If `barx=18,bary=100, sigma_(x)=14,sigma_(y)=20` and correlation `r_(xy)=0.8`, find the regression equation of y on x.

Answer

To find the regression equation of \( y \) on \( x \), we will follow these steps:

### Step 1: Write down the regression equation formula
The regression equation of \( y \) on \( x \) can be expressed as:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
where \( b_{yx} \) is the regression coefficient of \( y \) on \( x \), \( \bar{y} \) is the mean of \( y \), and \( \bar{x} \) is the mean of \( x \).

### Step 2: Calculate the regression coefficient \( b_{yx} \)
The regression coefficient \( b_{yx} \) can be calculated using the formula:
\[ b_{yx} = r_{xy} \cdot \frac{\sigma_y}{\sigma_x} \]
where:
- \( r_{xy} \) is the correlation coefficient,
- \( \sigma_y \) is the standard deviation of \( y \),
- \( \sigma_x \) is the standard deviation of \( x \).

Given:
- \( r_{xy} = 0.8 \)
- \( \sigma_y = 20 \)
- \( \sigma_x = 14 \)

Substituting the values:
\[ b_{yx} = 0.8 \cdot \frac{20}{14} \]
Calculating this gives:
\[ b_{yx} = 0.8 \cdot 1.42857 \approx 1.143 \]

### Step 3: Substitute the values into the regression equation
Now we substitute \( b_{yx} \), \( \bar{y} \), and \( \bar{x} \) into the regression equation:
- \( \bar{y} = 100 \)
- \( \bar{x} = 18 \)

The equation becomes:
\[ y - 100 = 1.143(x - 18) \]

### Step 4: Simplify the equation
Expanding the right side:
\[ y - 100 = 1.143x - 1.143 \cdot 18 \]
Calculating \( 1.143 \cdot 18 \):
\[ 1.143 \cdot 18 \approx 20.574 \]
Thus, we have:
\[ y - 100 = 1.143x - 20.574 \]
Adding 100 to both sides:
\[ y = 1.143x + 100 - 20.574 \]
Calculating \( 100 - 20.574 \):
\[ y = 1.143x + 79.426 \]

### Final Answer
The regression equation of \( y \) on \( x \) is:
\[ y = 1.143x + 79.426 \]