According to Example, the least-squares error for the best least-squares fit to the data p...

According to Example, the least-squares error for the best least-squares fit to the data points (1, 6), (4, 5), and (6, 14) is E = 22.13.

EXAMPLE Finding the Least-squares Line Find the least-squares line for the data points of Example 1.

SOLUTION The sums are calculated in Table and then used to determine the values of m and b.

TABLE

Therefore, the equation of the least-squares line is . With this line, the least-squares error can be shown to be about 22.13.

In practice, least-squares lines are obtained with graphing calculators, spread-sheets, computer software, or applets accessed via the Web.

Obtaining the Least-Squares Line with a Graphing Calculator On the TI-83/84 Plus graphing calculator screens in Fig, the data points are entered into lists, the least-squares line is calculated with the item LinReg(ax+b) of the STAT/CALC menu, and the data points and line are plotted with [STAT PLOT] and . The end of this section and Appendix D contain the details for obtaining least-squares lines with a graphing calculator.

Figure Obtaining a least-squares line with a TI-83/84 Plus.

Obtaining the Least-Squares Line with an Excel Spreadsheet Excel has special functions that calculate the slope and y-intercept of the least-squares line for a collection of data points. In Fig, the least-squares line of Example 2 is calculated and graphed in Excel. The end of this section shows how to obtain the graph in Fig.

Figure Obtaining a least-squares line with Excel.

The next example obtains a least-squares line and uses the line to make projections.

(a) Find the equation of the straight line through the two points (1, 6) and (6, 14).

(b) What is the least-squares error when the line in (a) is used to fit the three data points?