M <- 38/150
c <- 0.3
n <- 150
alpha <- 0.05

n*c*(1-c)

t0 <- (M-c)/sqrt(c*(1-c)/n)
t0

qnorm(alpha)


HH <- M-sqrt((M*(1-M)/n))*qnorm(alpha)
HH

p <- pnorm(t0)
p

X <- c(62, 54, 55, 60, 53, 58)
Y <- c(52, 56, 49, 50, 51)

shapiro.test(X)
shapiro.test(Y)

par(mfrow=c(1,2))
qqnorm(X, xlab='Teoreticky kvantil', ylab='Vyberovy kvantil')
qqline(X, col='red')

qqnorm(Y, xlab='Teoreticky kvantil', ylab='Vyberovy kvantil')
qqline(Y, col='blue')

hist(X)
hist(Y)


n1 <- length(X)
n2 <- length(Y)
s1 <- sd(X)
s2 <- sd(Y)
alpha <- 0.05


t0 <- s1^2/s2^2
t0

qf(alpha/2, n1-1, n2-1)
qf(1-alpha/2, n1-1, n2-1)

2*min(pf(t0, n1-1, n2-1), 1-pf(t0, n1-1, n2-1))

m1 <- mean(X)
m2 <- mean(Y)


n1 <- length(X)
n2 <- length(Y)

alpha=0.05

s1 <- sd(X)
s2 <- sd(Y)

s_h <- sqrt( ((n1-1)*s1^2 +(n2-1)*s2^2) / (n1+n2-2)  )
s_h

dh <- m1-m2-s_h*sqrt(1/n1+1/n2)*qt(1-alpha/2, n1+n2-2)
dh

hh <- m1-m2-s_h*sqrt(1/n1+1/n2)*qt(alpha/2, n1+n2-2)
hh

par(mfrow=c(1,1))
boxplot(X,Y)
points(1, mean(X), col='red', pch=19)
points(2, mean(Y), col='red', pch=19)