Final answer:
When a hypothesis test fails to reject a false null hypothesis, it is known as a Type II error. This error indicates that the test has not identified an existing effect or difference. The power of the test, which is inversely related to the probability of a Type II error, helps determine the test's effectiveness in correctly detecting an alternative hypothesis.
Step-by-step explanation:
A hypothesis test that fails to reject a false null hypothesis results in a Type II error. This type of error occurs when a decision is made not to reject the null hypothesis when, in reality, the null hypothesis is incorrect. It is the error of failing to detect an effect or difference when one actually exists. The probability of making a Type II error is denoted by β, and the power of the test, which is 1 - β, quantifies the likelihood that the test will correctly accept a true alternative hypothesis. As we want to minimize errors, both α (the probability of a Type I error, where a true null hypothesis is wrongly rejected) and β should be as small as possible. However, these probabilities are rarely zero due to the inherent variability in data and sampling methods.