The following results were obtained with respect to two variables x and y.
`Sigmax=15,Sigmay=25,Sigmaxy=83,Sigmax^(2)=55,Sigmay^(2)=135and n= 5`
(i) Find the regression coefficient `b_(xy)`.
(ii) Find the regrassion equation of x on y.

Answer

To solve the given problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Identify the given values
We have the following values:
- \( \Sigma x = 15 \)
- \( \Sigma y = 25 \)
- \( \Sigma xy = 83 \)
- \( \Sigma x^2 = 55 \)
- \( \Sigma y^2 = 135 \)
- \( n = 5 \)

### Step 2: Calculate the regression coefficient \( b_{xy} \)
The formula for the regression coefficient \( b_{xy} \) is given by:

\[ b_{xy} = \frac{\Sigma xy - \frac{\Sigma x \cdot \Sigma y}{n}}{\Sigma y^2 - \frac{(\Sigma y)^2}{n}} \]

Substituting the values into the formula:

1. Calculate \( \Sigma xy - \frac{\Sigma x \cdot \Sigma y}{n} \):
\[ \Sigma xy = 83 \]
\[ \Sigma x \cdot \Sigma y = 15 \cdot 25 = 375 \]
\[ \frac{375}{5} = 75 \]
\[ \Sigma xy - \frac{\Sigma x \cdot \Sigma y}{n} = 83 - 75 = 8 \]

2. Calculate \( \Sigma y^2 - \frac{(\Sigma y)^2}{n} \):
\[ \Sigma y^2 = 135 \]
\[ \frac{(\Sigma y)^2}{n} = \frac{25^2}{5} = \frac{625}{5} = 125 \]
\[ \Sigma y^2 - \frac{(\Sigma y)^2}{n} = 135 - 125 = 10 \]

Now, substitute these results back into the formula for \( b_{xy} \):
\[ b_{xy} = \frac{8}{10} = 0.8 \]

### Step 3: Find the regression equation of \( x \) on \( y \)
The regression equation of \( x \) on \( y \) is given by:
\[ x - \bar{x} = b_{xy}(y - \bar{y}) \]

First, we need to calculate \( \bar{x} \) and \( \bar{y} \):
1. Calculate \( \bar{x} \):
\[ \bar{x} = \frac{\Sigma x}{n} = \frac{15}{5} = 3 \]

2. Calculate \( \bar{y} \):
\[ \bar{y} = \frac{\Sigma y}{n} = \frac{25}{5} = 5 \]

Now substitute \( \bar{x} \), \( \bar{y} \), and \( b_{xy} \) into the regression equation:
\[ x - 3 = 0.8(y - 5) \]

### Step 4: Rearranging the equation
Distributing the \( 0.8 \):
\[ x - 3 = 0.8y - 4 \]
Adding 3 to both sides:
\[ x = 0.8y - 4 + 3 \]
\[ x = 0.8y - 1 \]

### Final Form of the Regression Equation
Rearranging gives us:
\[ 0.8y - x - 1 = 0 \]

### Summary of Results
1. The regression coefficient \( b_{xy} = 0.8 \).
2. The regression equation of \( x \) on \( y \) is \( 0.8y - x - 1 = 0 \).