#---------------------------#
#    Loesungen UeBlatt12    #
#---------------------------#


---------------
#  Aufgabe 1  #
---------------

N = 10000
n = 15
mu = -6
sigma = 2

T = rep(0,N)

for(k in 1:N)
{
  x = rnorm(n,mean=mu,sd=sigma)
  hatmu = sum(x)/n
  hats  = sqrt( sum((x-hatmu)^2)/(n-1) )

  T[k] = (hatmu-mu)/(hats/sqrt(n))
}

hist(T,prob=TRUE,xlim=c(-10,10),ylim=c(0,0.5),breaks=60)
curve(dt(x,df=n-1),col="red",add=TRUE)



---------------
#  Aufgabe 2  #
---------------

# 2a)

bevdata = read.table("C:/Users/detlef/OneDrive/hochschule/Vorlesungen/WS2223/Oekonometrie/Weltbevoelkerung.csv",header=TRUE,sep=";")
bevdata

bev = bevdata[,4]
bev
zeiten = bevdata[,3]
zeiten
plot(zeiten,bev)


# 2b+c)

zeiten[14]
zeiten2 = zeiten[1:14]
bev2 = bev[1:14]
zeiten2 = zeiten2 -1950
zeiten = zeiten -1950
logbev2 = log(bev2)


# Modell (1):
res2 = lm(bev2 ~ zeiten2)
a0 = res2$coef[1]
a1 = res2$coef[2]
a0
a1
bevfit1 = a0 + a1*zeiten

# Modell (2):
X = cbind(zeiten2,zeiten2^2)
res3 = lm(bev2 ~ X )
b0 = res3$coef[1]
b1 = res3$coef[2]
b2 = res3$coef[3]
b0
b1
b2
bevfit2 = b0 + b1*zeiten + b2*zeiten^2

# Modell (3):
res1 = lm(logbev2 ~ zeiten2)
beta0 = res1$coef[1]
beta1 = res1$coef[2]
B0 = exp(beta0)
r = beta1
B0
r
bevfit3 = B0 * exp(r*zeiten)

plot(zeiten,bev)
lines(zeiten,bevfit1,col="red")
lines(zeiten,bevfit2,col="blue")
lines(zeiten,bevfit3,col="green")