Exercise 1 (5 points)
For each of parts (a) through (d), indicate whether we would generally expect the performance of a flexible statistical learning method to be better or worse than an inflexible method. Justify your answer.
- The sample size \(n\) is extremely large, and the number of predictors \(p\) is small.
- The number of predictors \(p\) is extremely large, and the number of observations \(n\) is small.
- The relationship between the predictors and response is highly non-linear.
- The variance of the error terms, is extremely high.
Exercise 2 (5 points)
Describe the difference between a parametric and non-parametric statistical learning approach. What are the advantages of a parametric approach to regression or classification (as opposed to a noon-parametric approach)? What are its disadvantages?
Exercise 3 (5 points)
Carefully explain the the difference between the KNN classifier and KNN regression methods. Name a downside when using this model on very large data.
Exercise 4 (5 points)
Suppose we have a data set with five predictors, \(X1 =\) GPA, \(X2 =\) extracurricular activities (EA), \(X3 =\) Gender (1 for Female and 0 for Male), \(X4 =\) Interaction between GPA and EA, and \(X5 =\) Interaction between GPA and Gender. The response is starting salary after graduation (in thousands of dollars). Suppose we use least squares to fit the model, and get \(\beta_0 = 50\), \(\beta_1 = 20\), \(\beta_2 = 0.07\), \(\beta_3 = 35\), \(\beta_4 = 0.01\), \(\beta_5 = - 10\).
- Which answer is correct, and why?
- For a fixed value of EA and GPA, males earn more on average than females.
- For a fixed value of EA and GPA, females earn more on average than males.
- For a fixed value of EA and GPA, males earn more on average than females provided that the GPA is high enough.
- For a fixed value of EA and GPA, females earn more on average than males provided that the GPA is high enough.
- Predict the salary of a female with EA of 110 and a GPA of 4.0.
- True or false: Since the coefficient for the GPA/EA interaction term is very small, there is very little evidence of an interaction effect. Justify your answer.
Exercise 5 (10 points)
This question should be answered using the biomass
data set.
- Fit a multiple regression model to predict
HHV
using carbon
, hydrogen
and oxygen
.
- Provide an interpretation of each coefficient in the model. Be careful, note the values
Cruise
is able to take.
- Write out the model in equation form.
- For which the predictors can you reject the null hypothesis \(H_0: \beta_j = 0\)?
- On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome.
- How well do the models in (a) and (e) fit the data? How big was the effect of removing the predictor?