Least Squares Example
The scatter data below seems to have a linear relationship, so we will perform a linear regression.
x 1.5
1.5
2.0
2.5
2.5
3.0
3.5
3.5
4.0
y 23
24.5
25.0
30.0
33.5
40.0
40.5
47.0
49.0
Observation
x
y
x2
xy
y2
1
1.5
23.0
2.25
34.50
529.00
2
1.5
24.5
2.25
36.75
600.25
3
2.0
25.0
4.00
50.00
625.00
4
2.5
30.0
6.25
75.00
900.00
5
2.5
33.5
6.25
83.75
1122.25
6
3.0
40.0
9.00
120.00
1600.00
7
3.5
40.5
12.25
141.75
1640.25
8
3.5
47.0
12.25
164.50
2209.00
9
4.0
49.0
16.00
196.00
2401.00
Sum ofthe Column
24.0
312.5
70.50
902.25
11626.75
Now solving the two equations below simultaneously for m and b, we obtain the least squares estimates.
The equation of the estimated regression line is y = 10.603x + 6.449