The "LLM-as-an-annotator" and "LLM-as-a-judge" paradigms employ Large Language Models (LLMs) as annotators, judges, and evaluators in tasks traditionally performed by humans. LLM annotations are widely used, not only in NLP research but also in fields like medicine, psychology, and social science. Despite their role in shaping study results and insights, there is no standard or rigorous procedure to determine whether LLMs can replace human annotators. In this paper, we propose a novel statistical procedure, the Alternative Annotator Test (alt-test), that requires only a modest subset of annotated examples to justify using LLM annotations. Additionally, we introduce a versatile and interpretable measure for comparing LLM annotators and judges. To demonstrate our procedure, we curated a diverse collection of ten datasets, consisting of language and vision-language tasks, and conducted experiments with six LLMs and four prompting techniques. Our results show that LLMs can sometimes replace humans with closed-source LLMs (such as GPT-4o), outperforming the open-source LLMs we examine, and that prompting techniques yield judges of varying quality. We hope this study encourages more rigorous and reliable practices.