Mathematical Statistics Question (Power Function)

Question

Can someone explain to me why we would want to maximize the power function (the probability our parameter is part of our alternative hypothesis) if that minimizes Type II Error when Type I Error is regarded as worse?

Answer 1

We want to minimize both the probability of Type $1$ error $(\alpha)$ and Type $2$ error ($\beta$). We choose a level of $\alpha$, often but not always $0.05$. Given that we have chosen this, we would like to minimize the probability of Type $2$ error, which means to maximize the power.

That's why we call it "power"--it is the ability of the test to recognize that the null hypothesis is false when it is.