1) This question involves the use of simple linear regression on the Auto data s

1) This question involves the use of simple linear regression on the Auto data set.
(a) Use the lm() function to perform a simple linear regression with mpg as the response and horsepower as the predictor. Use the summary() function to print the results. Comment on the output.
3. Linear Regression
i. Is there a relationship between the predictor and the response?
ii. How strong is the relationship between the predictor and the response?
iii. Is the relationship between the predictor and the response positive or negative?
iv. What is the predicted mpg associated with a horsepower of 98? What are the associated 95 % confidence and prediction intervals?
(b) Plot the response and the predictor. Use the abline() function to display the least squares regression line.
(c) Use the plot() function to produce diagnostic plots of the least squares regression fit. Comment on any problems you see with the fit.
2) This question involves the use of multiple linear regression on the Auto data set.
(a) Produce a scatterplot matrix which includes all of the variables in the data set.
(b) Compute the matrix of correlations between the variables using the function cor(). You will need to exclude the name variable, cor() which is qualitative.
(c) Use the lm() function to perform a multiple linear regression with mpg as the response and all other variables except name as the predictors. Use the summary() function to print the results. Comment on the output. For instance: i. Is there a relationship between the predictors and the response? ii. Which predictors appear to have a statistically significant relationship to the response? iii. What does the coefficient for the year variable suggest?
(d) Use the plot() function to produce diagnostic plots of the linear regression fit. Comment on any problems you see with the fit. Do the residual plots suggest any unusually large outliers? Does the leverage plot identify any observations with unusually high leverage?
(e) Use the * and : symbols to fit linear regression models with interaction effects. Do any interactions appear to be statistically significant?
(f) Try a few different transformations of the variables, such as log(X), √ X, X2. Comment on your findings.
3) This problem focuses on the collinearity problem.
(a) Perform the following commands in R:
> set.seed(1)
> x1=runif (100)
> x2=0.5*x1+rnorm (100)/10
> y=2+2*x1+0.3*x2+rnorm (100) The last line corresponds to creating a linear model in which y is a function of x1 and x2. Write out the form of the linear model. What are the regression coefficients?
(b) What is the correlation between x1 and x2? Create a scatterplot displaying the relationship between the variables.
(c) Using this data, fit a least squares regression to predict y using x1 and x2. Describe the results obtained. What are βˆ0, βˆ1, and βˆ2? How do these relate to the true β0, β1, and β2? Can you reject the null hypothesis H0 : β1 = 0? How about the null hypothesis H0 : β2 = 0? 126
3. Linear Regression
(d) Now fit a least squares regression to predict y using only x1. Comment on your results. Can you reject the null hypothesis H0 : β1 = 0?
(e) Now fit a least squares regression to predict y using only x2. Comment on your results. Can you reject the null hypothesis H0 : β1 = 0?
(f) Do the results obtained in (c)–(e) contradict each other? Explain your answer.
(g) Now suppose we obtain one additional observation, which was unfortunately mismeasured.
> x1=c(x1, 0.1)
> x2=c(x2, 0.8)
> y=c(y,6)
Re-fit the linear models from (c) to (e) using this new data. What effect does this new observation have on the each of the models? In each model, is this observation an outlier? A high-leverage point? Both? Explain your answers.
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