Question

How is the rejection region defined, and how is that related to the p value? When do you reject or fail to reject the null hypothesis? Why do you think statisticians are asked to complete hypothesis testing? Can you think of examples in courts, in medicine, or in your area?

Answer 1

Definition:

The rejection region is used in hypothesis testing. Let T be a test statistic. Possible values of T can be divided into two regions, the acceptance region and the rejection region. If the value of T comes out to be in the acceptance region, the null hypothesis (the one being tested) is accepted, or at any rate not rejected. If T falls in the rejection region, the null hypothesis is rejected.
The terms 'acceptance region' and 'rejection region' may also refer to the subsets of the sample space that would produce statistics T that go into the acceptance region or rejection region as defined aboveThe "rejection region" is whatever tail(s) of the normal probability distribution
(or student t-distribution, or chi-square distribution) lies beyond the
critical point(s) indicated by the significance level.

For instance,

If you are conducting a "one-tailed test" where the alternate
hypothesis alleges that some average is (let's say) HIGHER than a particular
value, and you want to test it at alpha = 0.05, then the "rejection region" is
the highest 5% of the sample means that would result if the ACTUAL average
were exactly the particular value mentioned in the null hypothesis.

You might have a historical suggestion that the mean length of all whozeewhatzis
is 7.00 cm. If your alterate hypothesis says "The mean length of all whozeewhatzis is
greater than 7.00," and you are taking a sample of 40 whozeewhatzis to test
this, then you can use the sample standard deviation and the sample size to
construct a "standard error" for the sample mean. Just getting a sample mean
higher than 7.00 isn't good enough; it has to be at least a few standard errors
above the 7.00 cm. In the case of alpha=0.05, it would have to be at least 1.645
standard errors above the 7.00 cm. Every value larger than that lies in the "rejection
region." The z-score, as you must already know, is the number of standard errors
that separates the observed sample mean from the assumed null-hypothesis mean.

"Examples" - my own field was planetary science research. The validity of
newly-observed correlations is very often argued on the basis of some
statistical significance level (probability of Type I error). Another kind of
example that comes to mind is the sort of tests that have to be submitted to
FDA when seeking approval for a new drug. It has to be proven that using the
drug makes a significant difference to patients' well-being..