################
### OSMI CAS ###
################

#2.
prop.test(x = c(52,120), n = c(68,130), conf.level = 0.99) # odbracujemo H_0

#4.
2*(1-pf(1.28, 120, 120)) #p vrednost

#5.
2*(pf(0.7143, df1 = 24, df2 = 24))
pt(-0.433, df = 48)


#6.
A <- c(200,220,320,450,500,550,550,550,600,670,700,1000,1200,1200)
B <-c(230,230,300,300,300,400,450,450,500,500,500,800,800,800,1000,1000,1000,1500,1500,1500,1500)
# Prvo testiramo hipotezu o jednakosti disperzija
var.test(A, B, ratio = 1, alternative = "two.sided")
# p~0.2, pa mozemo smatrati da su disperzije jednake
t.test(A, B, mu = 0, var.equal = TRUE)
# p>0.05 => ne odbacujemo H0
# nema znacajnih razlika izmedju prosecnih plata u te dve firme 
# ILI
A <- c(200,220,320,450,500,550,550,550,600,670,700,1000,1200,1200)
B <-c(230,230,300,300,300,400,450,450,500,500,500,800,800,800,1000,1000,1000,1500,1500,1500,1500)
#prvo testiramo hipotezu o jednakosti disperzija
alfa <- 0.05
sn1 <- var(A)
sn2 <- var(B)
n1 <- length(A)
n2 <- length(B)

c1 <- qf((1/2)*alfa,n1-1,n2-1)    #trazimo levu granicu fiserove raspodele
c2 <- qf(1-(1/2)*alfa,n1-1,n2-1)  #trazimo desnu granicu fiserove raspodele
W <- c(c1,c2)                     #kriticna oblast
TN <- (sn1)/(sn2)                 #ako TN upadne izmedju c1 i c2 upada u kriticnu oblast
TN < c1 | TN > c2                 #ako je true onda upada u kriticnu oblast i odbacujemo H0 tj. nama je FALSE pa su disperzije jednake
TN1 <- (mean(A) - mean(B))/(sqrt(sn1/length(A) + sn2/length(B)))    #ako upadne u kriticnu oblast odbacujemo H0, tj da su prosecne plate jednake
Q <- (sn1/length(A) + sn2/length(B))^2/((sn1/length(A))^2/(length(A)-1)+(sn2/length(B))^2/(length(B)-1))  #stepeni slobode
c <- qt((1/2)*alfa,Q)
abs(TN1) < c                      #ne upada u kriticnu oblast. Prosecne plate su jednake, tj ne odbacujemo H0

#7.
# Prvo testiramo hipotezu o jednakosti disperzija
(F_kapa <- 1600/625)
(c1 <- qf(0.1/2, 24, 24))
(c2 <- qf(1-0.1/2, 24, 24))
F_kapa < c1 | F_kapa > c2 
# F_kapa jeste u kriticnoj oblasti => odbacujemo H0
# Disperzije nisu jednake
# broj stepeni slobode za Studentovu raspodelu
(Q <- (1600/25 + 625/25)^2/((1600/25)^2/24+(625/25)^2/24))
# Q>30, pa mozemo aproksimirati normalnom, inace bismo imali qt(0.05, df=40)
(c <- qnorm(0.05))
(t_kapa <- (570-600)/sqrt(1600/25 + 625/25))
t_kapa < c
# odbacujemo H0 tj. bolje je da uzme rezim rada B

#7.
(t_kapa <- 0.045/sqrt(0.0084)*sqrt(20))
(alfa <- 0.05)
(c <- qt(1-alfa, 19))
t_kapa > c    
# realizovana vrednost T upada u kriticnu oblast => odbacujemo H0

#9.
prvi <- c(3.4, 5.1, 3.5, 7.2, 7.5)
drugi <- c(2.8, 5.4, 3.3, 7.2, 7.8)
# pravimo uzorak razlika
d <- prvi - drugi
t.test(d, mu=0, alternative = "greater") # t.test za jedan uzorak
# drugi nacin:
t.test(prvi, drugi, mu = 0, paired = TRUE, alternative = "greater") 
# t.test za dva uzorka ali sa parametrom paired=TRUE sto sugerise na uparene podatke, 
# p>0.05 => ne odbacujemo H0 
# nema razlika izmedju srednjih vrednosti cena u dva supermarketa