xbar = c(rep(0,500))  
for(i in 1:500) {
  x = rnorm(100,5,2)    
  xbar[i] = mean(x) 
}


mean(xbar)
var(xbar)          
hist(xbar)
boxplot(xbar)


qqnorm(xbar)


for(i in 1:500) {
  x = rexp(100,.5)      
  xbar[i] = mean(x) 
}


hist(x)
hist(xbar)
mean(xbar)
var(xbar)
qqnorm(xbar)


for(i in 1:500) {
  x = rbinom(100,1,.5)  
  xbar[i] = mean(x) 
}
hist(x)
hist(xbar)
mean(xbar)
var(xbar)
qqnorm(xbar)        


x = rnorm(50, 5.501, .106)
xbar = mean(x)
ci.min = xbar - 1.96 * .106 / sqrt(50)
ci.max = xbar + 1.96 * .106 / sqrt(50)
print(c(ci.min,ci.max))


mean.in.ci = 0
for(i in 1:1000) {
  x = rnorm(50, 5.501, .106)
  xbar = mean(x)
  ci.min = xbar - 1.96 * .106 / sqrt(50)
  ci.max = xbar + 1.96 * .106 / sqrt(50)
  if(ci.min < 5.501 && ci.max > 5.501) mean.in.ci = mean.in.ci + 1
}
print(mean.in.ci)


mean.in.normci = 0
mean.in.tci = 0
for(i in 1:10000) {
  x = rnorm(50, 5.501, .106)
  xbar = mean(x)
  s = sqrt(var(x))
  ci.norm.min = xbar - 1.96 * s / sqrt(50)
  ci.norm.max = xbar + 1.96 * s / sqrt(50)
  ci.t.min = xbar - 2.0096 * s / sqrt(50)
  ci.t.max = xbar + 2.0096 * s / sqrt(50)
  if(ci.norm.min < 5.501 && ci.norm.max > 5.501) 
     mean.in.normci = mean.in.normci + 1
  if(ci.t.min < 5.501 && ci.t.max > 5.501) mean.in.tci = mean.in.tci + 1
}
print(c(mean.in.normci, mean.in.tci))