1.Based on our known "mtcars" dataset, we'll now build a model for predicting variable qsec, the time in seconds that it takes a car to drive 1/4 mile from a full stop. (

a) Select the optimal subset of independent variables for the prediction of the qsec using your favorite variable selection method. Show all methods used and justify your conclusion.

(b) Is there any indication of nonlinear relations? Explain how you reach this conclusion.

(c) Test significance of interaction between the weight (wt) and transmission (am, 0 = automatic, 1 = manual). (Hint: Use the "best" model according to part (a)).

2.A student fitted a linear regression function for a class assignment. The student plotted the residuals against responses and found positive relation. When the residuals were plotted against the fitted values , the student found no relation. (a) How could the differences arise? Which is the more meaningful plot? (b) Support your answer by deriving the sample covariance Cov( , ) and Cov( , ). Feel free to use any Formulae and any results that we derived in class.

3.An experiment was conducted to evaluate the effect of vitamin C on tooth growth. Sixty guinea pigs received various doses of vitamin C by one of two delivery methods, orange juice or ascorbic acid. Results of this experiment are in dataset ''ToothGrowth'' which is already loaded in R. You can look at it with commands attach(ToothGrowth), names(ToothGrowth), summary(ToothGrowth), ToothGrowth. The data set contains the following variables: len: Tooth length supp: supplement or delivery method (OJ = orange juice, VC = ascorbic acid) dose: Dose in milligrams/day (a) Fit a linear regression model that can be used to predict the tooth length based on the dose and the delivery method of vitamin C. (b) Is delivery method significant? (c) Is there a significant interaction between the dose and the delivery method, at = 5% level? (d) Write two regression equations explicitly, one equation for each delivery method. (e) What percent of the total variation of the tooth length is explained by this regression?