____Simulating size=100, 40000 normal samples, mu=456.2, sigma=2.441 we record how much of them provide Dmax greater than the observed Dmax= 0.12735 and D2max= 0.12157. This is the obvious way to get the respective p-values of DATA for 5% significance. We know yet that the relative frequencies are very lower than 0.05. Otherwise it was inconsistent with results of the respective test critical values (0.0888, 0.0847). Results, p-values , 5% significance level (routine <PONG>) ___________________Dmax ____D2max _______1st attempt __ 0.0005 ____0.0005___ _______2nd_________0.0005 ____0.0006___ _______3rd_________ 0.0005 ____0.0005___ _______4th_________ 0.0006 ____0.0006___
As expected, the sample under study is very (very) unlike to have been drawn from a normal population.
Luis A. Afonso REM "PONG" CLS DEFDBL A-Z RANDOMIZE TIMER PRINT : PRINT PRINT "__________________________ PONG " n = 100 DIM X(n), xx(n), diff(n), F(n) DEF fng (z, j) = -.5 * z * z * (2 * j + 1) / ((j + 1) * (2 * j + 3)) pi = 4 * ATN(1): c = 1 / SQR(2 * pi) FOR h = 1 TO n: F(h) = h / n: NEXT h REM mu = 456.2: sigma = 2.441 REM ally = 40000 REM FOR ju = 1 TO ally LOCATE 3, 50: PRINT USING "#########"; ally - ju meanX = 0: ssX = 0 high = 0: higher = 0 FOR i = 1 TO 100 aa = SQR(-2 * LOG(RND)) X(i) = mu + sigma * aa * COS(2 * pi * RND) meanX = meanX + X(i) / n ssX = ssX + X(i) * X(i) NEXT i ssdX = ssX - n * meanX * meanX varX = ssdX / (n - 1) mean = meanX sd = SQR(varX) REM REM FOR i = 1 TO n: X(i) = (X(i) - mean) / sd xx(i) = X(i): NEXT i REM PRINT REM mmajor = -1E+20: second = -1E+20 REM md = 0: soma2 = 0 REM REM ordered values: xx( ) REM FOR ii = 1 TO n: u = X(ii): w = 1 FOR jj = 1 TO n IF X(jj) < u THEN w = w + 1 NEXT jj: xx(w) = u NEXT ii REM REM "*********************" REM phi calculation
FOR tt = 1 TO n: z = xx(tt) zu = ABS(z): s = c * zu: ante = c * zu FOR j = 0 TO 4444444 xx = ante * fng(zu, j) s = s + xx ante = xx IF ABS(xx) < .00000049# THEN GOTO 20 NEXT j 20 IF z >= 0 THEN FF = .5 + s IF z < 0 THEN FF = .5 - s b = ABS(FF - F(tt - 1)) bb = ABS(F(tt) - FF): REM F(tt=0)=0 major = b IF bb > b THEN major = bb diff(tt) = major NEXT tt REM local difference = diff(tt) higher = -1 FOR ii = 1 TO n IF diff(ii) <= higher THEN GOTO 22 higher = diff(ii): llocal = ii 22 NEXT ii REM diff(llocal) = -2: high = -1 FOR i2 = 1 TO n IF diff(i2) <= high THEN GOTO 33 high = diff(i2): i1 = i2 33 NEXT i2 REM REM REM 5% significance level dataDmax = .12735: dataD2max = .12157 REM IF higher > dataDmax THEN f1 = f1 + 1 / ally IF high > dataD2max THEN f2 = f2 + 1 / ally NEXT ju PRINT PRINT " __________5% Dmax__________" PRINT USING " ##.#### "; f1 PRINT " __________5% D2max_________" PRINT USING " ##.#### "; f2 REM END