Question

An agent for a real estate company in a large city would like to be able to predict the monthly rental cost for​ apartments, based on the size of the​ apartment, as defined by square footage. A sample of eight apartments in a neighborhood was​ selected, and the information gathered revealed the data shown below. For these​ data, the regression coefficients are b 0 = 219.3999 and b 1 =0.9884.

Monthly Rent($) Size(Square Feet)
900 800
1,550 1,250
825 950
1,600 1,150
1,950 1,900
925 650
1,700 1,250
1,250 1,100

A. Determine the coefficient of​ determination, r2​, and interpret its meaning.

B. Determine the standard error of the​ estimate, Syx​, and interpret its meaning.

C. How useful is this model for predicting the monthly​ rent?

D. What other variables might explain the variation in monthly​ rent?

Answer 1

X	Y	XY	X²	Y²
800	900	720000	640000	810000
1250	1550	1937500	1562500	2402500
950	825	783750	902500	680625
1150	1600	1840000	1322500	2560000
1900	1950	3705000	3610000	3802500
650	925	601250	422500	855625
1250	1700	2125000	1562500	2890000
1100	1250	1375000	1210000	1562500
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
9050	10700	13087500	11232500	15563750

Sample size, n =	8
x̅ = Ʃx/n =	1131.25
y̅ = Ʃy/n =	1337.5
SSxx = Ʃx² - (Ʃx)²/n =	994688
SSyy = Ʃy² - (Ʃy)²/n =	1252500
SSxy = Ʃxy - (Ʃx)(Ʃy)/n =	983125

1. Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = 0.7758

r² measures the proportion of variation in monthly rent that can be explained by the variation
in apartment size.

-----------------------

2. Sum of Square error, SSE = SSyy -SSxy²/SSxx =    280803.0946

Standard error, S_yx = √(SSE/(n-2)) = 216.33427

S_yx measures the typical difference between an apartment's actual rent and the rent predicted
by the regression equation.

-----------------------

3. It is very useful for predicting the monthly rent because r² is close to 1 and S_YX is fairly small
compared to the actual rents.

-----------------------

4. The condition of the apartment.

Apartment building location.

Parking is available to tenants. etc.