# Question 1


# problematic 100 CI example
 


N <- 100
theta <- 1
xbar.vec <- rnorm(N,theta,sd=1)
s.vec    <- rchisq(N,df=20) / 20
ci.lngth <- qt(1 - 0.05/2,df=20)*s.vec
ind.cov  <- xbar.vec - ci.lngth < theta &  theta < xbar.vec + ci.lngth  
plot(1:N,xbar.vec,ylim=c(-3,6),xlab="Experiment",ylab="Theta",pch=3)
abline(h=theta,col=4,lwd=2)

segments((1:N)[ind.cov],(xbar.vec - ci.lngth)[ind.cov],(1:N)[ind.cov],(xbar.vec + ci.lngth)[ind.cov],col=3)
segments((1:N)[!ind.cov],(xbar.vec - ci.lngth)[!ind.cov],(1:N)[!ind.cov],(xbar.vec + ci.lngth)[!ind.cov],col=2)
sum(ind.cov)


# t.test checking valid coverge property


pcov.vec <- rep(NA,1000)

for(i in 1:1000)

{
N <- 100
theta <- 1
xbar.vec <- rnorm(N,theta,sd=1)
s.vec    <- rchisq(N,df=20) / 20
ci.lngth <- qt(1 - 0.05/2,df=20)*s.vec
ind.cov  <- xbar.vec - ci.lngth < theta &  theta < xbar.vec + ci.lngth  

pcov.vec[i] <- sum(ind.cov) / N
}

pcov.vec

t.test(pcov.vec)