Two-sample Student's t-test for Univariate Means

Given two univariate samples $x$ and $y$, it tests $$H_0 : \mu_x^2 \left\lbrace =,\geq,\leq \right\rbrace \mu_y^2\quad vs\quad H_1 : \mu_x^2 \left\lbrace \neq,<,>\right\rbrace \mu_y^2$$ using the procedure by Student (1908) and Welch (1947).

mean2.ttest(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  paired = FALSE,
  var.equal = FALSE
)

Arguments

x: a length-$n$ data vector.
y: a length-$m$ data vector.
alternative: specifying the alternative hypothesis.
paired: a logical; whether consider two samples as paired.
var.equal: a logical; if FALSE, use Welch's correction.

Value

a (list) object of S3 class htest containing:

statistic: a test statistic.
p.value: $p$-value under $H_0$.
alternative: alternative hypothesis.
method: name of the test.
data.name: name(s) of provided sample data.

References

Student (1908). “The Probable Error of a Mean.” Biometrika, 6(1), 1. ISSN 00063444.

Student (1908). “Probable Error of a Correlation Coefficient.” Biometrika, 6(2-3), 302--310. ISSN 0006-3444, 1464-3510.

Welch BL (1947). “The Generalization of `Student's' Problem when Several Different Population Variances are Involved.” Biometrika, 34(1/2), 28. ISSN 00063444.

Examples

# \donttest{
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  x = rnorm(57)  # sample x from N(0,1)
  y = rnorm(89)  # sample y from N(0,1)
  
  counter[i] = ifelse(mean2.ttest(x,y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.ttest'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))
#> 
#> * Example for 'mean2.ttest'
#> *
#> * number of rejections   : 45
#> * total number of trials : 1000
#> * empirical Type 1 error : 0.045
# }