#(1) 
x=c(53,44,45,55,59,51,72,56,50,58)
n=length(x)

#(a)
#X_1,...,X_n je nahodny vyber ze spojiteho rozdeleni s medianem \delta.
# H_0: \delta=60 ... je median (stredni hodnota) rovny 60?

#(b)
t.test(x,mu=60)

hist(x)
plot(density(x))

qqnorm(x)
qqline(x)

plot(ppoints(n,0),pnorm((sort(x) - mean(x))/sd(x)))
abline(0,1)

library(nortest)
library(normtest)
kurtosis.norm.test(x)
skewness.norm.test(x)
jb.norm.test(x)
shapiro.test(x)
lillie.test(x)
ad.test(x)
cvm.test(x)
pearson.test(x)


#(c)
Rplus=rank(abs(x-60))
t=sum(Rplus*sign(x-60))/(n+1)
t

R=1:n
W=sum(R)/(n+1)
k=2

for (i in 1:n){
X=combn(n,i)
K=length(X[1,])

for (j in 1:K){
b=X[,j]
W[k]=(sum(R[-b])-sum(R[b]))/(n+1)
k=k+1
}
}

a=table(W)/2^n
plot(a,type='h')
2*sum(W<=t)/2^n #presna p-hodnota

wilcox.test(x,mu=60,exact=TRUE) #exaktni Wilcoxonuv test

t2=t/sqrt(n*(2*n+1)/6/(n+1))

2*pnorm(t2)
wilcox.test(x,mu=60,exact=FALSE,cor=FALSE) #asymptoticky Wilcoxonuv test


#(d)
t=sum(qnorm(Rplus/2/(n+1)+1/2)*sign(x-60))

W=sum(qnorm(R/2/(n+1)+1/2))
k=2

for (i in 1:n){
X=combn(n,i)
K=length(X[1,])

for (j in 1:K){
b=X[,j]
W[k]=sum(qnorm(R[-b]/2/(n+1)+1/2))-sum(qnorm(R[b]/2/(n+1)+1/2))
k=k+1
}
}

a=table(W)/2^n
plot(a,type='h')

2*sum(W<=t)/2^n #presna p-hodnota


t2=t/sqrt(sum((qnorm((1:n)/2/(n+1)+1/2))^2))
2*pnorm(t2) #asymptoticka p-hodnota

library(snpar)
ns.test(x,q=60)      #van der Waerdenuv test

#(e)
t=sum(sign(x-60))
t

W=n
k=2

for (i in 1:n){
X=combn(n,i)
K=length(X[1,])

for (j in 1:K){
b=X[,j]
W[k]=length(R[-b])-length(R[b])
k=k+1
}
}

a=table(W)/2^n
plot(a,type='h')
2*sum(W<=t)/2^n #presna p-hodnota

U=sum(x>60)
binom.test(U,n)      #znamenkovy test jakozto spec. pripad binomickeho

library(BSDA)
SIGN.test(x, md=60)  #znamenkovy test

#asymptoticka verze
t2=t/sqrt(n)
2*pnorm(t2)   #asymptoticka p-hodnota


#(2)
x=c(14.4, 15.9, 14.4, 13.9, 16.6, 17.4, 18.6, 20.4, 20.4, 15.4, 15.4, 14.1)
y=c(20.4, 22.9, 19.4, 24.4, 25.1, 20.9, 24.6, 24.4, 24.9, 19.9, 21.4, 21.4)

#(a)
z=x-y
n=length(z)

hist(z)
plot(density(z))

qqnorm(z)
qqline(z)

plot(ppoints(n,0),pnorm((sort(z) - mean(z))/sd(z)))
abline(0,1)

library(nortest)
library(normtest)
kurtosis.norm.test(z)
skewness.norm.test(z)
jb.norm.test(z)
shapiro.test(z)
lillie.test(z)
ad.test(z)
cvm.test(z)
pearson.test(z)


#(b)
wilcox.test(x,y,paired=TRUE, alt="less")
wilcox.test(x-y,alt="less")

ns.test(x-y,q=0, alt="less")

U=sum((x-y)>0)
n=length(x)
binom.test(U,n,alt="less")
SIGN.test(x-y, alt="less")