Given regression co-efficient of y on x is 0.4 and r(x,y) = `(2)/( sqrt(10))` , then the regression co-efficient of x on y is

Options

A. 1
B. `0.2`
C. `sqrt(10)`
D. 2

Answer

To find the regression coefficient of \( x \) on \( y \) given the regression coefficient of \( y \) on \( x \) and the correlation coefficient, we can follow these steps:

### Step 1: Understand the given values
We are given:
- The regression coefficient of \( y \) on \( x \) (denoted as \( b_{y,x} \)) is \( 0.4 \).
- The correlation coefficient \( r(x,y) \) is \( \frac{2}{\sqrt{10}} \).

### Step 2: Use the relationship between regression coefficients and correlation coefficient
The relationship between the regression coefficients and the correlation coefficient is given by the formula:
\[ b_{y,x} \cdot b_{x,y} = r^2 \]
Where:
- \( b_{y,x} \) is the regression coefficient of \( y \) on \( x \).
- \( b_{x,y} \) is the regression coefficient of \( x \) on \( y \).
- \( r \) is the correlation coefficient.

### Step 3: Calculate \( r^2 \)
First, we need to find \( r^2 \):
\[ r = \frac{2}{\sqrt{10}} \implies r^2 = \left(\frac{2}{\sqrt{10}}\right)^2 = \frac{4}{10} = 0.4 \]

### Step 4: Substitute the known values into the equation
Now, we can substitute \( b_{y,x} \) and \( r^2 \) into the equation:
\[ 0.4 \cdot b_{x,y} = 0.4 \]

### Step 5: Solve for \( b_{x,y} \)
To find \( b_{x,y} \), we can rearrange the equation:
\[ b_{x,y} = \frac{0.4}{0.4} = 1 \]

### Conclusion
The regression coefficient of \( x \) on \( y \) is \( 1 \).