# Tutorial 8

Q1

x11 = c(30.9,31.9,31.3,32.1,30.9,31.3,31.3,32.1,30.3,32.2)
x12 = c(30.7,31.6,31.1,31.0,31.2,31.7,31.8,33.0,30.9,32.1)
x13 = c(30.9,31.6,31.0,31.7,30.5,31.4,31.8,31.7,30.8,32.2)
x14 = c(30.9,31.7,31.3,31.3,30.8,31.2,31.7,31.5,30.6,32.4)

x21 = c(31.5,31.2,31.3,30.4,30.7,29.8,31.4,30.9,31.1,31.3)
x22 = c(30.6,31.2,31.3,30.8,30.9,30.8,32.0,32.4,31.3,31.5)
x23 = c(30.8,31.1,31.5,30.4,30.9,30.9,31.7,31.8,31.2,31.6)
x24 = c(31.0,31.3,31.4,30.2,30.9,30.8,31.6,31.9,31.2,31.7)
X1=cbind(x11,x12,x13,x14)
X2=cbind(x21,x22,x23,x24)

(i)
S1 = var(X1)
S2= var(X2)
S = (S1+S2)/2

round(S1,4)
round(S2,4)
round(S,4)



(ii)
sd.inv = diag( sqrt(diag(S))^{-1} )
R = sd.inv%*%S%*%sd.inv
round(R,2)

(iii)

rho.j = apply(R,1,sum)
for (j in 1:4) {
  rho.j[j] = rho.j[j]-R[j,j]
}
rho.j = rho.j/3

rho = mean(rho.j)

a1 = (4-1)*(1-rho)^2
a2 = 2*(4-1)^2/4*sum((rho.j-rho)^2)

R.tmp = R
for (j in 1:4) {
  R.tmp[j,j] = rho
}
R.tmp = (R.tmp - rho)^2
a3 = apply(R.tmp, 1, sum)
a3 = sum(a3)

ep = a1/(a1-a2+a3)

Q2

l = c(-3,-1,1,3)
C1 = X1%*%l
C2 = X2%*%l

c1.m= mean(C1)
c1.sd = sqrt(var(C1)) 

c2.m= mean(C2)
c2.sd = sqrt(var(C2)) 

c(c1.m,c1.sd,c2.m,c2.sd)

c.m = (c1.m+c2.m)/2
c.sd = sqrt((c1.sd^2+c2.sd^2)/2 )

c(c.m, c.sd)


Q3

(i)
X1.m = apply(X1,2,mean)
X2.m = apply(X2,2,mean)
C = t(l)%*%(X1.m-X2.m)
C.se = sqrt( 20/10/10 * t(l)%*%S%*% l )
L1 = C/C.se

(ii)
L2 = (c1.m-c2.m)/c.sd /sqrt(20/10/10)