Given, `n = 5, Sigma x_(i) = 25, Sigma y_(i) = 20, Sigma x_(i) y_(i) = 90` and `Sigma x_(i)^(2) = 135`, find the regression coefficient of y on x.

Answer

To find the regression coefficient of \( y \) on \( x \), we will use the formula:

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

### Step 1: Identify the given values
- \( n = 5 \)
- \( \Sigma x_i = 25 \)
- \( \Sigma y_i = 20 \)
- \( \Sigma x_i y_i = 90 \)
- \( \Sigma x_i^2 = 135 \)

### Step 2: Calculate the numerator
We need to calculate \( \Sigma x_i y_i - \frac{\Sigma x_i \cdot \Sigma y_i}{n} \).

1. Calculate \( \frac{\Sigma x_i \cdot \Sigma y_i}{n} \):
\[ \frac{\Sigma x_i \cdot \Sigma y_i}{n} = \frac{25 \cdot 20}{5} = \frac{500}{5} = 100 \]

2. Now, calculate the numerator:
\[ \Sigma x_i y_i - \frac{\Sigma x_i \cdot \Sigma y_i}{n} = 90 - 100 = -10 \]

### Step 3: Calculate the denominator
Now we will calculate \( \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n} \).

1. Calculate \( \frac{(\Sigma x_i)^2}{n} \):
\[ \frac{(\Sigma x_i)^2}{n} = \frac{25^2}{5} = \frac{625}{5} = 125 \]

2. Now, calculate the denominator:
\[ \Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n} = 135 - 125 = 10 \]

### Step 4: Calculate the regression coefficient
Now we can substitute the values into the regression coefficient formula:
\[ b_{y|x} = \frac{-10}{10} = -1 \]

### Conclusion
The regression coefficient of \( y \) on \( x \) is:
\[ b_{y|x} = -1 \]