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 of finding the two lines of regression and estimating the value of \( y \) when \( x = 13.5 \), we will follow these steps:

### Step 1: Create a Table of Values
We start by organizing the given data into a table that includes \( x \), \( y \), \( x^2 \), \( y^2 \), and \( xy \).

| \( x \) | \( y \) | \( x^2 \) | \( y^2 \) | \( xy \) |
|---------|---------|-----------|-----------|----------|
| 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 |
| **Total** | | **1092** | **125** | **181** |

### Step 2: Calculate \( n \), \( \Sigma x \), \( \Sigma y \), \( \Sigma x^2 \), \( \Sigma y^2 \), and \( \Sigma xy \)
- \( n = 8 \) (number of observations)
- \( \Sigma x = 100 \)
- \( \Sigma y = 9 \)
- \( \Sigma x^2 = 1092 \)
- \( \Sigma y^2 = 125 \)
- \( \Sigma xy = 181 \)

### Step 3: Calculate the Means
- \( \bar{x} = \frac{\Sigma x}{n} = \frac{100}{8} = 12.5 \)
- \( \bar{y} = \frac{\Sigma y}{n} = \frac{9}{8} = 1.125 \)

### Step 4: Calculate the Regression Coefficients
1. **For the regression line of \( y \) on \( x \)**:
\[ b_{yx} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma x^2 - (\Sigma x)^2} \]
Substituting the values:
\[ b_{yx} = \frac{8 \cdot 181 - 100 \cdot 9}{8 \cdot 1092 - 100^2} = \frac{1448 - 900}{8736 - 10000} = \frac{548}{336} \approx 1.631 \]

2. **For the regression line of \( x \) on \( y \)**:
\[ b_{xy} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma y^2 - (\Sigma y)^2} \]
Substituting the values:
\[ b_{xy} = \frac{8 \cdot 181 - 100 \cdot 9}{8 \cdot 125 - 9^2} = \frac{1448 - 900}{1000 - 81} = \frac{548}{919} \approx 0.596 \]

### Step 5: Write the Regression Equations
1. **Equation of \( y \) on \( x \)**:
\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]
Substituting the values:
\[ y - 1.125 = 1.631(x - 12.5) \]
Rearranging gives:
\[ y = 1.631x - 19.6263 \]

2. **Equation 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:
\[ x = 0.596y + 11.830 \]

### Step 6: Estimate \( y \) when \( x = 13.5 \)
Using the regression equation of \( y \) on \( x \):
\[ y = 1.631(13.5) - 19.6263 \]
Calculating:
\[ y = 22.0365 - 19.6263 = 2.4102 \]

### Final Result
Thus, the estimated value of \( y \) when \( x = 13.5 \) is approximately \( 2.4102 \).