Question

12.21 In Problem 12.9 on page 424, an agent for a real estate company wanted to predict the monthly rent for one bedroom apartments, based on the size of the apartment (stored in ). Using the results of that problem, Rent-SilverSpring r2, a. determine the coefficient of determination, and interpret its meaning. b. determine the standard error of the estimate. c. How useful do you think this regression model is for predicting the monthly rent? d. Can you think of other variables that might explain the varia tion in monthly rent?

Size (Square feet)	Rent ($)
524	1110
616	1175
666	1190
830	1410
450	1210
550	1225
780	1480
815	1490
1070	1495
610	1680
835	1810
660	1625
590	1469
675	1395
744	1150
820	1140
912	1220
628	1434
645	1519
840	1105
800	1130
804	1250
950	1449
800	1168
787	1224
960	1391
750	1145
690	1093
840	1353
850	1530
965	1650
1060	1740
665	1235
775	1550
960	1545
827	1583
655	1575
535	1310
625	1195
749	1200
634	1185
641	1444
860	1385
740	1275
593	1050
880	1650
895	1340
692	1560

Answer 1

Solution:

Here, we have to construct the regression model for the prediction of the dependent variable Rent in $ based on the size (square feet) of the one bedroom apartment. Required regression model by using excel is given as below:

Regression Statistics
Multiple R	0.354314232
R Square	0.125538575
Adjusted R Square	0.106528544
Standard Error	186.0406563
Observations	48
ANOVA
	df	SS	MS	F	Significance F
Regression	1	228565.1932	228565.1932	6.603806955	0.013481438
Residual	46	1592111.786	34611.12578
Total	47	1820676.979
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	992.9926979	147.3668699	6.73823566	2.25238E-08	696.3585787	1289.626817
Size (Square feet)	0.493166782	0.19190957	2.569787336	0.013481438	0.10687286	0.879460704

For this regression model, the regression equation is given as below:

Rent ($) = 992.99 + 0.4932*Size (sq.ft.)

The correlation coefficient between two variables rent and size is given as 0.3543, which means there is a low positive linear relationship or association exists between the given two variables rent and size of an apartment.

Questions:

a. determine the coefficient of determination, and interpret its meaning.

The coefficient of determination or the value for R square for this regression model is given as 0.1255, which means only 12.55% of the variation in the dependent variable rent is explained by the independent variable size of the apartment.

b. determine the standard error of the estimate.

The standard error of the estimate is given as 186.0406563.

c. How useful do you think this regression model is for predicting the monthly rent?

The P-value for overall regression model is given as 0.013481438 which is less than 5% level of significance or alpha value 0.05, so we reject the null hypothesis that given regression model is not statistically significant. This means this regression model is useful for prediction of the monthly rent at 5% level of significance. (Please note that, P-value = 0.013481438 > α = 0.01, so this regression model is not useful at 1% level of significance.)

d. Can you think of other variables that might explain the variation in monthly rent?

Yes, there are several other variables that might explain the variation in monthly rent. The variables such as a location of the apartment, facilities available near the apartment, infrastructure of an apartment, etc. may cause the variation in the dependent variable monthly rent.