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.