# 9.2 b)
X1 <- c(62, 54, 55, 60, 53, 58) #dieta 1
X2 <- c(52, 56, 49, 50, 51)     #dieta2
m1 <- mean(X1)
m2 <- mean(X2)
n1 <- length(X1)
n2 <- length(X2)
c <-0
s1 <- sd(X1)
s2 <- sd(X2)
s.h2 <- ((n1-1)*s1^2+(n2-1)*s2^2)/(n1+n2-2) # vyzeny prumer vyberovych rozptylu
(s.h <- sqrt(s.h2))
(t0 <- ((m1-m2)-c)/(s.h*sqrt(1/n1+1/n2))) #testovaci statistika

alpha <- 0.05
(w1 <- qt(alpha/2, n1+n2-2))
(w2 <- qt(1-alpha/2, n1+n2-2))

dh <- m1-m2-s.h*sqrt(1/n1+1/n2)*qt(1-alpha/2, n1+n2-2) # dolni hranice IS 
hh <- m1-m2-s.h*sqrt(1/n1+1/n2)*qt(alpha/2, n1+n2-2)   # horni hranice IS

p1 <- pt(t0, n1+n2-2)                                  # P(T0<=t0)
p2 <- 1-pt(t0, n1+n2-2)                                # P(T0>t0)
2*min(p1, p2)                                          # p-hodnota

boxplot(X1, X2, names=c('dieta 1', 'dieta 2'),
        ylab='prirustky (v dkg)',
        main='Hmotnostni prirustky selat',
        col='blanchedalmond', border='coral4')
mtext('Krabicovy graf', line=0.4, cex=0.85, side=3)
points(c(m1,m2), pch=15, col='red')                    # zaznamenani prumeru do boxplotu


#-------------------------------------------------------------------
# 9.4
X <- c(1,1,0,1,1,0,0,0,0,0)
n <- length(X)
m <- mean(X)
alpha <- 0.05

theta <- mean(X)
n*theta*(1-theta)                          # podminka dobre aproximace < 9

dh <- m-sqrt(m*(1-m)/n)*qnorm(1-alpha/2)   # dolni hranice IS
hh <- m-sqrt(m*(1-m)/n)*qnorm(alpha/2)     # horni hranice IS

#------------------------------------------
# 9.4
X <- c(rep(1,38), rep(0,112))
m <- mean(X)
c <- 0.3
n <- length(X)
alpha <- 0.05

theta <- mean(X)
n*theta*(1-theta)                           # podminka dobre aproximace > 9
(t0 <- (m-c)/sqrt(c*(1-c)/n))               # testovaci statistika
(wh <- qnorm(alpha))                     

hh <- m-sqrt(m*(1-m)/n)*qnorm(alpha)        # horni hranice IS

pnorm(t0)                                   # p-hodnota