### 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 (C = 0.95) 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 = 1.96  #(for a 95% C.I. with normal dist.)
tstar = 2.042 #(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 R * C = 950

count/R # this number is near C = 0.95

# Confident Interval for the Confidence level C 

# count ~ Bin(R, C) appox N( R*C, R*(count/R)*(1-count/R) )
# C is the confidence level, C = 1 - alpha, C = 0.95, alpha = 0.5

count - zstar * sqrt( count * ( R - count) / R ) 
count + zstar * sqrt( count * ( R - count) / R ) 

# count/R  approx N( C, (count/R)*(1-count/R)/R )

count/R - zstar * sqrt( count * ( R - count) / R^3 ) 
count/R + zstar * sqrt( count * ( R - count) / R^3 )