Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

One-Sample t-Test - Small Samples

For small sample sizes, the mean shows a t-distribution (compare the case for large samples: in this case the means are distributed according to a normal distribution). The reason for this is that the estimate s of the standard deviation is not accurate enough. But again the assumption is that the distribution of the data is approximately normal. The only difference from the derivation for the large sample case is that, instead of the z-scores, we will use the t-scores, which we can obtain from statistical tables, or by using the distribution calculator. The t-distribution depends on the number of samples. Its parameter is called degrees of freedom, and it is equal to the number of samples minus 1: df = n - 1.

For a detailed description of how to perform a t-test, please have a look at the z-test section.

The prerequisite for the one-sample t-test is that the sample is normally distributed.