data(sleep)
x <- na.omit(sleep$extra[sleep$group == 1])
y <- na.omit(sleep$extra[sleep$group == 2])

alpha <- 0.05
ftest <- var.test(x, y)             # F-test for equality of variances
equal_var <- ftest$p.value >= alpha # TRUE -> Student's ; FALSE -> Welch's
ftest

# --- Unpaired / independent groups (default) ---
ttest <- t.test(x, y,
                var.equal   = equal_var,        # from the F-test above
                alternative = "two.sided",      # or "less", "greater"
                conf.level  = 1 - alpha,        # default 95%
                paired      = FALSE)
ttest

# reported quantities
mean(x); mean(y); mean(x) - mean(y)             # group means and difference
ttest$statistic                                 # t value
ttest$parameter                                 # degrees of freedom
ttest$p.value                                   # p-value
ttest$conf.int                                  # confidence interval
if (equal_var) "Student's t-test" else "Welch's t-test"

# --- Paired design (paired = TRUE) ---
# x[i] and y[i] are the two measurements on the same unit; the test uses the
# within-pair differences, so the F-test / equal-variance step does not apply
t.test(x, y,
       alternative = "two.sided",
       conf.level  = 1 - alpha,
       paired      = TRUE)

library(nortest)
library(tseries)
z <- na.omit(sleep$extra)
n <- length(z)

if (n >= 3 && n <= 5000) shapiro.test(z)        # Shapiro-Wilk
if (n >= 8)              nortest::ad.test(z)     # Anderson-Darling
if (n >= 3)              tseries::jarque.bera.test(z)  # Jarque-Bera

# Overall: each test flags "Not Normal" when its p-value < alpha;
# the overall verdict follows the majority, with ties resolved to "Not Normal".