###########
# Blatt 3 #
###########

#Aufgabe 2

library(MASS)

data(crabs)
attach(crabs)

#a
?crabs
head(crabs)
summary(crabs)

#a
boxplot(crabs[4:8])

max(RW)

points(2,max(RW), col = "magenta", pch = 19)


#b
mu_FL<-mean(FL)
mu_RW<-mean(RW)
mu_CL<-mean(CL)
mu_CW<-mean(CW)
mu_BD<-mean(BD)

sd_FL<-sd(FL)
sd_RW<-sd(RW)
sd_CL<-sd(CL)
sd_CW<-sd(CW)
sd_BD<-sd(BD)

hist(FL, freq = F)
lines((min(FL)-1):(max(FL)+1),dnorm((min(FL)-1):(max(FL)+1), mean = mu_FL, sd = sd_FL), col = "red")
legend(7,0.1,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(RW, freq = F)
lines((min(RW)-1):(max(RW)+1),dnorm((min(RW)-1):(max(RW)+1), mean = mu_RW, sd = sd_RW), col = "red")
legend(6,0.15,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(CL, freq = F, ylim = c(0, 0.06))
lines((min(CL)-1):(max(CL)+1),dnorm((min(CL)-1):(max(CL)+1), mean = mu_CL, sd = sd_CL), col = "red")
legend(12,0.045,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(CW, freq = F, ylim = c(0,0.05))
lines((min(CW)-1):(max(CW)+1),dnorm((min(CW)-1):(max(CW)+1), mean = mu_CW, sd = sd_CW), col = "red")
legend(15,0.04,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(BD, freq = F)
lines((min(BD)-1):(max(BD)+1),dnorm((min(BD)-1):(max(BD)+1), mean = mu_BD, sd = sd_BD), col = "red")
legend(6,0.1,"Dichte der Normalverteilung", col = "red", pch = 20)


#d
pairs(crabs[4:8], pch = 20)

cor(crabs[4:8])


#Aufgabe 5

x<-matrix(ncol = 1000, nrow = 100)

for(i in 1:100){
x[i,]<-rnorm(1000, mean = 2, sd = 5)
}

y<-matrix(ncol = 2, nrow = 100)

for(i in 1:100){
y[i,1]<-mean(x[i,]) - qnorm(0.975) * (5/sqrt(1000))
y[i,2]<-mean(x[i,]) + qnorm(0.975) * (5/sqrt(1000))
}

plot(0,0, type = "n", xlim = c(1,3), ylim = c(0,100))

for(i in 1:100){
lines(c(y[i,1],y[i,2]), c(i,i))
if(y[i,1] >= 2 || y[i,2] <= 2){lines(c(y[i,1],y[i,2]), c(i,i), col = "red")}
}


sum(y[,1] >= 2) + sum(y[,2] <= 2)