Residential real estate prices depend, in part, on property size and number of b

Question

Residential real estate prices depend, in part, on property size and number of bedrooms. The house size X1 (in hundreds of square feet), number of bedrooms X2, and house price Y (in thousands of dollars) of a random sample of houses in a certain county were observed. The data are listed in the following table (-0.05) House House size (X) Number of bedrooms (%) House price (Y) 2 20 80 95 18 25 19 17 104 110 175 85 89 PLease include SAS output and SAS codes with Answer. d) Determine r2Y X1, X2the squared multiple correlation between house price (Y) and the independent variables house size (X1) and number of bedrooms (X2)

Explanation / Answer

Here there are two independent variables and one dependent variable.

X1 : House size

X2 : number of bedrooms

Y : House price

Here we have to fir multiple regression.

We can fit multiple regression in MINITAB.

Steps :

ENTER data into MINITAB sheet --> STAT --> Regression --> Regression --> Response : Y --> Predictors : X1 and X2 --> Results : select second option --> ok --> ok

Output :

Regression Analysis: Y versus X1, X2

Analysis of Variance

Model Summary

Coefficients

Regression Equation

Intercept b0 = -16.1

Slope b1= 5.72  

b2 = -1.2

Interpretation : If we fixed X1 then one unit change in X2 will be 1.2 unit decrease in Y.

If we fixed X2 then one unit change in X1 willbe 5.72 unit increase in Y.

Here we can test two hypothesis.

i) Overall significance :

The hypothesis for the test is,

H0 : Bj = 0 Vs H1 : Bj not= 0

where Bj is the population slope for jth independent variable.

Assume alpha = level of significance = 0.05

Test statistic follows F-distribution.

Test statistic = 20.03

P-value = 0.008

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : Atleast one of the slope is differ than 0.

ii) Individual significance :

The hypothesis for the test is,

H0 : B = 0 Vs H1 : B not= 0

where B is the population slope for independent variable.

Assume alpha = level of significance = 0.05

Test statistic follows t-distribution.

Decision rule :

If P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : Independent variable is significant otherwise variable is significant.

Now from the output we can see that X1 is significant variable whereas X2 as insignificant variable.

R-sq = 86.38% = 0.8638

It expresses the proportion of variation in Y which is expressed by variation in X1 and X2.

Multiple R = 0.9535

It indicates that there is positive relationship between Y and X1 X2.

Source DF Adj SS Adj MS F-Value P-Value Regression 2 5733.32 2866.66 20.03 0.008 X1 1 1402.32 1402.32 9.80 0.035 X2 1 1.10 1.10 0.01 0.934 Error 4 572.39 143.10 Total 6 6305.71

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