If the regression coefficients of X on Y and Y on X are `-0.4, and -0.9` respectively, then the correlation coefficient between X and Y is …………

Answer

To find the correlation coefficient \( r \) between \( X \) and \( Y \) given the regression coefficients \( b_{XY} \) and \( b_{YX} \), we can follow these steps:

### Step 1: Identify the given values
We are given:
- The regression coefficient of \( X \) on \( Y \) (denoted as \( b_{XY} \)) is \( -0.4 \).
- The regression coefficient of \( Y \) on \( X \) (denoted as \( b_{YX} \)) is \( -0.9 \).

### Step 2: Use the formula for correlation coefficient
The formula for the correlation coefficient \( r \) in terms of the regression coefficients is:
\[ r = \sqrt{b_{XY} \cdot b_{YX}} \]

### Step 3: Substitute the values into the formula
Substituting the given values into the formula:
\[ r = \sqrt{(-0.4) \cdot (-0.9)} \]

### Step 4: Calculate the product
Calculating the product:
\[ (-0.4) \cdot (-0.9) = 0.36 \]

### Step 5: Take the square root
Now, we take the square root of \( 0.36 \):
\[ r = \sqrt{0.36} = 0.6 \]

### Step 6: Determine the sign of \( r \)
Since both regression coefficients \( b_{XY} \) and \( b_{YX} \) are negative, the correlation coefficient \( r \) will also be negative. Therefore:
\[ r = -0.6 \]

### Final Answer
The correlation coefficient between \( X \) and \( Y \) is \( -0.6 \).

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