It is given that `X=10, y=90 sigma_(x) =3. sigma_(y)=12` and `r_(xy) = 0.8` Then regression equation of X on Y is

Options

A. Y=3.2X+58
B. X=3.2Y+58
C. X=-8+0.2Y
D. Y=-8+0.2X

Answer

To find the regression equation of X on Y, we will use the formula for the regression line, which is given by:

\[ x - \bar{x} = r_{xy} \cdot \frac{\sigma_x}{\sigma_y} \cdot (y - \bar{y}) \]

Where:
- \( \bar{x} \) is the mean of X
- \( \bar{y} \) is the mean of Y
- \( \sigma_x \) is the standard deviation of X
- \( \sigma_y \) is the standard deviation of Y
- \( r_{xy} \) is the correlation coefficient between X and Y

### Step 1: Identify the given values
From the question, we have:
- \( \bar{x} = 10 \)
- \( \bar{y} = 90 \)
- \( \sigma_x = 3 \)
- \( \sigma_y = 12 \)
- \( r_{xy} = 0.8 \)

### Step 2: Substitute the values into the regression equation
Now, we substitute the given values into the regression equation:

\[ x - 10 = 0.8 \cdot \frac{3}{12} \cdot (y - 90) \]

### Step 3: Simplify the equation
First, calculate \( \frac{3}{12} \):

\[ \frac{3}{12} = \frac{1}{4} \]

Now substitute this back into the equation:

\[ x - 10 = 0.8 \cdot \frac{1}{4} \cdot (y - 90) \]

Now, calculate \( 0.8 \cdot \frac{1}{4} \):

\[ 0.8 \cdot \frac{1}{4} = 0.2 \]

So, we have:

\[ x - 10 = 0.2 \cdot (y - 90) \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to solve for \( x \):

\[ x = 10 + 0.2 \cdot (y - 90) \]

### Step 5: Distributing the 0.2
Distributing \( 0.2 \):

\[ x = 10 + 0.2y - 0.2 \cdot 90 \]

Calculating \( 0.2 \cdot 90 \):

\[ 0.2 \cdot 90 = 18 \]

So, we have:

\[ x = 10 + 0.2y - 18 \]

### Step 6: Final simplification
Now, simplify the equation:

\[ x = 0.2y - 8 \]

### Final Result
The regression equation of X on Y is:

\[ x = 0.2y - 8 \]