The t-Test

A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.

Student's $t$-distribution arises in estimation problems where the goal is to estimate an unknown parameter when the data are observed with additive errors.

Assumptions of a $t$-test depend on the population being studied and on how the data are sampled.

The $t$-test is the most powerful parametric test for calculating the significance of a small sample mean.

Two-sample t-tests for a difference in mean involve independent samples, paired samples, and overlapping samples.

Paired-samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice.

The following is a discussion on explicit expressions that can be used to carry out various $t$-tests.

The following is a discussion on explicit expressions that can be used to carry out various t-tests.

Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.

When the normality assumption does not hold, a nonparametric alternative to the $t$-test can often have better statistical power.

Cohen's $d$ is a method of estimating effect size in a $t$-test based on means or distances between/among means.