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.

翻译：鉴于一个积极的随机随机变数$X$, $X\ge0 a.s., 一个无效假设$H_0:E(X)\le\mu$, 一个无限大小为$X美元的随机抽样, 我们为H_0美元建造测试超边线, 也就是说, 如果满足了无边假设, 即肯定的超边线过程。 我们用测试超边线假设测试假设$H_ 0。 我们用测试超边假设构建测试超边线, 从而导致用超边线进行测试。 我们产生信任度较低的范围。 对于受约束的随机变量, 我们将技术扩展至H_ 0:E(X)\ ⁇ mu$的双面测试和建立信任间隔。 在财务审计中, 随机抽样被提议为收集足够证据, 以证明拒绝无边假设, 即财务报告中存在“ 物质” 错报。 我们工作的目标是提供一个数学背景, 通过重复随机抽样来代表这种证据收集过程, 同时确保预期的意义水平 。