#Soal No 2

x1 <- c(525,500,550,575,495,490,525,550)
x2 <- c(13,14,13,11,15,14,12,12)
obs <- c(1:8)
data <- cbind(obs,x1,x2)
data
##      obs  x1 x2
## [1,]   1 525 13
## [2,]   2 500 14
## [3,]   3 550 13
## [4,]   4 575 11
## [5,]   5 495 15
## [6,]   6 490 14
## [7,]   7 525 12
## [8,]   8 550 12
xbar <- apply(data[,2:3], 2, mean)
xbar
##     x1     x2 
## 526.25  13.00
cov_m <- cov(data[,2:3])
cov_m
##          x1         x2
## x1 926.7857 -35.000000
## x2 -35.0000   1.714286

##Selang Kepercayaan Ellips

library(ellipse)
## 
## Attaching package: 'ellipse'
## The following object is masked from 'package:graphics':
## 
##     pairs
n <- 8
p <- 2
plot(ellipse(cov_m, centre=xbar, level = 0.95, t=sqrt(((n-1)*p/(n*(n-p)))*qf(0.95,p,n-p))), type='l', main = "Ellips Kepercayaan 95%")
points(xbar[1], xbar[2])

##Selang Kepercayaan Simultan

T.ci = function(mu, Sigma, n, avec=rep(1,length(mu)), level=0.95){
p = length(mu)
cval = qf(level, p, n-p) * p * (n-1) / (n-p)
zhat = crossprod(avec, mu)
zvar = crossprod(avec, Sigma %*% avec) / n
const = sqrt(cval * zvar)
c(lower = zhat - const, upper = zhat + const)
}

#X1
T.ci(mu=xbar, Sigma=cov_m, n=n, avec=c(1,0),level=0.95)
##    lower    upper 
## 488.9635 563.5365
#X2
T.ci(mu=xbar, Sigma=cov_m, n=n, avec=c(0,1),level=0.95)
##    lower    upper 
## 11.39637 14.60363