set.seed(123456)

curve(dnorm(x,10,2), xlim = c(4, 16), ylim = c(0.0, 0.22))

par(mfrow=c(2,2))

ybar5<-numeric(10000)
for(j in 1:10000) {
  sample<-rnorm(5,10,2)
  ybar5[j] <-mean(sample)
}
mean(ybar5)
var(ybar5)
plot(density(ybar5), xlim = c(7, 13), ylim = c(0.0, 1.2))
curve(dnorm(x,10,sqrt(0.8)), add=T, lty=2)

ybar10<-numeric(10000)
for(k in 1:10000) {
  sample<-rnorm(10,10,2)
  ybar10[k] <-mean(sample)
}
mean(ybar10)
var(ybar10)
plot(density(ybar10),xlim = c(7, 13), ylim = c(0.0, 1.2))
curve(dnorm(x,10,sqrt(0.4)), add=T, lty=2)


ybar20<-numeric(10000)
for(m in 1:10000) {
  sample<-rnorm(20,10,2)
  ybar20[m] <-mean(sample)
}
mean(ybar20)
var(ybar20)
plot(density(ybar20),xlim = c(7, 13), ylim = c(0.0, 1.2))
curve(dnorm(x,10,sqrt(0.2)), add=T, lty=2)

ybar30<-numeric(10000)
for(n in 1:10000) {
  sample<-rnorm(30,10,2)
  ybar30[n] <-mean(sample)
}
mean(ybar30)
var(ybar30)
plot(density(ybar30),xlim = c(7, 13), ylim = c(0.0, 1.2))
curve(dnorm(x,10,sqrt(0.1333)), add=T, lty=2)

par(mfrow=c(1,1))
plot(density(ybar5), col="black", xlim = c(7, 13), ylim = c(0.0, 1.2))
lines(density(ybar10), col="red", xlim = c(7, 13), ylim = c(0.0, 1.2))
lines(density(ybar20), col="green", xlim = c(7, 13), ylim = c(0.0, 1.2))
lines(density(ybar30), col="blue", xlim = c(7, 13), ylim = c(0.0, 1.2))