Theory and Examples.Least squares and regression lines

Chapter 14, Problem 65E

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Problem 65E

Theory and Examples.

Least squares and regression lines When we try to fit a line y = mx + b to a set of numerical data points (x1, y1), (x2 , y2), … , (xn , yn) (Figure 14.48), 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

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

The line y = mx + 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.

FIGURE 14.48 To fit a line to noncollinear points, we choose the line that minimizes the sum of the squares of the deviations.