# Ficha 14

# 1.a)

nh <- 200
phobs <- 15/nh
ph0 <- 0.1
z <- (phobs-ph0)/sqrt(ph0*(1-ph0)/nh)
z <- round(z,2)
p.value <- pnorm(z)
p.value


# 1.b)

alfa <- 0.02
zc <- qnorm(alfa)
zc <- round(zc,2)
c <- ph0+zc*sqrt(ph0*(1-ph0)/nh)
c <- round(c,3)
c
ph1 <- 0.08
z <- (c-ph1)/sqrt(ph1*(1-ph1)/nh)
z <- round(z,2)
z
pnorm(z)


# 1.c)

alfa <- 0.05
nm <- 180
pmobs <- 9/nm
z <- qnorm(alfa/2)
z <- round(z,2)
pobs <- (nh*phobs+nm*pmobs)/(nh+nm)
c1 <- z*sqrt(pobs*(1-pobs)*(1/nh+1/nm))
c2 <- -c1
c1
c2
phobs-pmobs


# 2.a)

ni <- c(18,17,16,19)
nA <- sum(ni)
pA0 <- 0.4
pAobs <- sum(ni[1:2])/nA
alfa <- 0.01
zc <- qnorm(1-alfa)
zc <- round(zc,2)
c <- pA0+zc*sqrt(pA0*(1-pA0)/nA)
c


# 2.b)

z <- (pAobs-pA0)/sqrt(pA0*(1-pA0)/nA)
z <- round(z,2)
p.value <- pnorm(z)
1-p.value

# Verificação.
alfa <- 0.044
zc <- qnorm(1-alfa)
#zc <- round(zc,2)
c <- pA0+zc*sqrt(pA0*(1-pA0)/nA)
c
pAobs


# 2.c)

alfa <- 0.02
nB <- 60
pBobs <- 21/nB
#pAobs-pBobs = 0.15 > 0 => p.value calculado à direita.
pobs <- (nA*pAobs+nB*pBobs)/(nA+nB)
z <- (pAobs-pBobs)/sqrt(pobs*(1-pobs)*(1/nA+1/nB))
z <- round(z,2)
p.value <- 2*(1-pnorm(z))
p.value


# 3.a)

alfa <- 0.05 # Por defeito.
ni <- c(8,17,16,9)
nA <- sum(ni)
pAobs <- ni[2]/nA
pA0 <- 0.4
z <- (pAobs-pA0)/sqrt(pA0*(1-pA0)/nA)
z <- round(z,2)
p.value <- 2*pnorm(z)
p.value


# 3.b)

alfa <- 0.01
nB <- 50
pBobs <- 0.43
zc <- qnorm(alfa)
zc <- round(zc,2)
pobs <- (nA*pAobs+nB*pBobs)/(nA+nB)
c <- zc*sqrt(pobs*(1-pobs)*(1/nA+1/nB))
c


# 3.c)

pA_pB <- -0.06
c <- round(c,2)
z <- (c-pA_pB)/sqrt(pAobs*(1-pAobs)/nA+pBobs*(1-pBobs)/nB)
z <- round(z,2)
1-pnorm(z)


# 4.a)
mu <- 23.7
sig <- 9.34
z <- (15-mu)/sig
z <- round(z,2)
p0 <- pnorm(z)
p0 <- round(p0,4)
n <- 80
pobs <- 18/n
z <- (pobs-p0)/sqrt(p0*(1-p0)/n)
z <- round(z,2)
p.value <- 2*(1-pnorm(z))
p.value


# 4.b)

alfa <- 0.05
n1 <- n
p1obs <- pobs
n2 <- 100
p2obs <- 0.37
zc <- qnorm(alfa/2)
zc <- round(zc,2)
pobs <- (n1*p1obs+n2*p2obs)/(n1+n2)
c1 <- zc*sqrt(pobs*(1-pobs)*(1/n1+1/n2))
c2 <- -c1
c1
c2
p1obs-p2obs


# 4.c)

alfa <- 0.05
zc <- qnorm(alfa/2)
zc <- round(zc,3)
pobs <- (p1obs+p2obs)/2
n <- 2*pobs*(1-pobs)*(zc/(p1obs-p2obs))^2
floor(n)

# Confirmação
n1 <- 77
p1obs <- 0.225
n2 <- n1
p2obs <- 0.37
zc <- qnorm(alfa/2)
#zc <- round(zc,2)
pobs <- (n1*p1obs+n2*p2obs)/(n1+n2)
c1 <- zc*sqrt(pobs*(1-pobs)*(1/n1+1/n2))
c2 <- -c1
c1
c2
p1obs-p2obs