Least squares and regression lines When we try to fit a

Chapter 13, Problem 65E

(choose chapter or problem)

Least squares and regression lines When we try to fit a line \(y=m x+b\) to a set of numerical data points \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right), \ldots,\left(x_{\pi}, y_{n}\right)\) (Figure 13.48), we usually choose the line that minimizes the sum of the squares of the vertical distances from the points to the line. In theory, this means finding the values of m and b that minimize the value of the function

\(w=\left(m x_{1}+b-y_{1}\right)^{2}+\cdots+\left(m x_{n}+b-y_{n}\right)^{2}\).

Show that the values of m and b that do this are

\(m=\frac{\left(\Sigma x_{k}\right)\left(\Sigma y_{k}\right)-n \sum x_{k} y_{k}}{\left(\Sigma x_{k}\right)^{2}-n \Sigma x_{k}^{2}}\)

\(b=\frac{1}{n}\left(\Sigma y_{k}-m \Sigma x_{k}\right)\)

with all sums running from \(k=1\) to \(k=n\). Many scientific calculators have these formulas builtin, enabling you to find m and b with only a few keystrokes after you have entered the data.

         The line \(y=m x+b\) determined by these values of m and b is called the least squares line, regression line, or trend line for the data under study. Finding a least squares line lets you

1. summarize data with a simple expression,

2. predict values of y for other, experimentally untried values of x.

3. handle data analytically.

Equation Transcription:

Text Transcription:

y = mx + 5

(x_1,y_1)

(x_2,y_2), ... , (x_pi, y_n)

w = (mx_1 + b - y_1)^2 + times times times + (mx_n + b - y_n)^2

m = (sigma x_k)(sigma y_k)-n sigma x_k y_k/(sigma x_k)^2 - n sigma x_k^2

b = 1/n (sigma y_k - m sigma x_k)

k = 1

k = n