When one regression coefficient is positive, the other would also be ………..

Answer

To solve the question "When one regression coefficient is positive, the other would also be ………..", we can follow these steps:

### Step 1: Understand the relationship between regression coefficients and correlation coefficient
Regression coefficients are used to describe the relationship between two variables. The key property to note is that the sign of the regression coefficients is the same as the sign of the correlation coefficient.

**Hint:** Recall that correlation measures the strength and direction of a linear relationship between two variables.

### Step 2: Recall the formula for the correlation coefficient
The formula for the correlation coefficient (r) in terms of regression coefficients is given by:
\[ r = \sqrt{b_{xy} \cdot b_{yx}} \]
where \( b_{xy} \) is the regression coefficient of Y on X, and \( b_{yx} \) is the regression coefficient of X on Y.

**Hint:** Remember that the square root of a product is defined only when both factors are either positive or both are negative.

### Step 3: Analyze the implications of one regression coefficient being positive
If one regression coefficient (say \( b_{xy} \)) is positive, then for the product \( b_{xy} \cdot b_{yx} \) to be non-negative (which is necessary for the square root to be defined), the other regression coefficient \( b_{yx} \) must also be positive.

**Hint:** Think about how multiplying two numbers can yield a positive result.

### Step 4: Conclude the relationship
Since both regression coefficients must have the same sign for the correlation coefficient to be defined, if one regression coefficient is positive, the other must also be positive.

**Hint:** This conclusion is based on the properties of multiplication and the definition of the correlation coefficient.

### Final Answer
When one regression coefficient is positive, the other would also be positive.