#------------------
# Testovani hypotez
#------------------

# Prvni priklad - chlapci
# Nacteni dat
load("cviceni3.RData")
x <- data3$pr1$x

n <- length(x)
alpha <- 0.05

# prvni zpusob
ux <- (mean(x)-136.1)/6.4*sqrt(n)
kvantil <- qnorm(1-alpha)

# druhy zpusob
1 - pnorm(mean(x),mean=136.1,sd=6.4/sqrt(n))

# treti zpusob
D <- mean(x) - kvantil*6.4/sqrt(n)

# ctvrty zpusob - pro nezname sigma
library("TeachingDemos")
z.test(x,mu=136.1,alternative="greater",sd=6.4)

# Druhy priklad - hnojiva
# Nacteni dat
load("cviceni3.RData")
x <- data3$pr2$x
y <- data3$pr2$y

alpha <- 0.05
n1 <- length(x)
n2 <- length(y)
s1 <- sd(x)
s2 <- sd(y)
# a)
f <- (s1/s2)^2
kvantil1 <- qf(1-alpha/2,n1-1,n2-1)
kvantil2 <- 1/qf(1-alpha/2,n2-1,n1-1)
# b)
2*min(1-pf(var(x)/var(y),n1-1,n2-1),pf(var(x)/var(y),n1-1,n2-1))
# c)
D <- (s1/s2)^2/kvantil1
H <- (s1/s2)^2/kvantil2

# d)
var.test(x,y)

# I
prumer_x <- mean(x)
prumer_y <- mean(y)
s12 <- sqrt(((n1-1)*s1^2+(n2-1)*s2^2)/(n1+n2-2))
kvantil <- qt(1-alpha/2,n1+n2-2)
t <- (prumer_x - prumer_y)/s12*sqrt((n1*n2)/(n1+n2))

# II
p <- 2*(1-pt(t,n1+n2-2))

# III
D <- prumer_x - prumer_y - kvantil*s12*sqrt((n1+n2)/(n1*n2))
H <- prumer_x - prumer_y + kvantil*s12*sqrt((n1+n2)/(n1*n2))

# IV
t.test(x,y,var.equal=T)

# Treti priklad - listy
# Nacteni dat
load("cviceni3.RData")
x <- data3$pr3$x
y <- data3$pr3$y

z <- x - y
alpha <- 0.05
n <- length(z)
prumer <- mean(z)
odchylka <- sd(z)
kvantil <- qt(1-alpha/2,n-1)

# I
t <- prumer/odchylka*sqrt(n)

# II
p <- 2*(1-pt(t,n-1))

# III
D <- prumer - kvantil*odchylka/sqrt(n)
H <- prumer + kvantil*odchylka/sqrt(n)

# IV
t.test(x,y,paired=T)

# Ctvrty priklad - mince
# Nacteni dat
n <- 40
rub <- 22
lic <- n - rub
alpha <- 0.05
# vypocet
prumer <- rub/n 
odchylka <- sqrt(prumer*(1-prumer))
kvantil <- qnorm(1-alpha/2)

# I
u <- (prumer - 0.5)*sqrt(n)/odchylka

# II
p <- 2*(1-pnorm(u))

# III
D <- prumer - kvantil*odchylka/sqrt(n)
H <- prumer + kvantil*odchylka/sqrt(n)

# IV
binom.test(rub,n)


#-----------------------------------------------------------------------------------