1. Suppose we are interested in the linear model y;- = g + 61x\" +16%? + e.-. Also suppose the columns x1 and :2 of the design matrix for this model have mean I} and length 1. {That is. x'lxl = 1. and the same is true of :12.) Then if p is the correlation between it] and 312. we have the following: n U Cl 1 Cl Cl XX: 9 1p marrxrl= {l llpg 1f} =3 p 1 D willpf) Nilp?) (a) Determine what values of p will make the variance of 161 and 32 large. EXpiEin why. using what you know about the variance of the vector ,6. {No fair looking at the 'v'lF formulas.) {b} In our setup where the predictors have mean {l and length 1. show that SXX = 1. Use that to show that the 'v'lF formula on page 2'03 matches rm1 (above). 2. In a study on weight gain in rabbits. researchers randomly assigned E- rabbits to 1. 2. or 3 mg of one of dietary Supplement A or B (one rabbit to each level of each supplement). Consider the linear model Y = n + 61x1 +132X2+ e. where x1 is the dosage level of the supplement. and x2 is a dummy variable indicating the type of supplement used. (a) Compute the variance inflation factor for variable x1. You should be able to do this completely without the use of statistical software. Explain. using the word "orthogonal." why the variance ination factor is the value computed. {b} Now suppose the researcher used levels 1. 2. and 3 for supplement A. and levels 2. 3. and 4 for supplement B. Use software if desired. What is the variance inflation factor for variable 11 in this case? Is it larger or smaller than in part {a} above? vvhy? {c} Now consider a linear model (not for our rabbits) Y = g +5111 +16%; + ..+ .6po + e. Under what circumstances would the variance ination factors for all p variables be equal to 1? (Hint: To get started in part (a). it is perfectly reasonable to use R and the vif(} function. You'll have to invent a response vector y; try: 3;