Find the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).
Solution:
To find an the equation y = a + bx by the method of least squares the normal equations are
\(\sum y\) = na +b\(\sum x\)
\(\sum xy\) = a\(\sum x\) + b\(\sum x^2\)
Given data
| x | 2 | 3 | 5 | 6 |
|---|---|---|---|---|
| y | 3 | 2 | 1 | 0 |
The required data for normal equations are
| x | 2 | 3 | 5 | 6 | \(\sum x = 16\) |
|---|---|---|---|---|---|
| y | 3 | 2 | 1 | 0 | \(\sum y = 6\) |
| x2 | 9 | 4 | 1 | 36 | \(\sum x^2 = 50\) |
| xy | 6 | 6 | 5 | 0 | \(\sum xy = 17\) |
∴ The normal equations are
6 = 4a + 16b --- (1)
17 = 16a + 50b --- (2)
Solving these equations for a and b
Multiplying (1) with 4 and subtracting from (2)
24 = 16a + 64b
17 = 16a + 50b
⇒ 7 = 14 b
b = 7/14 = 0.5
b= 0.5
Substitute b = 0.5 in equation (2)
17 = 16a + 50(0.5)
17 = 16a + 25
17 - 25 = 16a
-8 = 16 a
a = -8/16
a = - 1/2 = -0.5
Therefore, the equation of the line for the given data is is y = a+ bx
⇒ y = (-0.5) + (0.5)x
Find the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).
Summary:
The Equation of line ⇒ y = (-0.5) + (0.5)x is the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).
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