint**: Recall that a negative correlation means that as one variable increases, the other variable tends to decrease.

2. **Regression Coefficients**:
There are two regression coefficients:
- \( b_{xy} \): slope of the regression line of y on x.
- \( b_{yx} \): slope of the regression line of x on y.

**Hint**: Remember that regression coefficients represent the change in one variable in response to a change in another variable.

3. **Relationship Between Correlation and Regression Coefficients**:
The relationship between the correlation coefficient and the regression coefficients can be expressed as:
\[ r^2 = b_{xy} \cdot b_{yx} \]
The sign of r gives us information about the signs of the regression coefficients.

**Hint**: Think about how the product of two numbers can be negative.

4. **Analyzing the Signs**:
- If \( r \) is negative, then \( r^2 \) is positive.
- For the product \( b_{xy} \cdot b_{yx} \) to be positive (since \( r^2 \) is positive), both \( b_{xy} \) and \( b_{yx} \) must either be both positive or both negative.
- If \( r \) is negative, it implies that both regression coefficients must be negative.

**Hint**: Consider the implications of a negative product and how it relates to the signs of the individual components.

5. **Conclusion**:
Since we established that if the correlation coefficient \( r \) is negative, then both regression coefficients \( b_{xy} \) and \( b_{yx} \) must be negative. Therefore, the slope of the respective regression line is negative.

**Final Answer**: The sign of the slope of the respective regression line is **negative**.