#1)a)
conf=function(x,level){
  return(c(mean(x)-qt(1-(1-level)/2, length(x)-1)*(sqrt((1/(length(x)-1))*sum((x-mean(x))^2)))/(sqrt(length(x))), mean(x)+qt(1-(1-level)/2, length(x)-1)*(sqrt((1/(length(x)-1))*sum((x-mean(x))^2)))/(sqrt(length(x)))))
}

level=0.95
counter=0
for(i in 1:100000){
  x=rnorm(100,5,20)
  KI=conf(x,level)
  if(KI[1]<5 & KI[2]>5){
    counter=counter+1
  }
}
print(counter/100000)

#2)a)
conf1=function(x,level){
  q=qnorm(1-((1-level)/2))
  n=length(x)
  return(c(mean(x)+(q^2)/(2*n)-sqrt((mean(x)*q^2)/n+q^4/(4*n^2)), mean(x)+(q^2)/(2*n)+sqrt((mean(x)*q^2)/n+q^4/(4*n^2))))
}

conf2=function(x,level){
  q=qnorm(1-((1-level)/2))
  n=length(x)
  return(c(max(0, mean(x)-q/sqrt(n)*sqrt(mean(x))), max(0, mean(x)+q/sqrt(n)*sqrt(mean(x)))))
}


level=0.95
counter1=0
counter2=0
länge1=0
länge2=0
for(i in 1:100000){
  x=rpois(1000,5)
  KI1=conf1(x,level)
  if(KI1[1]<5 & KI1[2]>5){
    counter1=counter1+1
  }
  KI2=conf2(x,level)
  if(KI2[1]<5 & KI2[2]>5){
    counter2=counter2+1
  }
  länge1=länge1+KI1[2]-KI1[1]
  länge2=länge2+KI2[2]-KI2[1]
}
print(counter1/100000)
print(counter2/100000)
print(länge1/100000)
print(länge2/100000)