##############################################################
#               AULA 20 - Testes de Hipoteses                #
##############################################################

rm(list = ls()); cat('\014')

#######################################################################

# Exemplo: Teste para a média de duas populações Normais (teste t de Student)

# AMOSTRA

# Mulheres
n1 = 9782
xbarra = 36.883459
s1 = 13.532427

# Homens
n2 = 20380
ybarra = 39.184004
s2 = 12.873243

var.test(idades ~ sexo, data, alternative = "two.sided")
var.test(idadem, idadeh, alternative = "two.sided")

Fcal = s1^2 / s2^2
Fcal
## [1] 1.105034

pvalor = pf(Fcal, n1 - 1, n2 - 1)
pvalor
## [1] 1

# Erro tipo 1 = alfa
alfa = 0.05

# Graus de liberdade
gl = n1 + n2 - 2

# Valor de Ttab para o erro tipo 1
qt(alfa/2, gl)
## [1] -1.960043

qt(1 - alfa/2, gl)
## [1] 1.960043

S2p = ((n1-1)*s1^2 + (n2-1)*s2^2) / (n1 - 1 + n2 - 2)
S2p
## [1] 171.371

ttab = (xbarra - ybarra) / sqrt(S2p*(1/n1+1/n2))
ttab
## [1] -14.28723

pvalor = 2*pt(ttab, n1 - 1 + n2 - 2)
pvalor
## [1] 3.719862e-46

t.test(x, y = NULL, alternative = c("two.sided", "less", "greater"),
       mu = 0, paired = FALSE, var.equal = FALSE, conf.level = 0.95, ...)

t.test(formula, data, subset, na.action, ...)

require(graphics)
sleep

plot(extra ~ group, data = sleep)

with(sleep, mean(extra[group == 1]))
## [1] 0.75

with(sleep, mean(extra[group == 2]))
## [1] 2.33

with(sleep, sd(extra[group == 1]))
## [1] 1.78901

with(sleep, sd(extra[group == 2]))
## [1] 2.002249



# Forma tradicional

with(sleep, t.test(extra[group == 1], extra[group == 2]))
##
## Welch Two Sample t-test
##
## data: extra[group == 1] and extra[group == 2]
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -3.3654832 0.2054832
## sample estimates:
## mean of x mean of y
## 0.75 2.33


# Formula tipo regressão

t.test(extra ~ group, data = sleep)
##
## Welch Two Sample t-test
##
## data: extra by group
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means between group 1 and group 2 is not e## 95 percent confidence interval:
## -3.3654832 0.2054832
## sample estimates:
## mean in group 1 mean in group 2
## 0.75 2.33


set.seed(12345)
x = rnorm(10, 5, 1)
y = rnorm(10, 5, 2)

mean(x)
## [1] 4.867056

mean(y)
## [1] 5.571956

s1 = sd(x)
s2 = sd(y)
n1 = length(x)
n2 = length(y)

# Teste igualdade de variancias

var.test(x, y, alternative = "two.sided")
##
## F test to compare two variances
##
## data: x and y
## F = 0.23356, num df = 9, denom df = 9, p-value = 0.0413
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 0.05801292 0.94030993
## sample estimates:
## ratio of variances
## 0.2335597

Fcal = s1^2/s2^2
Fcal
## [1] 0.2335597

pvalor = 2*pf(Fcal, n1 - 1, n2 - 1)
pvalor
## [1] 0.04130414

# Variancias são diferentes, uma vez que pvalor < alfa. Portanto, usaremos o
# teste de igualdade de medias para variancias desconhecidas e diferentes

t.test(x, y, alternative = "two.sided", var.equal = FALSE)
##
## Welch Two Sample t-test
##
## data: x and y
## t = -1.1921, df = 12.987, p-value = 0.2546
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -1.9825002 0.5727008
## sample estimates:
## mean of x mean of y
## 4.867056 5.571956

nu = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1-1) +(s2^2/n2)^2/(n2-1))
nu
## [1] 12.9866

2*pt(-1.1921, nu)
## [1] 0.2545508


# DADOS 1
set.seed(12345)
x = rnorm(10, 5, 1)
y = rnorm(10, 4, 1)

mean(x)
## [1] 4.867056

mean(y)
## [1] 4.285978

sd(x)
## [1] 0.8136554

sd(y)
## [1] 0.8418052

length(x)
## [1] 10

length(y)
## [1] 10

t.test(x, y, alternative = "two.sided", var.equal = FALSE)
##
## Welch Two Sample t-test
##
## data: x and y
## t = 1.5695, df = 17.979, p-value = 0.134
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.1968018 1.3589580
## sample estimates:
## mean of x mean of y
## 4.867056 4.285978


# DADOS 2
x = rnorm(50, 5, 1)
y = rnorm(50, 4, 1)

mean(x)
## [1] 5.20829

mean(y)
## [1] 4.27182

sd(x)
## [1] 1.234496

sd(y)
## [1] 1.046154

length(x)
## [1] 50

length(y)
## [1] 50

t.test(x, y, alternative = "two.sided", var.equal = FALSE)
##
## Welch Two Sample t-test
##
## data: x and y
## t = 4.0922, df = 95.432, p-value = 8.935e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.4821888 1.3907509
## sample estimates:
## mean of x mean of y
## 5.20829 4.27182

t.test(x, y, alternative = "two.sided", var.equal = TRUE)