Question

Give an example of a situation where you would not want to select a very small significance level.

Short answer

A situation where we would not want to select a very small significance level would be during tests for a serious disease, where a false positive (Type I error) could lead to unnecessary stress, expensive treatments, and negative impacts on a patient's quality of life. In this scenario, it's important to minimize the chances of a Type I error, hence, a larger significance level would be more suitable.

Step-by-step solution

Understand the Scenario

First, it's necessary to understand the implications of both type I and type II errors in different situations. In some contexts, a Type I error can have much more serious consequences than a Type II error, so it's in those cases we would want to avoid a very small significance level. The cost of making a Type I error should be carefully considered.

Evidence for an Example

An example can be of a medical test for a serious disease, where a false positive (Type I error) can lead to unnecessary stress, costly treatments and negative impact on patient's quality of life. Here, the null hypothesis is that the patient does not have the disease and we would not want to reject this falsely at a small significance level.

Formulate Conclusion

The selection of the significance level in the test is not always about minimizing errors, but also about minimizing the adverse consequences of these errors. In scenarios involving grave consequences, like the disease example, a higher significance level might be more appropriate to reduce the risk of Type I error.