# 10.1.1

library(MASS)
head(faithful)

install.packages("diptest")
library(diptest)

par(mfrow=c(1,2))
with(faithful,plot(waiting,eruptions,main="Faithful scatterplot",ylab="Eruption duration (mins)",xlab="Inter-ruption interval (mins)"))
with(faithful,hist(eruptions,xlab="Eruption duration (mins)"))


with(faithful,dip(waiting))
[1] 0.04143689
with(faithful,dip(eruptions))
[1] 0.09238103

### We have a significant indication of bimodality in the intervals bet´ween eruptions
### we can use chi-squared splitting at the  medians to see if there is a significant association

topleft<-length(which(faithful$waiting < median(faithful$waiting) & faithful$eruptions > median(faithful$eruptions)))
bottomleft<-length(which(faithful$waiting < median(faithful$waiting) & faithful$eruptions < median(faithful$eruptions)))
topright<-length(which(faithful$waiting > median(faithful$waiting) & faithful$eruptions > median(faithful$eruptions)))
bottomright<-length(which(faithful$waiting > median(faithful$waiting) & faithful$eruptions < median(faithful$eruptions)))

chisq.test(matrix(c(topleft,bottomleft,topright,bottomright),nrow=2,byrow=FALSE))

#	Chi-squared test for given probabilities
#
# data:  matrix(c(bottomleft, topleft, bottomright, topright), nrow = TRUE,     byrow = FALSE)
# X-squared = 122.28, df = 3, p-value < 2.2e-16