Question

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

x	1	3	4
y	4	3	7

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =  +  x

(b) Now suppose you are given these (x, y) data pairs.

x	4	3	7
y	1	3	4

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =  +  x

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

Yes

No    

(d) Solve your answer from part (a) for x (rounded to three digits after the decimal).
x =  +  y

Do you get the least-squares equation of part (b) with the symbols x and y exchanged?

Yes

No    

(e) In general, suppose we have the least-squares equation y = a + 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).

Switching x and y values sometimes produces the same least-squares equation and sometimes it is different.

In general, switching x and y values produces a different least-squares equation.   

In general, switching x and y values produces the same least-squares equation.

Answer 1

(a)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	1	4	1	16	4
	3	3	9	9	9
	4	7	16	49	28
Total	8	14	26	74	41

(b)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	4	1	16	1	4
	3	3	9	9	9
	7	4	49	16	28
Total	14	8	74	26	41

(c)

Yes

(d)

y'= 0.786*x+ 2.571

x = (y' - 2.571) / 0.786 = 0.1272*y' - 3.271

No

(e)

In general, switching x and y values produces the same least-squares equation.