  Given a positive random variable $X$, $X\ge0$ a.s., a null hypothesis
$H_0:E(X)\le\mu$ and a random sample of infinite size of $X$, we construct test
supermartingales for $H_0$, i.e. positive processes that are supermartingale if
the null hypothesis is satisfied. We test hypothesis $H_0$ by testing the
supermartingale hypothesis on a test supermartingale. We construct test
supermartingales that lead to tests with power 1. We derive confidence lower
bounds. For bounded random variables we extend the techniques to two-sided
tests of $H_0:E(X)=\mu$ and to the construction of confidence intervals. In
financial auditing random sampling is proposed as one of the possible
techniques to gather enough evidence to justify rejection of the null
hypothesis that there is a 'material' misstatement in a financial report. The
goal of our work is to provide a mathematical context that could represent such
process of gathering evidence by means of repeated random sampling, while
ensuring an intended significance level.
