. Show that Var (YE) = Var (ei) in the simple linear regression model. (Yes, this should be that simple.) What did you assmne? Prove that Cov{n.X,bY) = abCov(X, Y), for any two random variables X21\" and constants (1,3). T. A statistics professor has been involved in a collaborative research project with two entomologists. The statistics part of the project involves tting regression models to large data sets. Together they have written and submitted a manuscript to art entomology journal. The manuscript contains a number of scatter plots with each showing an estimated regression line (based on a valid model] and 1.3 Exercises 43 associated individual 95% condence intervals for the regression function at each .r value. as well as the observed data. A referee has asked the following question: I don't understand how 95% of the observations fall outside the 95% Cl as depicted in the gures. Briey explain how it is entirely possible that 95% of the observations fall outside the 95'3- Cl as depicted in the Figures. 8. Show that the least. squares criterion applied to the \"interceptonly\" model: yiz0+ei;i:11-":nr results in the estimator B\" = g for [30. llint: Follow the steps: (a) 1Write down your design matrix, X. {It won't be the same as any we've used in class. Double check: does 3; = X .3 + e. give back the set of equations listed above? Notice this model has no predictor variable. The number of columns in the design matrix should always equal the number of parameters in the model. (b) Use the formula [1' = {X'X)_]X'y to get the least squares estimator.