In both problems, you have a random pair (X, Y) and Y^ is the best linear predictor of Y based on X.

a) We Say that the regression line passes through the point of averages. Show this by setting X=x and finding the corresponding value of Y^.

b) Find E(Y^) .

c) The differenceYY^ is called a residual. It is the difference between the actual and fitter values of Y. Find the expectation of the residual and confirm that the answer justifies the following statement: "No matter what the shape of the scatter diagram, the average of the residual is 0."

d) Find Var(Y^) .

e) We have the following statement: SDofySDoffittedvalues=r

Justify this statement, by using the result from (d) to show that: Var(Y)Var(Y^)=r2

This result is whyr2 is sometimes called "the proportion of variability explained by the linear model".