```Date: Dec 5, 2012 10:32 AM
Author: Luis A. Afonso
Subject: p-values ranges under Null Hypotheses

p-values ranges under Null HypothesesUnder Null Hypotheses true p-values do follow a Uniform Distribution when the Test Statistics is continuous. Given a set n of p-values the probability that the range be < x is easily calculated: for example:_____ n=8 provides ______x_____P (range<x) __ _____0.500____.0352_______0.525____.0475_______0.550____.0632_______0.575____.0826__Therefore if we observe a range of 0.525 or less we reject uniformity at 4.75% confidence level, H0 is false; on contrary 0.550 or more there is not sufficient evidence to reject uniformity.The former case, low range set, if we are dealing with a right side hypotheses test, induces that the values are anomalous near 1: the observed statistical p-values are oddly away from 0, not sufficiently dispersed in [0, 1] which happens if the null hypotheses is really true. Luis A. Afonso        REM "U-range"        CLS        DEFDBL A-Z        INPUT " sample size          "; n        PRINT : PRINT : PRINT : CLS        PRINT : PRINT        PRINT " UNIFORM distribution: Dudewicz, Mishra, p.285 "        PRINT " Modern Mathematical Statistics, Aug. 1987     "        PRINT " prob. Range < x      "REM     DEF fnu (n, x) = n * (n - 1) * (x ^ (n - 1) / (n - 1) - x ^ n / n)REM        PRINT "    x   prob.    "        FOR x = .5 TO 1.025 STEP .025        COLOR 7        u = fnu(n, x)        IF u < .055 THEN COLOR 14        IF u > .1 THEN GOTO 4        PRINT USING "  .### .####        "; x; fnu(n, x)4       NEXT x        PRINT " sample size =   "; n        END
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