#exercicio 7
x1=c(844, 1678, 1573, 3025, 653, 1991, 2187, 1262, 4624, 5249, 4258, 2943, 5092, 4496)
x2=c(67,57,39,54,46,62,69,74,69,76,80,79,82,85)
x3=c(5.75, 6.05, 5.45, 5.7, 5.55, 5, 6.4, 6.1, 6.05, 6.15, 5.55, 6.4, 6.55, 6.5)
x0=matrix(1,14,1);x0
X=as.matrix(cbind(x0, x1, x2, x3))
Y = c(88, 80, 42, 37, 37, 20, 20, 18, 18, 4, 2, 2, -2, -7)
dados=cbind(Y,X[,-1]);dados
#########################################
#EXERCICIO
#Ry1.3 
ry=residuals(lm(Y~x3))
r1=residuals(lm(x1~x3))
cor(ry,r1) #resolucao rapida no R
cor.test(ry,r1) 

#fazendo passo a passo
#matriz sigma (var-cov)
sigma=cov(dados); sigma

S11 = as.matrix(sigma[1:2, 1:2]); S11
S12 = as.matrix(sigma[1:2, 4]); S12
S21 = t(as.matrix(sigma[4, 1:2])); S21
S22 = as.matrix(sigma[4, 4]); S22

S1.2 = S11 - (S12%*%solve(S22)%*%S21); S1.2

Ry1.3 = S1.2[1,2] / sqrt(S1.2[1,1]*S1.2[2,2]) ; Ry1.3