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 = 1.071x + 0.143(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 = 0.357x + 1.595 (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 = 0.143 + 1.071xfor x. Do you get the least-squares equation ofpart (b) with the symbols x and y exchanged?
(e) In general, suppose we have the least-squares equation y = a + bxfor a setof 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 andy exchanged)? Explain using parts (a) through (d).
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