Achievement tests are commonly seen in educational or employment settings. a lar

Question

Achievement tests are commonly seen in educational or employment settings. a large population of let Y_1, Y_2, and Y_3 represent scores for different parts of an exam. Suppose that E(Y_1) = 12, Var (Y_1) = 4, E(Y_2) = 16, Var(Y_2) = 9, and R(Y_3) = 20, Var(Y_3) = 16. Suppose additionally that Y_1 and Y_2 are independent Cov(Y_1, Y_3) = 0.8, and Cov(Y_2, Y_3) = -6.7. Two different summary measures are computed to assess a subject's performance: U_1 = 0.5Y_1 - 2Y_2 and U_2 = 3Y_1 - Y_3. Find expected values and standard deviations of measures U1 and U2

Explanation / Answer

Expected value of U1 =E(U1) =0.5*E(Y1)-2*E(Y2) =0.5*12-2*16=-26

std deviation of U1= (0.52*Var(Y1)+22*Var(Y2)-2*0.5*2Cov(Y1,Y2))1/2=(0.25*4+4*9)1/2=6.0828

Expected value of U2 =3*E(Y1)-E(Y3)=3*12-20=16

std deviation of U2= (32*Var(Y1)+Var(Y3)-2*3*1Cov(Y1,Y3))1/2=(9*4+16-2*3*1*0.8)1/2=6.8702

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