### Eksempler Kap. 8

# Eks. 8.1 DBC Tensile styrke i titanlegeringer
n=31;xmean=1064;sigma=55
stderrmean=sigma/sqrt(n)
stderrmean
round(xmean+c(0,-1,1)*stderrmean,1)
round(xmean+c(0,-1,1)*2*stderrmean,1)
round(xmean+c(0,-1,1)*1.96*stderrmean,1)

# Konfidensintervall for mu=E(Xi) for normalfordelte Xi og kjent varians
n=10
mu=0
sigma=2
x=rnorm(n,mu,sigma)
c(mean(x),mean(x)-1.96*sigma/sqrt(n),mean(x)+1.96*sigma/sqrt(n))

plot(c(mean(x)-1.96*sigma/sqrt(n),mean(x)+1.96*sigma/sqrt(n)),c(1,1),type="l",xlim=c(-2.7,2.7),ylim=c(0,100),xlab="mu",ylab="Interval nummer")

for (sim in 2:100){
  x=rnorm(n,mu,sigma)
  lines(c(mean(x)-1.96*sigma/sqrt(n),mean(x)+1.96*sigma/sqrt(n)),c(sim,sim))
}
abline(v=mu)


# Tilnærmet konfidensintervall for p=E(Xi) for binære Xi, Y = ant suksesser ~ Bin(n,p)
n=100;p=0.3
y=rbinom(1,n,p)
phat=y/n
se=sqrt(phat*(1-phat)/n)
c(phat,phat-1.96*se,phat+1.96*se)

y=rbinom(100,n,p)
phat=y/n
se=sqrt(phat*(1-phat)/n)
# Merk y, yhat, phat og se er vektorer
plot(c(phat[1]-1.96*se[1],phat[1]+1.96*se[1]),c(1,1),type="l",xlim=c(0.1,0.5),ylim=c(0,100),xlab="p",ylab="Simulering")
for (sim in 2:100) lines(c(phat[sim]-1.96*se[1],phat[sim]+1.96*se[sim]),c(sim,sim))
abline(v=p)
title("Hundre 95% KI for binomisk p")