Download Method of Least Squares Predicting Y from X

Method of Least Squares
Predicting Y from X
X
1
2
3
4
5
6
Y
2
3
4
4
5
7
2 = 1b +a
3 = 2b +a
4 = 3b +a
4 = 4b +a
5 = 5b +a
7 = 6b +a
y = bx + a
No values of “b” and “a” will satisfy simultaneously all six equations.
Obtain two equations: 1st and 2nd Normal Equations.
1st = multiplying each equation by its “b” coefficient and summing.
2nd = multiplying each equation by tis “a” coefficient and summing.
1st Normal
1b + 1a = 2
4b + 2a = 6
9b + 3a = 12
16b + 4a = 16
25b + 5a = 25
36b + 21a = 42
91b + 21a = 103
2nd Normal
1b + 1a = 2
2b + 1a = 3
3b + 1a = 4
4b + 1a = 4
5b + 1a = 5
6b + 1a = 7
21b + 6a = 25
Thus: b and a for the least squared solution are derived by solving the 1st and 2nd
Normal Equations simultaneously.
91b + 21a = 103
21b + 6a = 25
Divide 1st by 91d and 2nd by 21
b + .2857a = 1.1905
b + .2308a = 1.1319
and subtract 1st from the 2nd
.0549a = 0.0586
a = 1.067
Substitute the a value into the 1st equation and solve for b
b = 1.1318 - (.2308)(1.067)
= .886
y = .886x + 1.065