The influence of a single observation on some standard test statistics

The ideas of influence are now well known, and influence functions have been investigated widely, especially in the contexts of regression analysis and multivariate analysis. However, there seems to be no published account of the influence of single observations on simple one-sample tests for means and variance. The usual test statistics, based on normality assumptions, for a mean when the variance is known, and for the variance, display an obvious behaviour when an extra observation is added to a data set. However, the t-test for a single mean has a more interesting, and less predictable, pattern of behaviour. The t-statistic also demonstrates how the theoretical influence function can sometimes be misleading when used to estimate the effect of inserting or deleting a single observation.