T-tests with models close to the normal distribution

The t-distribution is a very usual distribution for several test statistics because a normal distribution is frequently assumed as underlying model. Even in some tests based on robust statistics, such as the test based on the sample trimmed mean, a t-distribution is used as distribution for the standardized sample trimmed mean if the underlying model is normal. Nevertheless, it is necessary to have a deeper understanding of the behaviour of these kind of tests and the computations of their key elements, such as the p-value and the critical value, with small samples, when the underlying model is close but different from the normal distribution. In this paper, we obtain good analytic approximations, with small samples, for the p-value and the critical value of a t-test (i.e., a test with a t-distribution for the test statistic under a normal model), studying its behaviour when the underlying distribution is close but different from the normal model. We conclude the paper with a discussion on some robustness properties of t-tests.