p vs. alpha or the Test Statistic vs. the CV
As you know, most researchers who set up and test a null hypothesis hope that they will be able to "reject the null." There are, of course, a few exceptions to this general state-of-affairs . . . but they constitute a tiny minority. In the vast majority of cases, the researcher hopes that he/she will be able to REJECT THE NULL.
In order for the typical researcher to get what he/she wants, the sample evidence must turn out to be inconsistent with what we'd expect to happen if the null were true. That should make sense. If the sample data turned out to be CONSISTENT with Ho, the researcher would have no scientific grounds to reject the null. It's only when the sample data turn out to be INCONSISTENT with Ho that the researcher can logically reject his/her null.
To determine whether the sample evidence is INCONSISTENT with Ho, the researcher will do one of two things.
Note that in the first of the two approaches, the researcher will REJECT THE NULL if p turns out to be SMALLER than the level of significance. If, on the other hand, the researcher uses the second approach, he/she will REJECT THE NULL if the calculated value is LARGER than the critical value. So . . . in the first approach, the researcher is hoping for a small p while in the second approach he/she is hoping for a large calculated value.
Finally, you need to realize that it doesn't matter which of the two approaches the researcher uses. Both procedures lead to the exact same decision about Ho. In other words, if the researcher is able to REJECT THE NULL because p is SMALLER than alpha, then it will be the case (guaranteed) that the sample data would lead to a calculated value that's LARGER than the critical value. Or, if the researcher must make a fail-to-reject decision because p is LARGER than alpha, then it'll be the case (guaranteed) that the calculated value, if it's computed, will be SMALLER than the critical value.
Copyright © 2012
Schuyler W. Huck