One-sided significance test: essay - key question

Question

For an essay on the one-sided significance test, I have considered the following structure:

First, generally describe the one-sided test (such as an introduction, setting up the hypotheses, decision rule), then describe the errors more accurately, then write an example and finally make a comparison between the two-sided significance test and the alternative test with the one-sided significance test.

Is that a good structure or could we improve it? The structure should be such that the level is always higher, it should be divided into three requirement areas.

In addition, I am looking for a key question. I thought something like, "How do we know how much we're wrong if we don't reject $H_0$?", and describe in the answer the probability of error? Or is that wrong? Or is that not suitable as a key question?

Answer 1

I would suggest the usual way to approach hypothesis-tests

• Formulation of the hypothesis $H_0$.
• Formulation of the alternative hypothesis $H_1$
• Choosing a maximum probability for the error of first kind denoted $\alpha$
• Choosing a critical range $B$ with the property that an occurence in $B$ has probability at most $\alpha$ assuming $H_0$ is true.
• Deciding whether the observed value lies in $B$. If it is, we assume $H_1$. If it is not, we have no reason to reject $H_0$.

As a key-question the probability to make an error of the second kind is a good one, but the exact probability distribution should be given so that the probability can actually be calculated.

Another possibility for a key question : Would it affect the decision, if we used a two-sided test rather than a one-sided-test ?