A Type 2 error, also known as a false negative, occurs in statistical hypothesis testing when a null hypothesis is not rejected when it is, in fact, false. In other words, it's the error of failing to reject a false null hypothesis.

Example

Let's consider a pharmaceutical company testing a new drug. The null hypothesis (H0) might be that the new drug has no effect on patients. The alternative hypothesis (H1) would be that the drug does have an effect on patients.

A Type 2 error would occur if the clinical trials conclude that the drug has no effect (i.e., they fail to reject the null hypothesis), when in reality, the drug does have an effect (i.e., the null hypothesis is false).

Context in Multi-arm Experiments

In the context of multi-arm experiments, as mentioned in the provided text, there's a trade-off between Type I and Type 2 errors. If you want to be cautious and maintain your Type I error rates at 5%, you can use a Bonferroni correction, but you should realize that you're increasing your Type 2 error rates.

Significance

The probability of making a Type 2 error is often denoted by β. The power of a test, which is the probability of correctly rejecting a false null hypothesis, is calculated as 1 - β. Therefore, the higher your risk of a Type 2 error, the lower the power of your test.