Given that the observations are :
`(9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8)`. Find the two lines of regression and estimate the value of y when x = 13.5.

Answer

To solve the problem, we will follow these steps:

### Step 1: Organize the Data
We will create a table to summarize the given observations and calculate the necessary values.

| x | y | x² | y² | xy |
|----|----|------|------|------|
| 9 | -4 | 81 | 16 | -36 |
| 10 | -3 | 100 | 9 | -30 |
| 11 | -1 | 121 | 1 | -11 |
| 12 | 0 | 144 | 0 | 0 |
| 13 | 1 | 169 | 1 | 13 |
| 14 | 3 | 196 | 9 | 42 |
| 15 | 5 | 225 | 25 | 75 |
| 16 | 8 | 256 | 64 | 128 |

### Step 2: Calculate Summations
Now, we will calculate the summation of x, y, x², y², and xy.

- \( \Sigma x = 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100 \)
- \( \Sigma y = -4 - 3 - 1 + 0 + 1 + 3 + 5 + 8 = 9 \)
- \( \Sigma x^2 = 81 + 100 + 121 + 144 + 169 + 196 + 225 + 256 = 1292 \)
- \( \Sigma y^2 = 16 + 9 + 1 + 0 + 1 + 9 + 25 + 64 = 125 \)
- \( \Sigma xy = -36 - 30 - 11 + 0 + 13 + 42 + 75 + 128 = 181 \)

### Step 3: Calculate Means
Next, we will calculate the means of x and y.

- \( \bar{x} = \frac{\Sigma x}{n} = \frac{100}{8} = 12.5 \)
- \( \bar{y} = \frac{\Sigma y}{n} = \frac{9}{8} = 1.125 \)

### Step 4: Calculate Regression Coefficients
Now we will calculate the regression coefficients \( b_{yx} \) and \( b_{xy} \).

**For \( b_{yx} \):**
\[ b_{yx} = \frac{\Sigma xy - \frac{1}{n}(\Sigma x)(\Sigma y)}{\Sigma x^2 - \frac{1}{n}(\Sigma x)^2} \]
Substituting the values:
\[ b_{yx} = \frac{181 - \frac{1}{8}(100)(9)}{1292 - \frac{1}{8}(100^2)} \]
Calculating:
\[ b_{yx} = \frac{181 - 112.5}{1292 - 1250} = \frac{68.5}{42} \approx 1.63 \]

**For \( b_{xy} \):**
\[ b_{xy} = \frac{\Sigma xy - \frac{1}{n}(\Sigma x)(\Sigma y)}{\Sigma y^2 - \frac{1}{n}(\Sigma y)^2} \]
Substituting the values:
\[ b_{xy} = \frac{181 - \frac{1}{8}(100)(9)}{125 - \frac{1}{8}(9^2)} \]
Calculating:
\[ b_{xy} = \frac{181 - 112.5}{125 - 10.125} = \frac{68.5}{114.875} \approx 0.596 \]

### Step 5: Form the Regression Equations
**1. Regression Line of Y on X:**
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
Substituting the values:
\[ y - 1.125 = 1.63(x - 12.5) \]
Rearranging gives:
\[ y = 1.63x - 19.25 \]

**2. Regression Line of X on Y:**
\[ x - \bar{x} = b_{xy}(y - \bar{y}) \]
Substituting the values:
\[ x - 12.5 = 0.596(y - 1.125) \]
Rearranging gives:
\[ x = 0.596y + 11.83 \]

### Step 6: Estimate the Value of y when x = 13.5
Using the regression equation of Y on X:
\[ y = 1.63(13.5) - 19.25 \]
Calculating:
\[ y = 22.005 - 19.25 = 2.755 \]

### Final Answer
The estimated value of y when x = 13.5 is approximately **2.755**.