If `barx =18, bary=100`, Var(X)=196, Var(Y) = 400 and p(X,Y) = 0.8, find the regression lines. Extimate the value of y when x = 70 and that of x when y = 90.

Answer

To solve the problem step by step, we will find the regression lines and estimate the required values.

### Given Data:
- Mean of X, \( \bar{x} = 18 \)
- Mean of Y, \( \bar{y} = 100 \)
- Variance of X, \( \text{Var}(X) = 196 \) (thus, \( \sigma_x = \sqrt{196} = 14 \))
- Variance of Y, \( \text{Var}(Y) = 400 \) (thus, \( \sigma_y = \sqrt{400} = 20 \))
- Correlation coefficient, \( r = 0.8 \)

### Step 1: Find the Regression Coefficient \( b_{yx} \)
The regression coefficient of Y on X is given by:

\[ b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} \]

Substituting the known values:

\[ b_{yx} = 0.8 \cdot \frac{20}{14} = 0.8 \cdot 1.4286 \approx 1.1428 \]

### Step 2: Write the Regression Equation of Y on X
The regression line of Y on X is given by:

\[ y - \bar{y} = b_{yx} (x - \bar{x}) \]

Substituting the values:

\[ y - 100 = 1.1428 (x - 18) \]

Expanding this:

\[ y - 100 = 1.1428x - 20.5714 \]

Thus,

\[ y = 1.1428x + 79.4286 \]

### Step 3: Estimate the Value of Y when X = 70
Substituting \( x = 70 \) into the regression equation:

\[ y = 1.1428(70) + 79.4286 \]

Calculating:

\[ y = 80.996 + 79.4286 \approx 160.424 \]

### Step 4: Find the Regression Coefficient \( b_{xy} \)
The regression coefficient of X on Y is given by:

\[ b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y} \]

Substituting the known values:

\[ b_{xy} = 0.8 \cdot \frac{14}{20} = 0.8 \cdot 0.7 = 0.56 \]

### Step 5: Write the Regression Equation of X on Y
The regression line of X on Y is given by:

\[ x - \bar{x} = b_{xy} (y - \bar{y}) \]

Substituting the values:

\[ x - 18 = 0.56 (y - 100) \]

Expanding this:

\[ x - 18 = 0.56y - 56 \]

Thus,

\[ x = 0.56y - 38 \]

### Step 6: Estimate the Value of X when Y = 90
Substituting \( y = 90 \) into the regression equation:

\[ x = 0.56(90) - 38 \]

Calculating:

\[ x = 50.4 - 38 = 12.4 \]

### Final Results:
- The regression equation of Y on X: \( y = 1.1428x + 79.4286 \)
- The estimated value of Y when X = 70: \( y \approx 160.424 \)
- The regression equation of X on Y: \( x = 0.56y - 38 \)
- The estimated value of X when Y = 90: \( x = 12.4 \)