T-tests are used, most frequently, when testing to see if the mean value of a population, μ, equals some pre-specified value, a, based on a random sample for which the standard deviation, σ, is estimated from the sample rather than known. A similar test can also be applied to the difference between two means, again with unknown standard deviations and for tests of proportions. T-tests may also be applied to determine whether the coefficients in regression models are significantly different from zero.
If the population standard deviations are known, or excellent estimates for these exist, a z-test for mean values should be used. Deviation from Normality is not always a serious problem, but if a sample distribution appears to be substantially non-Normal then wither the data must be transformed or an alternative test, such as the Wilcoxon rank-sum test, should be used. It is common to test for Normality before conducting t-tests.