### This is a program by Matthew Berman. 
### Suppose we take a sample of size n = 31 from a Normal population with mean 2 
### and standard deviation 3, where both mu and sigma are both unknown. To calculate a 95% Confidence 
### Interval for mu we use the standard t-interval formula. To show that the 95% 
### confidence level is correct we simulate random samples from the population and 
### compute the C.I.'s for each sample and check the coverage. We should expect to 
### see 95% of the intervals cover mu. 

####################################################################### 

# R is number of samples
R <- 1000 
# N is number of observations per sample
N <- 31
# 31 was chosen so that the t-statistic will have 30 df

mu <- 2    #arbitrary value of mu
sigma <- 3 #arbitrary value of sigma

count <- 0
zstar <- qnorm(0.975) 		#(for a 95% C.I. with normal dist.)
tstar <- qt(0.975,df = 30)	#(for a 95% C.I. with t dist., 30 df)

for(i in 1:R){
x <- rnorm(N,mu,sigma)
xbar <- mean(x)
xsd <- sqrt(var(x))
UCL <- xbar + tstar * xsd / sqrt(N)
LCL <- xbar - tstar * xsd / sqrt(N)
if( mu > LCL && mu < UCL) count <- count + 1 ;
}

count #this number is near 950

count/R #this number is near 0.95