Predicting Prices of Used Cars (Regression Trees). The file ToyotaCorolla.jmp contains the data on used cars (Toyota Corolla) on sale during late summer of 2004 in The Netherlands. It has 1436 records containing details on 38 attributes, including Price, Age, Kilometers, HP, and other specifications. The goal is to predict the price of a used Toyota Corolla based on its specifications. (The example in Section 9.8 is a subset of this dataset).

Data preprocessing. Split the data into training (50\%), validation (30\%), and test $(20 \%)$ datasets.

a. Run a regression tree with the output variable Price and input variables Age_08_04, KM, Fuel_Type, HP, Automatic, Doors, Quarterly_Tax, Mfg_Guarantee, Guarantee_Period, Airco, Automatic_Airco, CD_Player, Powered_Windows, Sport_Model, and Tow_Bar. Set the minimum split size to 1 , and use the split button repeatedly to create a full tree (hint, use the red triangle options to hide the tree and the graph). As you split, keep an eye on RMSE and RSquare for the training, validation and test sets.

i. Describe what happens to the RSquare and RMSE for the training, validation and test sets as you continue to split the tree.

ii. How does the performance of the test set compare to the training and validation sets on these measures? Why does this occur?

iii. Based on this tree, which are the most important car specifications for predicting the car's price?

iv. Refit this model, and use the Go button to automatically split and prune the tree based on the validation RSquare. Save the prediction formula for this model to the data table.

v. How many splits are in the final tree?

vi. Compare RSquare and RMSE for the training, validation and test sets for the reduced model to the full model.

vii. Which model is better for making predictions? Why?