When attributes are rare and few or none are observed in the selected sample from a finite universe, sampling statisticians are increasingly being challenged to use whatever methods are available to declare with high probability or confidence that the universe is near or completely attribute-free. This is especially true when the attribute is undesirable. Approximations such as those based on normal theory are frequently inadequate with rare attributes. For simple random sampling without replacement, an appropriate probability distribution for statistical inference is the hypergeometric distribution. But even with the hypergeometric distribution, the investigator is limited from making claims of attribute-free with high confidence unless the sample size is quite large using nonrandomized techniques. In the hypergeometric setting with rare attributes, exact randomized tests of hypothesis a, re investigated to determine the effect on power of how one specifies the null hypothesis. In particular, specifying the null hypothesis as zero attributes does not always yield maximum possible power. We also consider the hypothesis specification question under complex sampling designs including stratified random sampling and two-stage cluster sampling (one case involves random selection at first stage and another case involves probability proportional to size without replacement selection at first stage). Also under simple random sampling, this article defines and presents a simple algorithm for the construction of exact randomized'' upper confidence bounds which permit one to possibly report tighter bounds than those exact bounds obtained using nonrandomized'' methods.