Find the equation of the regression line of y on x, if the observation (x,y) are as follows :
`(1, 4), (2, 8), (3, 2), (4, 12), (5, 10), (6, 14), (7, 16), (8, 6), (9, 18)`
Also, find the estimated value of y when x = 14.

Answer

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

### Step 1: List the Observations
The observations provided are:
\[ (1, 4), (2, 8), (3, 2), (4, 12), (5, 10), (6, 14), (7, 16), (8, 6), (9, 18) \]

### Step 2: Calculate the Necessary Summations
We need to calculate the following:
- \( \Sigma x \)
- \( \Sigma y \)
- \( \Sigma xy \)
- \( \Sigma x^2 \)
- \( \Sigma y^2 \)

**Calculations:**
1. **Sum of \( x \)**:
\[ \Sigma x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]

2. **Sum of \( y \)**:
\[ \Sigma y = 4 + 8 + 2 + 12 + 10 + 14 + 16 + 6 + 18 = 90 \]

3. **Sum of \( xy \)**:
\[ \Sigma xy = (1 \cdot 4) + (2 \cdot 8) + (3 \cdot 2) + (4 \cdot 12) + (5 \cdot 10) + (6 \cdot 14) + (7 \cdot 16) + (8 \cdot 6) + (9 \cdot 18) \]
\[ = 4 + 16 + 6 + 48 + 50 + 84 + 112 + 48 + 162 = 490 \]

4. **Sum of \( x^2 \)**:
\[ \Sigma x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = 285 \]

5. **Sum of \( y^2 \)**:
\[ \Sigma y^2 = 4^2 + 8^2 + 2^2 + 12^2 + 10^2 + 14^2 + 16^2 + 6^2 + 18^2 \]
\[ = 16 + 64 + 4 + 144 + 100 + 196 + 256 + 36 + 324 = 1140 \]

### Step 3: Calculate the Regression Coefficient \( b_{yx} \)
The formula for the regression coefficient \( b_{yx} \) is:
\[ b_{yx} = \frac{\Sigma xy - \frac{\Sigma x \cdot \Sigma y}{n}}{\Sigma x^2 - \frac{(\Sigma x)^2}{n}} \]
Where \( n \) is the number of observations (which is 9).

**Substituting the values:**
\[ b_{yx} = \frac{490 - \frac{45 \cdot 90}{9}}{285 - \frac{45^2}{9}} \]
\[ = \frac{490 - 450}{285 - 225} = \frac{40}{60} = \frac{2}{3} \]

### Step 4: Calculate \( \bar{x} \) and \( \bar{y} \)
\[ \bar{x} = \frac{\Sigma x}{n} = \frac{45}{9} = 5 \]
\[ \bar{y} = \frac{\Sigma y}{n} = \frac{90}{9} = 10 \]

### Step 5: Write the Equation of the Regression Line
The equation of the regression line is given by:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
Substituting the values:
\[ y - 10 = \frac{2}{3}(x - 5) \]
Rearranging gives:
\[ y = \frac{2}{3}x + \left(10 - \frac{2}{3} \cdot 5\right) \]
\[ y = \frac{2}{3}x + \left(10 - \frac{10}{3}\right) = \frac{2}{3}x + \frac{30}{3} - \frac{10}{3} = \frac{2}{3}x + \frac{20}{3} \]

### Step 6: Find the Estimated Value of \( y \) when \( x = 14 \)
Substituting \( x = 14 \) into the regression equation:
\[ y = \frac{2}{3}(14) + \frac{20}{3} = \frac{28}{3} + \frac{20}{3} = \frac{48}{3} = 16 \]

### Final Result
The equation of the regression line of \( y \) on \( x \) is:
\[ y = \frac{2}{3}x + \frac{20}{3} \]
And the estimated value of \( y \) when \( x = 14 \) is:
\[ y = 16 \]