Why are non-conservative tests desired over conservative ones? In my textbook, it says that P(Type I Error) =  α is a non-conservative test and my professor noted that this one is desired over the conservative test with P(Type I Error)  ≤ α. Why is that the case? Wouldn't we want the probability that we reject $H_0$ when it is true to be less than the significance level?
Edit: Conservative tests will detect a correct violation of the null hypothesis less often than a non-conservative tests (taken from the book).

When you set $\alpha$, you are saying that you accept a type I error rate of $\alpha$. Once you deem this acceptable, get as low of a type II error rate as you can!
Sure, you might be able to get a lower type I error rate, but that will sacrifice power, and you already decided that a type I error rate of $\alpha$ is acceptable.
