A linear regression function is shown as follows: Y = Bo + BI*X + H, H ~ N(0, o') The data is shown in the following table: Table 1 X 4 5 1 V 4 5 2 2Choosing Bo & B, to minimize E(Y; -B. -B,X,)? (2) i= 1 Using take the first derivative of equation (2) calculus w.r.t Bo & Bi: =>-2*) (Y - Bo - B*X) = 0 X3 *'d +"d*u = A3 -2*E X (Y - Bo - B*X) = 0 => EXY = B, * EX + B,*EX- ( 2b ) SXY E ( x; - X ) * ( Y; - Y)~ B, = 1=1 (3) E(x; - x)2 SXX i= 1 B . = Y - B, * X (4)3 Y ; = B + B, * X (5) The numerator part is called the residual sum of squares; RSS and e; = Yi- Y; ; e; is termed residual. n -2 (6)Estimated Bo and Pi are denote by (0, a vector of two components. B E(P) = E( .. OLS estimator is unbiased (7) B Var( ) has the smallest variance among the linear (8) B estimator ... OLS estimator is efficient.Var(P) = Var(B.) Cov( B., B.) Cov(Bo,B.) Var(B,) EX -EX; n* EX,' -(EX; )2 -EX; n (2) SXX = [(X; - X)= = _(x2 - 2X; * X+X ) = [x2 - 2*X* [X; + EX- = EX2 . 2*. EX; i* EX; +n*X' = yx2 . 2(2x; )= n n n = EX.2 - 2*. (EX;) Ex 2 (2X; )2 n n n Let's focus on the estimated variance of Bs: Var(B ) = 0'*- EX.2 =0*. SXX + (EX;) = 2* SXX +n* x2 n * SXX n * SXX + n SXX (10)a) Report and interpret the estimates of Bo and B. b) List the predicted y (9) and the corresponding residuals in a table c) Find the estimated variance &. d) Find the test statistic (i.e., t-value) for B, where the null and alternative is B, = 0 vs. B, # 0 e) Find the p-value of the test statistic for B, f) What's your hypothesis testing conclusion if the rejection region is set at 5%? g The coefficient of the determination (R3) and explain what it means. h) Show the data points, linear regression line, residuals in a figure