##################################

#The t distribution: 
x <- (-100):100/10
plot(x, dt(x, 1), type="l")
lines(x, dt(x, 3), col="red")

lines(x, dt(x, 10), col="yellow")
lines(x, dt(x, 100), col="cyan")
lines(x, dt(x, 1000), col="blue")

lines(x, dnorm(x))

###################################

#Example showing the connection of the t distr.
#with the normal and chi-square: 
sample5 <- rnorm(10000)/sqrt(rchisq(10000, 5)/5)
x <- (-20):20
hist(sample5, breaks=x-0.5, prob=TRUE)
points(x, dt(x, 5))

###################################

#Examples of F-distributions: 
x <- 1:100/20
plot(x, df(x, 1, 1), type="l")
lines(x, df(x, 1, 3))
lines(x, df(x, 1, 100))
lines(x, df(x, 3, 1))
lines(x, df(x, 3, 10))
lines(x, df(x, 3, 100))
lines(x, df(x, 10, 1))
lines(x, df(x, 10, 10))
lines(x, df(x, 10, 100))

###################################

#Examples of differences between t tests:
data1 <- c(24, 34, 25, 43, 12, 54, 32, 35)
data2 <- c(42, 12, 29, 13, 31, 27, 19)

t.test(data1, data2)

help(t.test)

t.test(data1, data2, var.equal=T)
t.test(data1, data2, var.equal=T, conf.level=0.99)
t.test(data1, data2, var.equal=T, 
   alternative="greater")

##################################

#A randomization test replacing a t-test: 

t.test(data1, data2)

A <- matrix(c(data1, data2), 10000, 15, byrow=T)
B <- apply(A, 1, function(x) {sum(sample(x, 8))})
C <- sum(data1)+sum(data2) - B
D <- B/8 - C/7
hist(D)
diff <- mean(data2)-mean(data1)
abline(v=diff)
sum(diff>D)/length(D)

##################################

#Another randomization test, for paired data: 

dat1 <- rpois(15, 4)
dat2 <- dat1 + rt(15, 3) +1
t.test(dat1, dat2, paired=T)

diffs <- dat1 - dat2

A <- matrix(diffs, 10000, 15, byrow=T)
A <- A*matrix((runif(10000*15)<0.5)*2-1, 10000, 15)
B <- apply(A, 1, mean)
hist(B)
mean(diff)