###########
# Blatt 5 #
###########

#aufgabe 2

#a
library(MASS)

data(crabs)
attach(crabs)

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

#b
boxplot(crabs[4:8])

max(RW)

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


#c
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)
curve(dnorm(x, mean = mu_FL, sd = sd_FL), col = "red", add = T)
legend(7,0.1,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(RW, freq = F)
curve(dnorm(x, mean = mu_RW, sd = sd_RW), col = "red", add = T)
legend(6,0.15,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(CL, freq = F, ylim = c(0, 0.06))
curve(dnorm(x, mean = mu_CL, sd = sd_CL), col = "red", add = T)
legend(12,0.045,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(CW, freq = F, ylim = c(0,0.05))
curve(dnorm(x, mean = mu_CW, sd = sd_CW), col = "red", add = T)
legend(15,0.04,"Dichte der Normalverteilung", col = "red", pch = 20)

hist(BD, freq = F)
curve(dnorm(x, mean = mu_BD, sd = sd_BD), col = "red", add = T)
legend(6,0.1,"Dichte der Normalverteilung", col = "red", pch = 20)

par(mfrow = c(2,3))
qqnorm(FL, pch = 19)
qqnorm(RW, pch = 19)
qqnorm(CL, pch = 19)
qqnorm(CW, pch = 19)
qqnorm(BD, pch = 19)


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

cor(crabs[4:8])

#aufgabe 3
x<-matrix(ncol = 1000, nrow = 100)

x[1:100,]<-replicate(100,rnorm(1000, mean = 2, sd = 5))

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

y[,1]<-apply(x,1, mean) - qnorm(0.975) * (5/sqrt(1000))
y[,2]<-apply(x,1, mean) + qnorm(0.975) * (5/sqrt(1000))

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

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

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")}
}