Question

Car manufacturers produced a variety of classic cars that continue to increase in value. Suppose the following data is based upon the Martin Rating System for Collectible Cars, and shows the rarity rating (1–20) and the high price ($1,000) for 15 classic cars.

Model	Rating	Price ($1,000)
A	16	125.0
B	13	70.0
C	17	140.0
D	19	1,325.0
E	19	3,975.0
F	18	1,025.0
G	14	87.0
H	17	425.0
I	19	2,675.0
J	16	350.0
K	16	250.0
L	18	350.0
M	17	425.0
N	18	1,625.0
O	15	77.5

b) Develop an estimated multiple regression equation with x = rarity rating and x² as the two independent variables. (Round b₀ and b₁ to the nearest integer and b₂ to one decimal place.)

ŷ =

c) Consider the nonlinear relationship shown by equation (16.7):E(y) = β₀β₁^xUse logarithms to develop an estimated regression equation for this model. (Round b₀ to three decimal places and b₁to four decimal places.)

log(ŷ) =

Answer 1

Solution

we will solve it by using excel and the steps are

Enter the Data into excel

Model	Rating	Rating^2	Price
A	16	256	125
B	13	169	70
C	17	289	140
D	19	361	1,325.00
E	19	361	3,975.00
F	18	324	1,025.00
G	14	196	87
H	17	289	425
I	19	361	2,675.00
J	16	256	350
K	16	256	250
L	18	324	350
M	17	289	425
N	18	324	1,625.00
O	15	225	77.5

Click on Data tab

Click on Data Analysis

Select Regression

Select input Y Range as Range of dependent variable.

Select Input X Range as Range of independent variable

click on labels if your selecting data with labels

click on ok.

So this is the output of Regression in Excel.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8
R Square	0.7
Adjusted R Square	0.7
Standard Error	664.0
Observations	15.0
ANOVA
	df	SS	MS	F	Significance F
Regression	2.0	12698545.4	6349272.7	14.4	0.0
Residual	12.0	5290849.8	440904.2
Total	14.0	17989395.2
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	34041.9	13568.8	2.5	0.0	4477.9	63605.8	4477.9	63605.8
Rating	-4596.9	1677.2	-2.7	0.0	-8251.3	-942.5	-8251.3	-942.5
Rating^2	154.4	51.4	3.0	0.0	42.4	266.3	42.4	266.3

Price =34042 -4597*Rating+154.4*Rating^2

c) Consider the nonlinear relationship shown by equation (16.7):E(y) = β₀β₁^xUse logarithms to develop an estimated regression equation for this model.

take log(price) and log(Rating)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8716
R Square	0.7596
Adjusted R Square	0.7411
Standard Error	0.6654
Observations	15.0000
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	18.1908	18.1908	41.0854	0.0000
Residual	13.0000	5.7558	0.4428
Total	14.0000	23.9467
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-22.4232	4.4363	-5.0545	0.0002	-32.0073	-12.8391	-32.0073	-12.8391
log(Rating)	10.0919	1.5744	6.4098	0.0000	6.6905	13.4933	6.6905	13.4933

Log(Price ) = -22.423+10.0919*log(Rating)