To find the line of best fit (also known as the least squares regression line) for the given data points, we need to calculate the slope $m$ and y-intercept $b$ using the formulas:

$m = \frac{n \sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}$

$b = \frac{\sum{y_i} - m \sum{x_i}}{n}$

Where:

• $n$ is the number of data points.
• $x_i$ and $y_i$ are the x and y values of the data points.

Given points:

(4, 4), (6, 1), (7, -1), (8, 1)

Step 1: Calculate necessary sums

• $\sum{x_i} = 4 + 6 + 7 + 8 = 25$
• $\sum{y_i} = 4 + 1 - 1 + 1 = 5$
• $\sum{x_i^2} = 4^2 + 6^2 + 7^2 + 8^2 = 16 + 36 + 49 + 64 = 165$
• $\sum{x_i y_i} = (4 \times 4) + (6 \times 1) + (7 \times -1) + (8 \times 1) = 16 + 6 - 7 + 8 = 23$

Step 2: Calculate the slope $m$

$m = \frac{4 \times 23 - 25 \times 5}{4 \times 165 - 25^2} = \frac{92 - 125}{660 - 625} = \frac{-33}{35} \approx -0.943$

Step 3: Calculate the y-intercept $b$

$b = \frac{5 - (-0.943) \times 25}{4} = \frac{5 + 23.575}{4} = \frac{28.575}{4} \approx 7.144$

Step 4: Write the equation of the line

The equation of the line of best fit is:

$y = -0.943x + 7.144$

Would you like to explore any of the steps further? Let me know if you have any questions!

Here are some related questions you might find interesting:

1. How do you interpret the slope of a regression line in a real-world context?
2. How do outliers affect the line of best fit?
3. How do you calculate the coefficient of determination (R²) for this line?
4. What is the difference between a line of best fit and interpolation?
5. How do you calculate the predicted y-value for a given x-value using the line of best fit?

Tip: Always check for outliers before calculating the line of best fit, as they can significantly skew the results.