To solve this least-squares regression problem using the given dataset:

Dataset:

We need to calculate the regression line equation:
$\hat{y} = mx + b$
where $m$ is the slope and $b$ is the y-intercept.

Steps to Compute:

1. Input the data into a TI-84 calculator:

   • Go to STAT > EDIT.
   • Enter the $x$-values in L1 and the $y$-values in L2.

2. Find the regression equation:

   • Press STAT > CALC > LinReg(ax+b) (option 4).
   • Select L1 and L2 as the data lists, then hit ENTER.

3. Manual Calculation (if needed):

   • Use these formulas for slope ($m$) and intercept ($b$): $m = \frac{n\sum(xy) - \sum(x)\sum(y)}{n\sum(x^2) - (\sum(x))^2}$ $b = \frac{\sum(y) - m\sum(x)}{n}$ where $n$ is the number of data points.

Let me calculate these values for you now. numpy as np

Given data points

x = np.array([10, 6, 8, 14, -7, -1, 7, -9]) y = np.array([3, 1, 30, 31, 0, 3, -3, -15])

Number of points

n = len(x)

Calculations for regression

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x_squared = np.sum(x ** 2)

Slope (m)

m = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x ** 2)

Intercept (b)

b = (sum_y - m * sum_x) / n

m, bThe regression line equation is:
$\hat{y} = 1.3515x + 1.5199$
(Rounded to 4 decimal places as requested.)

Would you like a step-by-step breakdown of the calculations or have any other questions?

Here are five related questions to explore:

1. What is the interpretation of the slope $m$ in this context?
2. How would the regression line change if an outlier were added to the dataset?
3. Can you calculate the correlation coefficient $r$ for this data?
4. What does the y-intercept $b$ represent for these values?
5. How can you use this regression line to predict values for $y$ given specific $x$?

Tip: Always plot the data points and the regression line to visually confirm the accuracy of your model.