Critical Thinking: Exchange x and y in Least-Squares Equation

(a) Suppose you are given the following (x, y) data pairs:

x 1 3 4 y 2 1 6 Show that the least-squares equation for these data is y 5 0.066 1 1.393x

(rounded to three digits after the decimal)

(b) Now suppose you are given these (x, y) data pairs:
x 2 1 6 y 1 3 4 Show that the least-squares equation for these data is y 5 0.971 1 0.494x (rounded to three digits after the decimal).

(c) In the data for parts

(a) and (b), did we simply exchange the x and y values of each data pair?

(d) Solve y 5 0.066 1 1.393x for x. Do you get the least-squares equation of part

(b) with the symbols x and y exchanged?

(e) In general, suppose we have the least-squares equation y 5 a 1 bx for a set of data pairs (x, y). If we solve this equation for x, will we necessarily get the least-squares equation for the set of data pairs (y, x) (with x and y exchanged)? Explain using parts

(a) through (d).