It is given that `barX=10, barY=90, sigma_(X)=3, sigma_(Y)=12 and r_(XY)=0.8.` The 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`

Explanation

To find the regression equation of X on Y, we can use the formula:

\[ X - \bar{X} = r_{XY} \cdot \frac{\sigma_X}{\sigma_Y} \cdot (Y - \bar{Y}) \]

Where:
- \(\bar{X} = 10\) (mean of X)
- \(\bar{Y} = 90\) (mean of Y)
- \(\sigma_X = 3\) (standard deviation of X)
- \(\sigma_Y = 12\) (standard deviation of Y)
- \(r_{XY} = 0.8\) (correlation coefficient between X and Y)

### Step 1: Substitute the values into the formula

Substituting the given values into the regression equation formula:

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

### Step 2: Simplify the fraction

Calculate \(\frac{3}{12}\):

\[ \frac{3}{12} = 0.25 \]

Now substitute this back into the equation:

\[ X - 10 = 0.8 \cdot 0.25 \cdot (Y - 90) \]

### Step 3: Calculate \(0.8 \cdot 0.25\)

Now calculate \(0.8 \cdot 0.25\):

\[ 0.8 \cdot 0.25 = 0.2 \]

Now substitute this back into the equation:

\[ X - 10 = 0.2 \cdot (Y - 90) \]

### Step 4: Distribute \(0.2\)

Distributing \(0.2\) gives:

\[ X - 10 = 0.2Y - 18 \]

### Step 5: Solve for X

Now, add \(10\) to both sides to solve for \(X\):

\[ X = 0.2Y - 18 + 10 \]

This simplifies to:

\[ X = 0.2Y - 8 \]

### Final Regression Equation

Thus, the regression equation of X on Y is:

\[ X = 0.2Y - 8 \]

---