A common problem in experimental work is to obtain a mathematical relationship
\(y=f(x)\) between two variables \(x\) and \(y\) by "fitting" a curve to points in
the plane that correspond to experimentally determined values of \(x\) and \(y,\)
say
$$
\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n},
y_{n}\right)
$$
The curve \(y=f(x)\) is called a mathematical model of the data. The general
form of the function \(f\) is commonly determined by some underlying physical
principle, but sometimes it is just determined by the pattern of the data. We
are concerned with fitting a straight line \(y=m x+b\) to data. Usually, the
data will not lie on a line (possibly due to experimental error or variations
in experimental conditions), so the problem is to find a line that fits the
data "best" according to some criterion. One criterion for selecting the line
of best fit is to choose \(m\) and \(b\) to minimize the function
$$
g(m, b)=\sum_{i=1}^{n}\left(m x_{i}+b-y_{i}\right)^{2}
$$
This is called the method of least squares, and the resulting line is called
the regression line or the least squares line of best fit. Geometrically,
\(\left|m x_{i}+b-y_{i}\right|\) is the vertical distance between the data point
\(\left(x_{i}, y_{i}\right)\) and the line \(y=m x+b\)
These vertical distances are called the residuals of the data points, so the
effect of minimizing \(g(m, b)\) is to minimize the sum of the squares of the
residuals. In these exercises, we will derive a formula for the regression
line.
The purpose of this exercise is to find the values of \(m\) and \(b\) that produce
the regression line.
(a) To minimize \(g(m, b),\) we start by finding values of \(m\) and \(b\) such that
\(\partial g / \partial m=0\) and \(\partial g / \partial b=0 .\) Show that these
equations are satisfied if \(m\) and \(b\) satisfy the conditions
$$
\left(\sum_{i=1}^{n} x_{i}^{2}\right) m+\left(\sum_{i=1}^{n} x_{i}\right)
b=\sum_{i=1}^{n} x_{i} y_{i}
$$
\(\left(\sum_{i=1}^{n} x_{i}\right) m+n b=\sum_{i=1}^{n} y_{i}\)
(b) Let \(\bar{x}=\left(x_{1}+x_{2}+\cdots+x_{n}\right) / n\) denote the
arithmetic
average of \(x_{1}, x_{2}, \ldots, x_{n} .\) Use the fact that
$$
\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2} \geq 0
$$
to show that
$$
n\left(\sum_{i=1}^{n} x_{i}^{2}\right)-\left(\sum_{i=1}^{n} x_{i}\right)^{2}
\geq 0
$$
with equality if and only if all the \(x_{i}\) 's are the same.
(c) Assuming that not all the \(x_{i}\) 's are the same, prove that the
equations in part (a) have the unique solution
$$
\begin{aligned} m=& \frac{n \sum_{i=1}^{n} x_{i} y_{i}-\sum_{i=1}^{n} x_{i}
\sum_{i=1}^{n} y_{i}}{n \sum_{i=1}^{n} x_{i}^{2}-\left(\sum_{i=1}^{n}
x_{i}\right)^{2}} \\ b=& \frac{1}{n}\left(\sum_{i=1}^{n} y_{i}-m
\sum_{i=1}^{n} x_{i}\right) \end{aligned}
$$
[Note: We have shown that \(g\) has a critical point at these values of \(m\) and
\(b\). In the next exercise we will show that \(g\) has an absolute minimum at
this critical point. Accepting this to be so, we have shown that the line \(y=m
x+b\) is the regression line for these values of \(m \text { and } b .]\)
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