I have a sample that is collected to verify the accuracy of a new random number generator. Applying the goodness of fit test to check if this sample comes from the Standard Normal Distribution and Uniform Distribution, I found out that both of these accept null hypothesis although they are contradicting to each other. Is it theoretically possible? Why?
Hypothesis tests don't accept the null hypothesis. Instead, they fail to reject the null hypothesis. That's not the same thing. For instance, if your sample size is small enough, it's certainly possible to fail to reject both null hypotheses. All that the tests are saying is that the hypotheses "this data came from a standard normal" and "this data came from a uniform distribution" both seem plausibly consistent with the data, and that's not impossible.