Question

Suppose the following data were collected from a sample of 1515 houses relating selling price to square footage and the architectural style of the house. Which of the following is the best equation to use relating the selling price of a house to square footage and the style of the house?

Copy Data

Housing Prices

Selling Price	Square Footage	Colonial (1 if house is Colonial style, 0 otherwise)	Ranch (1 if house is Ranch style, 0 otherwise)	Victorian (1 if house is Victorian style, 0 otherwise)
391430391430	23032303	00	11	00
381002381002	20532053	11	00	00
403539403539	20132013	00	00	11
405271405271	25522552	00	00	11
406578406578	31313131	00	00	11
471858471858	36593659	00	11	00
392188392188	23322332	00	11	00
475616475616	35883588	11	00	00
401742401742	18431843	00	00	11
404836404836	26562656	11	00	00
333709333709	13371337	11	00	00
393618393618	23892389	11	00	00
365651365651	17991799	00	11	00
404239404239	23212321	00	00	11
375624375624	19461946	00	11	00

Answer 1

Regression analysis is a basic method used in statistical analysis of data. It’s a statistical method which allows estimating the relationships among variables. One needs to identify dependent variable which will vary based on the value of the independent variable. For example, the value of the house (dependent variable) varies based on square feet of the house (independent variable). Regression analysis is very useful tool in predictive analytics.

E(Y | X) = f(X, β)

Y = f(X) = ?0 + ?1 * X

?0 is the intercept of the line

?1 is the slope of the line

Linear regression algorithm is used to predict the relationship(line) among data points. There can be many different (linear or nonlinear) ways to define the relationship. In the linear model, it is based on the intercept and the slope. To find out the most optimal relationship, we need to train the model with the data.

Before applying the linear regression model, we should determine whether or not there is a relationship between the variables of interest. A scatterplot is a good starting point to help in determining the strength of the relationship between two variables. The correlation coefficient is a valuable measure of association between variables. Its value varies between -1 (weak relationship) and 1 (strong relationship).

Once we determine that there is a relationship between variables, next step is to identify best-fitting relationship (line) between the variables. The most common method is the Residual Sum of Squares (RSS). This method calculates the difference between observed data (actual value) and its vertical distance from the proposed best-fitting line (predicted value). It squares each difference and adds all of them.

The MSE (Mean Squared Error) is a quality measure for the estimator by dividing RSS by total observed data points. It is always a non-negative number. Values closer to zero represent a smaller error. The RMSE (Root Mean Squared Error) is the square root of the MSE. The RMSE is a measure of the average deviation of the estimates from the observed values. This is easier to observe compare to MSE, which can be a large number.

RMSE (Square root of MSE) = √ (MSE)

The additional number of variables will add more dimension to the model.

Y = f(X) = ?0 + ?1 * X1 + ?1 * X2 + ?1 * X3

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 3 1.48677E+22 4.95589E+21 19.17 0.000
Square Footage 1 1.47522E+22 1.47522E+22 57.08 0.000
Colonial (1 if house is Colonia 1 1.65815E+20 1.65815E+20 0.64 0.440
Ranch (1 if house is Ranch styl 1 1.12558E+20 1.12558E+20 0.44 0.523
Error 11 2.84317E+21 2.58470E+20
Total 14 1.77108E+22

Model Summary

S R-sq R-sq(adj) R-sq(pred)
1.60770E+10 83.95% 79.57% 65.77%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 2.85798E+11 17251859972 16.57 0.000
Square Footage 4994 661 7.55 0.000 1.00
Colonial (1 if house is Colonia -740537218 924569907 -0.80 0.440 1.33
Ranch (1 if house is Ranch styl -610158292 924612666 -0.66 0.523 1.33

Regression Equation

Selling Price = 285797688046 + 4994 Square Footage
- 740537218 Colonial (1 if house is Colonia
- 610158292 Ranch (1 if house is Ranch styl

Fits and Diagnostics for Unusual Observations

Obs Selling Price Fit Resid Std Resid
5 4.06578E+11 4.42185E+11 -3.56063E+10 -2.64 R