2. Use the data set cars_kaggle.csv. a. Create 4 probit models to predict if the person did (Purchased = 1) or did not (Purchased =0) buy a car. Use only Annual Salary (x1) as the explanatory, use only Age (x2) as the explanatory, use Annual Salary and Age, use Annual Salary, age and the interaction between Age and Annual Salary. b. List out all four models in the form of probit (ft) = a + Bx1 + Bx2... c. Fill in the following table (remember residual deviance from R is deviance) Model AIC Deviance P-value P-value Comparison Difference Pair for for comparing model to saturated Model 1 Model 2 Model 3 Model 4 comparing model to Null model (1 and 3) (2 and 3) (3 and 4) in Deviance for Nested models d. Which is the best probit model? Explain. e. Repeat parts a, b and c using logit models instead of probit models. P-value comparing models f. Which is the best logit model? g. Using the best probit model and best logit model from parts d and f, create the ROC curves. h. Using the AIC and ROC curve, which model between the best logit and best probit fits the data better? The best model to predict if a person will pass a test is a logit model that uses the hours spent on homework and the type of lecture used but not including an interaction term. This model is better than the null model (p-value = 0.003) but is still missing terms when compared to the saturated model (p- value = 0.312). However, it outperformed the other models based on difference in deviances and more area under the ROC curve. The AIC was a little bit higher than the probit model with the same explanatory variables, but they were very close, and the ROC curve was quite a bit better for the logit model. If you chose Model 1: Salary 50000 75000 100000 150000 Age 25 35 45 55 i. Summarize your results in a form like the following paragraph and fill in the predicted probabilities using the chosen model. If you chose Model 2: 50000 75000 100000 150000 Predicted Probability of a purchase If you chose Model 3 or 4: Salary Age 50000 75000 100000 150000 Predicted Probability of a purchase 35 35 35 35 45 45 45 45 Predicted Probability of a purchase