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
x | y | xy | x2 |
1 | 6 | 6 | 1 |
4 | 5 | 20 | 16 |
6 | 14 | 84 | 36 |
∑x = 11 | ∑y = 25 | ∑xy = 110 | ∑x2 = 53 |
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 projec-tions.
(a) Find the equation of the straight line through the two points (4, 5) and (6, 14).
(b) What is the least-squares error when the line in (a) is used to fit the three data points?
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