#Aufgabe 3

x<-c(20,16,34,23,27,32,18,22)
y<-c(64,61,84,70,88,92,72,77)

#3a)
plot(x,y)

#3b)

modell<-lm(y~x)

abline(modell$coefficient[1], modell$coefficient[2])

#3c)

#Berechne Teststatistik T 
#zu 2b)

gamma<-0.99
S_xx<-var(x)


S_squared <- function(x, y, modell) {
	S<-0

	for(i in 1:length(y)){
		S<-S+(y[i]-modell$coefficient[1]- modell$coefficient[2]*x[i])^2
	}
	return(S / (length(y)-2))
}



T<- (abs(modell$coefficient[2]))* (sqrt((length(x)-1)*S_xx))/(sqrt(S_squared(x,y,modell)))
T

qt(1-(1-gamma)/2, length(x)-2)


#zu 2c)

LU <- function(x, y, modell, x0, gamma) {
	t<- qt(1-(1-gamma)/2, length(x)-2)
	L<- modell$coefficient[1]+modell$coefficient[2]*x0 - t* sqrt(S_squared(x,y,modell)) * sqrt(1/length(x)+((x0-mean(x))^2)/((length(x-1))*var(x)))
	U<- modell$coefficient[1]+modell$coefficient[2]*x0 + t* sqrt(S_squared(x,y,modell)) * sqrt(1/length(x)+((x0-mean(x))^2)/((length(x-1))*var(x)))
	
	return(cbind(L,U))
}

 LU(x,y,modell,30,0.90)

#zu 2d)

Prognose <- function(x, y, modell, x0, gamma) {
	t<- qt(1-(1-gamma)/2, length(x)-2)
	L<- modell$coefficient[1]+modell$coefficient[2]*x0 - t* sqrt(S_squared(x,y,modell)) * sqrt(1+1/length(x)+((x0-mean(x))^2)/((length(x-1))*var(x)))
	U<- modell$coefficient[1]+modell$coefficient[2]*x0 + t* sqrt(S_squared(x,y,modell)) * sqrt(1+1/length(x)+((x0-mean(x))^2)/((length(x-1))*var(x)))
	
	return(cbind(L,U))
}

Prognose(x,y,modell,30,0.90)