Question

(a) Suppose you are given the following (x, y) data pairs. x: 1 2 6 y: 4 3 9 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 9 y 1 2 6 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).

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

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

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

Answer 1

a)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.940634162
R Square	0.884792627
Adjusted R Square	0.769585253
Standard Error	1.5430335
Observations	3
ANOVA
	df	SS	MS	F	Significance F
Regression	1	18.2857143	18.2857143	7.68	0.220463
Residual	1	2.38095238	2.38095238
Total	2	20.6666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	1.904761905	1.52455339	1.2493901	0.42970499	-17.4665256	21.2760494
x	1.142857143	0.41239305	2.77128129	0.220463	-4.09709337	6.38280766

y^ = 1.9048 + 1.1429 x

b)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.940634162
R Square	0.884792627
Adjusted R Square	0.769585253
Standard Error	1.27000127
Observations	3
ANOVA
	df	SS	MS	F	Significance F
Regression	1	12.3870968	12.3870968	7.68	0.220463
Residual	1	1.61290323	1.61290323
Total	2	14
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-1.129032258	1.66058551	-0.6799001	0.61986902	-22.2287717	19.9707072
x	0.774193548	0.27936303	2.77128129	0.220463	-2.77545035	4.32383745

y^ = -1.129 + 0.7742 * x

c)

yes

d)

x^ = -1.129 + 0.7742 * y