Question

x-axis= 0.3, 0.31, 0.28, 0.32, 0.34, 0.29, 0.33, 0.26, 0.3, 0.27, 0.27

y-axis = 0.17, 0.19, 0.21, 0.22, 0.28, 0.26, 0.32, 0.17, 0.19, 0.19, 0.17

what is the equation of the least squares best fit line?

Answer 1

@amoodarya @BulletWithButterflyWings @CGGURUMANJUNATH @djcool31 @electrokid @Falco276 @grant330sims @Hero

Answer 2

@ash2326

Answer 3

the question asks for a "linear least square fit" i.e., find the equation of a straight line that fits the data with a least squared error.
1. let the equation of the straight line be
\(y=ax+b\)
where a and b are arbitrary contacts to find.
2. each of the given (x,y) points should ideally lie on this line
\[\begin{align}
0.17&=a(0.3)+b& 0.19&=a(0.31)+b&0.21&=a(0.28)+b&\cdots
\end{align}\]
3. but there is an error in fitting the points on the line
\[\begin{align}
\delta_1&=0.17-(0.3a+b)\\
\delta_2&=0.19-(0.31a+b)\\
\delta_3&=0.21-(0.28a+b)\\
&\vdots
\end{align}\]
4. For a least-square fit, the square of these errors should be minimized i.e,
\[\begin{align}\Delta(a,b)&=\delta_1^2+\delta_2^2+\delta_3^2+\cdots\\
\Delta(a,b)&=(0.17-0.3a-b)^2+(0.19-0.31a-b)^2+(0.21-0.28a-b)^2+\cdots
\end{align}\]
5. you can use the partial derivatives with respect to "a" and "b", set them to zer and solve the two equations to get "a" and "b" values.

NOTE: use the following only if asked and is for your verification.
It so happens that for a straight line fit, the formulae for "a" and "b" can be expressed as:
\[\begin{align}
a&=\frac{\sum y\times \sum x^2-\sum x\times \sum xy}{n\sum x^2-\left(\sum x\right)^2}\\
b&=\frac{\sum n\times \sum xy-\sum x\times \sum y}{n\sum x^2-\left(\sum x\right)^2}
\end{align}\]

Answer 4

correction in the last equation\[b=\frac{\color{red}{n\times \sum xy}-\sum x\times \sum y}{n\sum x^2-\left(\sum x\right)^2}\]