#==========================================================
#                 P R I K L A D    6.1
#==========================================================

data <- read.delim('01-one-sample-mean-skull-mf.txt')
data <- na.omit(data)
skull.BM <- data[data$sex == 'm', 'skull.B']
length(skull.BM)

# H0: Data pochazi z normalniho rozdeleni.
# H1: Data nepochazi z normalniho rozdeleni.
# hladina vyznamnosti alpha = 0.05 
# (neni-li uvedena hladina vyznamnosti v zadani, volime ji sami. Nejcastejsi volbou byva alpha = 0.05)
library(nortest)
lillie.test(skull.BM) # p-hodnota = 0.07662

# Protoze p-hodnota = 0.07622 > alpha = 0.05, H0 nezamitame na hladine vyznamnosti alpha = 0.05.
# Data pochazi z normalniho rozdeleni.

#==========================================================
#                 P R I K L A D    6.2
#==========================================================
# Histogram
b     <- seq(123, 150, by = 3)
centr <- seq(124.5, 148.5, by = 3)

par(mar = c(4, 5, 1, 1))
hist(skull.BM, col = 'dodgerblue', prob = T, breaks = b, density = 30,
     axes = F, xlab = 'nejvetsi sirka mozkovny (mm) - muzi', 
     ylab = 'relativni cetnosti', main = '', ylim = c(0, 0.085))
box(bty = 'o')

axis(side = 1, centr)
axis(side = 2, las = 1)

x <- seq(min(skull.BM) - 10, max(skull.BM) + 10, length = 512)
y <- dnorm(x, mean = mean(skull.BM), sd = sd(skull.BM))
lines(x, y, col = 'dodgerblue3', lwd = 2)

# Q-Q graf
qqnorm(skull.BM, main = '', las = 1,
       xlab = 'teoreticky kvantil',
       ylab = 'vyberovy kvantil', pch = 21,
       col = 'dodgerblue3')
qqline(skull.BM, col = 'dodgerblue4', lwd = 2)


#==========================================================
#                 P R I K L A D    7.1
#==========================================================

# H0: mu = 136.402
# H1: mu != 136.402 (oboustranna alternativa)

m <- mean(skull.BM)
s <- sd(skull.BM)
c <- 136.402
n <- length(skull.BM)
alpha <- 0.05 # hladina vyznamnosti alpha = 0.05 (zvolena podle zadani)

# Testovani kriticky oborem
(t0 <- (m - c) / (s / sqrt(n)))  # testovaci statistika t0 
qt(alpha / 2, n - 1)     # -1.9711
qt(1 - alpha / 2, n - 1) #  1.9711

# t0 = 2.3856; kriticky obor W = (-infty ; -1.9711> U <1.9711 ; infty)
# Protoze t0 nalezi do W, H0 zamitame na hladine vyznamnosti alpha = 0.05.


# Teestovani intervalem spolehlivosti
(dh <- m - s / sqrt(n) * qt(1 - alpha / 2, n - 1)) # dolni hranice
(hh <- m - s / sqrt(n) * qt(alpha / 2, n - 1)) # horni hranice
# IS = (136.5381 ; 137.8322)
# Protoze c = 136.402 nenalezi do IS, H0 zamitame na hladine vyznamnosti alpha = 0.05.

# Testovani p-hodnotou
(pval <- 2 * min(pt(t0, n - 1), 1 - pt(t0, n - 1))) # p-hodnota

# Protoze p-hodnota = 0.01791 < alpha = 0.05, H0 zamitame na hladine vyznamnosti alpha = 0.05.
# Zaver testovani: Všemi tremi zpusoby H0 zamitame.

# Interpretace vysledku: Mezi stredni hodnotou nejvetsi sirky mozkovny 
# u muzu staroveke a novoveke egyptske populace existuje statisticky vyznamny rozdil.

# Krabicovy diagram
par(mar = c(2,4,1,1))
boxplot(skull.BM, ylab = 'nejvetsi sirka mozkovny (mm)', col = 'mintcream', 
        las = 1, border = 'darkblue')
points(mean(skull.BM), pch = 19, col = 'blue')
points(136.402, pch = 19, col = 'red')
legend('topright', col = c('blue', 'red'), pch = 19, 
       legend = c('staroveka pop.', 'novoveka pop.'), 
       bty = 'n')

#==========================================================
#                 P R I K L A D    7.2
#==========================================================
# H0: sigma^2 >= 6.411^2 
# H1: sigma^2  < 6.411^2 (levostranna alternativa)
# alpha = 0.1 (ze zadani)

data <- read.delim('01-one-sample-mean-skull-mf.txt')
data <- na.omit(data)
skull.BM <- data[data$sex == 'm', "skull.B"]

m <- mean(skull.BM)
s <- sd(skull.BM)
c <- 6.411^2
n <- length(skull.BM)
alpha <- 0.1

# Testovani kritickym oborem
(t0 <- (n-1) * s ^ 2 / c) # tetsovaci statistika
qchisq(alpha, n - 1)      # kriticky obor W
# W = (0; 188.9801); t0 = 121.7635; t0 nalezi do W -> H0 zamitame.

# Testovani Intervalem spolehlivosti
(HH <- (n - 1) * s ^ 2 / qchisq(alpha, n - 1)) # horni hranice IS
# IS = (-infty ; 26.4947); 6.411^2 = 41.10092 nenalezi do IS -> H0 zamitame.

# Testovani p-hodnotou
(pval <- pchisq(t0, n - 1)) # p-hodnota
# p-hodnota = 4.35 x 10^(-8), alpha = 0.1; p-hodnota < alpha -> H0 zamitame.

# Interpretace vysledku: Rozptyl nejvetsi sirky mozkovny muzu staroveke 
# egyptske populace je statisticky vyznamne mensi nez rozptyl nejvetsi sirky mozkovny 
# muzu novoveke egyptske populace.

#==========================================================
#                 P R I K L A D    7.3
#==========================================================

# H0: sigma >= 6.411 -> H0: sigma^2 >= 6.411^2
# H1: sigma  < 6.411 -> H1: sigma^2  < 6.411^2 (levostranna alternativa)
# alpha = 0.1 (ze zadani)

# Test o smerodatne odchylce prevedeme na test o rozptylu a vsimneme si, 
# ze zadani vede na uplne stejne reseni jako v prikladu 7.2.
# Vypocet by byl tedy totozny s vypoctem uvedenym v prikladu 7.2.

# Zaver: Smerodatna odchylka nejvetsi sirky mozkovny muzu staroveke 
# egyptske populace je statisticky vyznamne mensi nez smerodatna odchylka nejvetsi sirky mozkovny 
# muzu novoveke egyptske populace.