3240_F21_A2 v Submit your completed assignment to Crowdmarlr by 11:59 pm on Wednesday October 20. Submit 3 pdf documents, one {or each question number. Each part to each question is worth 5 marks, I've included some R code that may be helpful at the end of this document. 1. Consider the PCB / egg shell thickness data from Assignment #1. (a) Find a 95% condence interval for the true mean eggshell thickness if PCB concentration is 200 ppm. (b) Find a 95% prediction interval for the true mean eggshell thickness if PCB concentration is 200 ppm. (0) What proportion ol the variance in eggshell thickness can be explained by the linear relationship with PCB concentration? 2. Table 311 in Montgomery, Peck, and Vining's Linear Regression Analysis contains data on Pinot Noir wine. We'll look at a subset of this data, contained in a le called 324D_F21_wine. We'll investigate a possible relationship between the quality of the wine and several variables: Clarity, Aroma, Body, Flavor. and Oakiness. Run a multiple regression of Quality on Clarity, Aroma, Body, Flavor, and Oakiness. (3) Give an interpretation of the coeicient of Oakiness (in the context of the problem at hand). (b) Create appropriate residual plots and include them in your submission. Do any of the residual plots show problems with the model? (c) Carry out a test of the null hypothesis that none of the variables are linearly related to Quality. Give appropriate hypotheses (in words and symbols), test statistic, pvalue, and conclusion. (You don't need to carry out any calculations; you can use information hrom the output.) (d) Carry out a test ol the null hypothesis that Oalriness is not linearly related to Quality, given the other variables are in the model. (In other words, after adjusting for the other variables.) Give an appropriate conclusion in the context of the problem at hand. That's it for the computer portion of this assignment. 3. Show that the hat matrix is symmetric and idempotent. (That is, show that H' = H and that HR = H.) 4. Show that the vector of observed residuals can be expressed as e = (I 11):, 5. Prove or disprove: 1/12 = We 6. (a) Show that in the simple linear regression setting, the usual F test statistic of Ho: r : 0 can be written as R2( 2) n _ F 1122 (h) In a simple linear regression with n = 300, how large rnust R1 be in order for there to be signicant evidence against the null hypothesis from a), at the 5% signicance level, 7. Suppose we run a simple linear regression and find the estimated oovarianoe matrix of the parameter estimators is: l] u- in