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To solve this problem and compute the least-squares regression equation, we will follow these steps:

### Steps for the Calculation:

1. **Write down the data points**:  
   The data is given as:  
   \[
   x = [8, 4, 6, 12, -9, -3, 5, -11]
   \]  
   \[
   y = [3, 3, 31, 36, -2, 3, -2, -14]
   \]

2. **Compute key values for the least-squares formula**:
   The formula for the regression line is:
   \[
   \hat{y} = mx + b
   \]
   where:
   - \(m\) is the slope, given by:
     \[
     m = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}
     \]
   - \(b\) is the y-intercept, given by:
     \[
     b = \frac{\sum y_i - m \sum x_i}{n}
     \]

3. **Calculate the required summations**:
   We'll compute:
   - \(\sum x_i\)
   - \(\sum y_i\)
   - \(\sum x_i^2\)
   - \(\sum x_i y_i\)

4. **Substitute the values and compute \(m\) and \(b\)**.

Let me calculate these values for you. Data points
x = [8, 4, 6, 12, -9, -3, 5, -11]
y = [3, 3, 31, 36, -2, 3, -2, -14]

# Number of data points
n = len(x)

# Calculate necessary summations
sum_x = sum(x)
sum_y = sum(y)
sum_x2 = sum([xi**2 for xi in x])
sum_xy = sum([xi * yi for xi, yi in zip(x, y)])

# Calculate slope (m) and y-intercept (b)
m = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2)
b = (sum_y - m * sum_x) / n

m, bThe least-squares regression equation is:

\[
\hat{y} = 1.5063x + 4.9906
\]

Here:
- The slope (\(m\)) is approximately \(1.5063\).
- The y-intercept (\(b\)) is approximately \(4.9906\).

Would you like a detailed explanation of the calculations, or have any specific questions?

### 5 Questions for Exploration:
1. How does the least-squares method minimize error in regression?
2. What is the significance of the slope (\(m\)) in the context of this dataset?
3. How does the y-intercept (\(b\)) influence the regression line?
4. Could you use this regression line to make predictions for other \(x\) values?
5. What other tools, besides a TI-84 calculator, can compute regression equations?

### Tip:
Always check the correlation coefficient to assess the strength of the relationship between \(x\) and \(y\) when performing regression analysis.

Compute the least-squares regression equation for the given data set. Use a TI-84 calculator. Round the slope and y-intercept to at least four decimal places.

Learn how to compute the least-squares regression equation for a given dataset using a TI-84 calculator. This example walks you through finding the slope and y-intercept using statistical formulas. Key data points include x = [8, 4, 6, 12, -9, -3, 5, -11] and y = [3, 3, 31, 36, -2, 3, -2, -14].

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