Find the equation of the least squares line

Question

Find the equation of the least-squares line for the following data points.
P1(1, 8), P2(2, 6), P3(5, 6), P4(7, 4), P5(10, 1)

Answer 1

Auburn,

 

Make a table with each row being a different data point, and the columns being x, y, xy, & x² and fill in this table by making these preliminary computations.

 

Let  N = number of data points (5 here).

      m = slope of the regression line.

      b  = the y-intercept of that line when x = 0.

 

Then,

           m = { [N ∑xy - (∑x) (∑y)] / [ N ∑(x²) - (∑x)²]} = -0.685

 

           b  = (1/N)[ ∑y - m(∑x)]  =  + 8.425

 

Thus, the best-fit regression line is  y = -0.685 x + 8.425

 

You can work thru the data and verify this.

Answer 2

The equation is y = M*x + C, where

 

M = (XY - X*Y) / (XX - X*X) and C = Y - M*X

 

with

 

XX = average x^2 = (1 + 4 + 25 + 49 + 100)/5 = 179/5,

XY = average x*y = (8 + 12 + 30 + 28 + 10)/5 = 88/5,

X = average x = (1 + 2 + 5 + 7 + 10)/5 = 5,

Y = average y = (8 + 6 + 6 + 4 + 1)/5 = 5.

 

Thus, M = (88 - 125)/(179 - 125) = -37/54 and C = 5 - (-37/54)*5 = 455/54.

 

So, y = (-37/54)*x + 455/54, or 37*x + 54*y = 455.