Question

A table has the points (0, 2.50) (2, 4.03) (4, 5.56) (6, 7.09) what is the equation for the line of best fit?

Answer 1

To find the equation for the line of best fit, we can use the least squares method.

Step 1: Calculate the mean of the x-values and the mean of the y-values.
Mean of x: (0 + 2 + 4 + 6) / 4 = 3
Mean of y: (2.50 + 4.03 + 5.56 + 7.09) / 4 = 4.795

Step 2: Calculate the sum of the products of the differences between each x-value and the mean of x, and each y-value and the mean of y.
Sum of (x - mean of x)(y - mean of y):
(0 - 3)(2.50 - 4.795) + (2 - 3)(4.03 - 4.795) + (4 - 3)(5.56 - 4.795) + (6 - 3)(7.09 - 4.795) =
-3(-2.295) + (-1)(-0.765) + (1)(0.765) + (3)(2.295) =
6.885 + 0.765 + 0.765 + 6.885 = 14.3

Step 3: Calculate the sum of the squared differences between each x-value and the mean of x.
Sum of (x - mean of x)^2:
(0 - 3)^2 + (2 - 3)^2 + (4 - 3)^2 + (6 - 3)^2 =
9 + 1 + 1 + 9 = 20

Step 4: Calculate the slope of the line of best fit.
Slope: sum of (x - mean of x)(y - mean of y) / sum of (x - mean of x)^2:
14.3 / 20 = 0.715

Step 5: Calculate the y-intercept of the line of best fit.
y-intercept: mean of y - slope * mean of x:
4.795 - 0.715 * 3 = 4.795 - 2.145 = 2.65

Therefore, the equation for the line of best fit is y = 0.715x + 2.65.