In class we mentioned for a simple linear regression model, the corresponding σ 2 is estimated as ˆσ 2 = pn i=1(yi − yˆi) 2 n − 2 . show the estimator is unbiased by following the steps below. (a) show that cov(yi , yˆi) = σ 2 [ 1 n + (xi−x) 2 sxx ]. (b) show that var(yi − yˆi) = σ 2 [1 − 1 n − (xi−x) 2 sxx ]. (c) show that e( pn i=1(yi − yˆi) 2 ) = (n − 2)σ 2 . hint: cov(ax + by, cw + dv ) = accov(x, w) + adcov(x, v ) + bccov(y, w) + bdcov(y, v ), where a, b, c, d are constants and x, y, w, v are random variables.

Given: ad¯¯¯¯¯ is an altitude. prove: ab2+ac2=cb2 right triangle a b c with right angle a. point d lies on side b c and segment a d is drawn. angle a d c is a right angle. drag and drop a reason into each box to correctly complete the two-column proof. statement reason ad¯¯¯¯¯ is an altitude, and ∠bac is a right angle. given ∠adb and ∠adc are right angles. definition of altitude ∠bac≅∠bda ? ∠bac≅∠adc ? ∠b≅∠b ? ∠c≅∠c reflexive property of congruence △abc∼△dba ? △abc∼△dac aa similarity postulate abbd=cbab ? ab2=(cb)(bd) cross multiply and simplify. acdc=cbac polygon similarity postulate ac2=(cb)(dc) cross multiply and simplify. ab2+ac2=ab2+(cb)(dc) addition property of equality ab2+ac2=(cb)(bd)+(cb)(dc) substitution property of equality ab2+ac2=(cb)(bd+dc) ? bd+dc=cb segment addition postulate ab2+ac2=cb2 substitution property of equality