The univariate t-test

The t-test is a basic tool in statistics used to compare the means of two samples. Its most strict version assumes that both samples are distributed according to a Gaussian distribution and have the same (unknown) variance.

In statistics, we refer to the null hypothesis as the hypothesis we want to test, and the alternative hypothesis as the one we use when the null is rejected. They are usually referred to as H₀ and H₁.

The t-test tests the equality of means from two populations, and it can be formulated in three possible ways:

• H₀: The means of the two populations are the same versus H₁: in that the means of the populations are different.
• H₀: The mean of population 1 is greater or equal than the one from population 2 versus H₁: the mean of population 1 is smaller than population 2.
• H₀: The mean of population 1 is smaller or equal than the one from population...