Find the line of best fit for the points (1,-3), (-1, 4), (0,2), and (1,1).
To find the line of best fit for a given set of points, you need to perform linear regression analysis. Here's how you can do it step by step:
1. Write down the given points: (1, -3), (-1, 4), (0, 2), and (1, 1).
2. Calculate the average of the x-values and the average of the y-values. In this case:
- Average of x-values: (1 + (-1) + 0 + 1)/4 = 0
- Average of y-values: (-3 + 4 + 2 + 1)/4 = 1
3. Calculate the differences between each x-value and the average x-value, and each y-value and the average y-value. These differences are represented as (x - x_avg) and (y - y_avg). This will help with the further calculations. Here are the differences for each point:
- (1 - 0, -3 - 1) -> (1, -4)
- (-1 - 0, 4-1) -> (-1, 3)
- (0 - 0, 2 - 1) -> (0, 1)
- (1 - 0, 1 - 1) -> (1, 0)
4. Calculate the slope of the line (m) using the formula: m = Σ (x - x_avg)(y - y_avg) / Σ (x - x_avg)^2
- Applying the formula, using the differences calculated earlier:
m = [(1 * -4) + (-1 * 3) + (0 * 1) + (1 * 0)] / [(1 * 1) + (-1 * -1) + (0 * 0) + (1 * 1)]
m = (-4 - 3 + 0 + 0) / (1 + 1 + 0 + 1)
m = -7 / 3
m = -2.33 (rounded to two decimal places)
5. Calculate the y-intercept (b) using the formula: b = y_avg - m * x_avg
- Applying the formula:
b = 1 - (-2.33 * 0)
b = 1
6. Write out the equation of the line of best fit using the calculated slope and y-intercept.
- The equation is y = mx + b, where m is the slope and b is the y-intercept.
- Plugging in the values, we get: y = -2.33x + 1
So, the line of best fit for the given set of points is y = -2.33x + 1.