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Appendix C C-29<br />
Then, for any line y mx b we have<br />
e 1 2 [m(1) b] e 2 4 [m(2) b] e 3 5 [m(3) b]<br />
2 m b 4 2m b 5 3m b<br />
Now, by definition, for the least-squares line we want to find the values for m<br />
and b so that the quantity E e 2 1 e 2 2 e 2 3 is as small as possible. Using the<br />
expressions we’ve just calculated for e 1 , e 2 , and e 3 , we have<br />
E e 2 1 e 2 2 e 2 3<br />
(2 m b) 2 (4 2m b) 2 (5 3m b) 2<br />
(1)<br />
Equation (1) expresses E as a function of two variables, m and b. In all the work<br />
we’ve done previously on maximum-minimum problems, we were always<br />
able to express one variable in terms of the other, and then apply our knowledge<br />
of quadratics. In this case, though, m and b are not related, so that strategy<br />
doesn’t work here. In fact, in general, the methods of calculus are required<br />
to handle a two-variable maximum-minimum problem. Therefore to make this<br />
problem accessible within our context, we’ll assume the result (from calculus)<br />
that in this problem b turns out to be 23. Go ahead now and make the substitution<br />
b 23 in equation (1). After simplifying, you’ll have a quadratic<br />
function, and you know how to find the value of m that minimizes the function.<br />
Then, with both m and b known, you have determined the least-squares line<br />
y mx b. As a visual check, graph the data set along with the least-squares<br />
line. See whether things look reasonable. Finally, use a graphing utility to compute<br />
the least-squares line and check your answer.<br />
Exercises<br />
In each case you are given a data set and the value for b in the least-squares<br />
line y mx b. Find the equation of the least-squares line using the method<br />
indicated above. Check your answer using an appropriate graphing utility.<br />
Then graph the data set along with the least-squares line. In Exercise 2, in<br />
reporting the final answer, round both b and m to two decimal places.<br />
1. (Easy numbers) x 1 2 3 b 3<br />
y 2 5 11<br />
2. (“Actual” numbers) The data in the following table are from a study relating<br />
air pollution and incidence of respiratory disease. The data were collected<br />
in five cities over the years 1955, 1957, and 1958. The subjects were women<br />
working in RCA factories. In the table, x denotes the average concentration
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