The geometric mean of negative regression coefficients is ______

Answer

To solve the question regarding the geometric mean of negative regression coefficients, we can follow these steps:

### Step 1: Understand the Regression Coefficients
The regression coefficients of \( y \) on \( x \) is denoted as \( b_{yx} \) and the regression coefficient of \( x \) on \( y \) is denoted as \( b_{xy} \).

### Step 2: Recall the Relationship Between Correlation and Regression Coefficients
The relationship between the correlation coefficient \( r \) and the regression coefficients is given by:
\[ r^2 = b_{xy} \cdot b_{yx} \]

### Step 3: Consider the Signs of the Coefficients
If both regression coefficients are negative, then both \( b_{xy} \) and \( b_{yx} \) are negative. This means that the correlation coefficient \( r \) will also be negative.

### Step 4: Express the Geometric Mean
The geometric mean of two numbers \( a \) and \( b \) is given by:
\[ \text{Geometric Mean} = \sqrt{ab} \]
In our case, we want to find the geometric mean of \( b_{xy} \) and \( b_{yx} \):
\[ \text{Geometric Mean} = \sqrt{b_{xy} \cdot b_{yx}} \]

### Step 5: Substitute the Relationship with Correlation Coefficient
From the relationship established in Step 2, we can substitute \( b_{xy} \cdot b_{yx} \) with \( r^2 \):
\[ \text{Geometric Mean} = \sqrt{b_{xy} \cdot b_{yx}} = \sqrt{r^2} = |r| \]

### Step 6: Determine the Sign
Since both regression coefficients are negative, the geometric mean will also be negative. Therefore, we can express the geometric mean of the negative regression coefficients as:
\[ \text{Geometric Mean} = -|r| \]

### Final Answer
Thus, the geometric mean of the negative regression coefficients is:
\[ -\sqrt{b_{xy} \cdot b_{yx}} = -|r| \]