Question

A Realtor is interested in modeling the selling price of houses based on the square footage and the age of the house. The data was collected in the two largest cities in Arkansas and is presented here.

          Square footage X1                   Age in years X2   style           Selling price Y

                   775                                37               Traditional 28,000

                   700                                49               Traditional 34,000

                   720                                54               Traditional 34,500

                   864                                37               Rambler      39,900

                   650                                35               Traditional 40,000

                   780                                79               Victorian    41,500

                   900                                48               Traditional 42,500

                   816                                35               Rambler      53,500

                  1800                              17               Victorian    57,000

                   1340                              66               Victorian    59,000

                   1800                              18               Rambler      59,500

                   1124                              34               Traditional 62,000

                   2880                              24               Victorian    68,500

                   1480                              75               Rambler      72,500

                   1652                              94               Victorian    70,000

                   2088                              71               Victorian    73,112

                   1700                              34               Traditional 76,780

                   1262                              78               Rambler      77,350

                   1500                              54               Victorian    85,590

                   1200                              35               Victorian    79,900

                   650                               45               Traditional 48,100

We need two indicator variables for the style of the house. I will choose Traditional as the base category.

I rambler = { 1 if house is a rambler; 0 if not Ivictor = { 1 if house is victorian; 0 if not

When entering the data do not use the commas

1. Plot y vs. x1 and y vs. x2.   Do you see any curvature in these 2 plots? If so what can be suggested about the variables? Now what model do you think needs to be used.

2. Suppose someone wishes to use the regression model

Y= B0+B1X1+B2X2+B3X1^2+B5X1X2+B6Irambler+B7Ivictor+E

a) Write Down the prediction equation.

b) Interpret R2.

c) Test if the regression model is useful. (F-test)

Answer 1

1)

There seems to some linear curve.

Graph shows that the data is distributed independent without any curvature.

2) Using Excel:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.820985
R Square	0.674016
Adjusted R Square	0.534308
Standard Error	11801.35
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	6	4031480929	671913488.1	4.824476239	0.007212744
Residual	14	1949805195	139271799.6
Total	20	5981286124
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3439.088	30244.23111	0.11371054	0.911081829	-61428.33623	68306.51192
X1	62.42002	32.17639197	1.939932185	0.072804956	-6.591478554	131.4315154
X2	-94.0395	479.2688606	-0.196214598	0.84726185	-1121.969016	933.8899223
X1^2	-0.01555	0.007485261	-2.077782585	0.056609328	-0.031607032	0.000501543
X1*X2	0.118454	0.305604947	0.387604802	0.704137596	-0.537003475	0.773911364
Rambler	3081.234	7524.897006	0.409471859	0.688389479	-13058.06531	19220.53244
Victorian	3291.8	8165.320837	0.403144039	0.692931624	-14221.07096	20804.6718

b) R-squared value: 0.674016

The proporiton of variance explained by regression equation is 0.674016

c)

ANOVA
	df	SS	MS	F	Significance F
Regression	6	4031480929	671913488.1	4.824476239	0.007212744
Residual	14	1949805195	139271799.6
Total	20	5981286124

Test statistic is significant and can conlude there is atleast one variable good for prediction.