Question

We reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region. Why?

Short answer

The simple justification for this is that the absence of evidence supporting the falsity of a statement does not provide enough to support its truthfulness.

Step-by-step solution

Given information

The statement is given that "We reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region".

Explaining the given statement

The null hypothesis is an assertion assumed to be true until proven otherwise by facts. When a test fails to reject the null hypothesis, it is important to remember that this does not prove it.

The simple justification for this is that the absence of evidence supporting the falsity of a statement does not provide enough to support its truthfulness. For example, there are frequently many null hypotheses for the mean's true value that can't reject for a given collection of data. Hence, we reject the null hypothesis when the test statistic falls in the rejection region, but we do not accept the null hypothesis when the test statistic does not fall in the rejection region.