Unexpectedly, the theoretically best reject-region of T-test is bounded.

For NHST vs , theoretically, is s.t. , rather than zero. Nevertheless, pratically a large t, rejecting both and , should not be counted as any evidence to retain or reject .

To verify the shape of --

Classic Neyman-Pearson approach demo

It notes here that N-P approach does not utilize the information in the accurate p value. Actually, at the time N-P approach was firstly devised, the accurate p value was not available usually. Now almost all statistic softwares provide accurate p values and the N-P approach becomes obsolete. Wilkinson & APA TFSI (1999) recommended to report the accurate p value rather than just significance/insignificance, unless p is smaller than any meaningful precision.


Wilkinson, L. & APA TFSI (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604.



The tail(s) of p value

For any given vs , the p value of any given point x is , Where

-- See R. Weber's Statistics Note (Chap 6.2 & 7.1)

I made some wrong comment on the pdf Null Ritual (Gigerenzer, Krauss, & Vitouch, 2004) Where three types of significance level (rather than p value) were discussed. I had written the comment to note that the chapter had ignored the role of in definition of p value. In almost every textbook, the two-tail p vs single-tail p are differentiated. Usually, the two-tail p is defined by like .

Here I demonstrate a three-tail p value case on R platform.

f=0.5 * dchisq(abs(z),df=5);
## is * binomial(-1 vs 1) ##

Do you agree the region nearby zero under the "V" curve (which is below the horizontal line) should be the 3rd tail? I think so, if only includes all other possible distributions in the same shape.

You'll also agree there will be two asymmetrical tails if includes just two asymmetrical curves, for example, and () while is the standardized normal distribution.