# Lab. 11 - 22 maggio 2025

############
#Esercizio 1
############
rm(list=ls())
dati=c(63,65,94,37,83,95,70,96,47,29,52,38,47,79,66,25,48,80,52,49)
summary(dati)
# (a)
t.test(dati, alternative="two.sided", conf.level=0.95)$conf.int
# (b)
t.test(dati, alternative="less",mu=70,conf.level=0.95)
# (c)
# p-value = 0.07 > 0.05 e quindi accetto l'ipotesi nulla (nota: 70 è nell'IC)
# Del resto: t=-1.89 e
qt(0.05,df=19)
# è uguale a -1.729133 --> rifiuto H0
# Infatti p.value<0.05
pt(-1.8948,19)
# (d)
# t = -1.8948, df = 19, n=20, p-value = 0.037, mean of x = 60.75.
# (e)
# da calcolare
# S2=n*(x.med-theta.0)^2/t^2
S2=20*((60.75-70)^2)/((-1.8948)^2)
S2
# (f) 
t.test(dati, alternative="less",mu=70,conf.level=0.99)




############
#Esercizio 2
############
rm(list=ls())
x.a=c(100,125,135,128,140,142,128,137,156,142,134,122,100,98,101)
x.b=c(95,87,100,75,110,85,95,98,96,100,107)
# (a)
mean(x.a)
mean(x.b)
# (b)
var(x.a)
var(x.b)
# (c)
t.test(x.a)$conf.int
t.test(x.b)$conf.int
# (d)
t.test(x.a,conf.level = 0.8)$conf.int
t.test(x.b,conf.level = 0.8)$conf.int
# (e)
t.test(x.b,alternative = "less",mu=100)
# (f)
t.test(x.a,x.b)$conf.int
# (g)
t.test(x.a,x.b,conf.level = 0.98)$conf.int
# (h)
t.test(x.a,x.b,alternative = "two.sided")
# rifiuto HO: mu.a=mu.b, ovvero mu=0 perché l'intervalli non contiene zero
# (i)
t.test(x.a,x.b,alternative = "two.sided",mu=40)
# accetto HO: vdi p-value e IC
t.test(x.a,x.b,alternative = "two.sided",mu=40)$p.value
(j)
t.test(x.a,x.b,alternative = "less",mu=45)
# rifiuto H0
t.test(x.a,x.b,alternative = "less",mu=45)$p.value



############
#Esercizio 3
############
rm(list=ls())
data()
data(women)
attach(women)
women
# a)
summary(height)
summary(weight)
# b)
help(shapiro.test)
shapiro.test(height)   # W = 0.96359, p-value = 0.7545 (accetto H_0)
shapiro.test(weight)   # W = 0.96036, p-value = 0.6986 (accetto H_0)
#c)
t.test(height)$conf.int   # 62.52341 67.47659
t.test(weight)$conf.int   # 128.1504 145.3162
# d)
t.test(height, conf.level=0.85)$conf.int   #  63.24128 66.75872
t.test(weight, conf.level=0.85)$conf.int   #  130.6383 142.8284
# e)
t.test(height,conf.level=0.99,mu=68, alternative="less")
# p-value = 0.01053 < 0.05: rifiuto al livello 0.05 ma accetto a livello 0.01
# f) p-value = 0.01053, t = -2.5981
# g) 
nn=15
tt=-2.5981
x.med=65
mu.0=68
S2=nn*((x.med-mu.0)^2)/(tt^2)
S2



############
#Esercizio 4
############
rm(list=ls())
# help(poisson.test)
poisson.test(x=27,T=7)    # 2.541881 5.611940
# intervallo asintotico 
# x.med=poisson.test(x=27,T=7)$estimate
x.med=27/7
x.med
n=7
alpha=0.05
z=qnorm(1-alpha/2)
L=x.med-z*sqrt(x.med/n)
U=x.med+z*sqrt(x.med/n)
c(L,U)                   #2.402247   5.312039 



############
#Esercizio 5
############
rm(list=ls())
x.a=c(100,125,135,128,140,142,128,137,156,142,134,122,100,98,101)
x.b=c(95,87,100,75,110,85,95,98,96,100,107)
IC.var=function(dati, conf.lev){
  S2=var(dati)
  n=length(dati)
  alpha=1-conf.lev
  q1=qchisq(alpha/2,n-1)
  q2=qchisq(1-alpha/2,n-1)
  L=(n-1)*S2/q2
  U=(n-1)*S2/q1
  #return(c(L,U))
  risultato=c("L"=L,"U"=U, "S2"=S2)
  print(risultato)
}
IC.var(x.a,0.95)
IC.var(x.b,0.95)