Answer All questions Work Independently A real estate expert was interested in d
ID: 3126983 • Letter: A
Question
Answer All questions
Work Independently
A real estate expert was interested in developing a regression model that relates the selling price (in thousand of dollars) of properties to characteristics of the properties. Data were available on 30 properties that were sold recently. The expert developed a long list of possible explanatory variables. After a careful screening, it was decided that the following four characteristics should be considered.
Variable
Description
x1
Property taxes (annual taxes in dollars)
x2
House size (floor area in square feet)
x3
Lot size (in acres)
x4
Attractiveness Index
Regression Analysis: Selling Price versus Taxes, House, Lot, Attract
The regression equation is
Selling Price = 11.8 - 0.0233 Taxes + 0.109 House + 44.4 Lot + 2.99 Attract
Predictor Coef SE Coef T P
Constant 11.83 66.32 0.18 0.860
Taxes -0.02331 0.02056 -1.13 0.268
House 0.10948 0.02442 4.48 0.000
Lot 44.40 21.76 2.04 0.052
Attract 2.9926 0.6589 4.54 0.000
S = 32.13 R-Sq = 72.1% R-Sq(adj) = 67.7%
Analysis of Variance
Source DF SS MS F P
Regression 4 66827 16707 16.18 0.000
Residual Error 25 25815 1033
Total 29 92641
a.Write out the general Multiple linear regression model for this problem with the variables
where Y = selling price of properties
b.Write out the estimated (least-squares) regression line for this problem.
c.Use the estimated regression line to predict the average selling price of 2900 square-foot homes on a 2.5-acre lot with $6000 in annual property taxes and an attractive index of 45.
d.What is the b3 slope estimate in terms of this problem?
e.What is the correlation coefficient determined by the multiple linear regression model using taxes, house size, lot size, and attractiveness as predictors?
f.What percentage of variation in selling price is explained by the multiple linear regression model using taxes, house size, lot size, and attractiveness as predictors?
g. Explained how the adjusted R was calculated?
Variable
Description
x1
Property taxes (annual taxes in dollars)
x2
House size (floor area in square feet)
x3
Lot size (in acres)
x4
Attractiveness Index
Explanation / Answer
here the response variable is Y=selling price of properties
and the predictors are X1=Property taxes (annual taxes in dollars)
X2=House size (floor area in square feet)
X3=Lot size (in acres)
X4=Attractiveness Index
a) so the general Multiple linear regression model is
Y=b0+b1*X1+b2*X2+b3*X3+b4*X4+e
where b0,b1,b2,b3,b4 are parameters that are to be estimated using LEAST SQUARE METHOD.
and e is the error term.
assumption is that e~N(0,sigma2)
b) the least square regression line as found using MINITAB using least square method is [as provided in the question]
Y = 11.8 - 0.0233*X1 + 0.109*X2 + 44.4*X3 + 2.99*X4 [answer]
c) we need to predict the average selling price of 2900 square-foot homes on a 2.5-acre lot with $6000 in annual property taxes and an attractive index of 45.
i.e, we need to predict Y given that X2=2900 X1=6000 X3=2.5 X4=45 using the estimated linear regression equation.
so Y=11.8 - 0.0233*6000 + 0.109*2900 + 44.4*2.5 + 2.99*45=433.65 thousand dollars [answer]
d) using the equation the b3 slope estimate is=the coefficient of X3=44.4 [answer]
e) in the question it is given that R2=72.1%=0.721
so the correlation coefficient is sqrt(R2)=sqrt(0.721)=0.849 [answer]
f) the percentage of variation in selling price is explained by he multiple linear regression model using taxes, house size, lot size, and attractiveness as predictors is determined by adjusted R2
so R-Sq(adj) = 67.7%
so 67.7% of variation in selling price is explained by he multiple linear regression model using taxes, house size, lot size, and attractiveness as predictors [answer]
g) adjusted R2 was calculated using the formula
adj-R2=R2-(1-R2)*p/(n-p-1)
where R2=0.721 p=total number of explanatory variables=4 n=total number of observations=30
so adj-R2=0.721-(1-0.721)*4/(30-4-1)=0.677 [answer]