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Luis A. Afonso

Posts: 4,273
From: LIsbon (Portugal)
Registered: 2/16/05

Testing differences on a normal sample pair
Posted: Nov 5, 2012 1:08 PM

Plain Text

Testing differences on a normal sample pair

We intent this time to show how the P-values are distributed when one tests the difference of mean normal Populations. To this purpose samples X~N (0, 1).n and Y~N (0, 1): n are simulated and the statistics is:
T = (xhat ? yhat - D)/s
__s^2 = (ssdX + ssdY)/(n+n-2) * (1/nX + 1/nY)
It´s known that T follows a Student T with nX + nY - 2 df. Lacking an algorithm to provide the inverse T we switched to normal which is sufficiently accurate since the sample sizes are large, n=100, because student t with 100+100-2= 198 df is very lose to normal standard.
See 26.2.10 M. Abramowitz, I. Stegun, Handbook of Mathematical Functions. A recurrence formula (program WATCH) my own, was used.

Results and Discussion

P-values are shown (not in full) distributed in classes each one 0.05 amplitude. One get only 2% *rejections* from [0, 1], and any (as expected) when D= 0.
We counted
1- alpha(D= -0.5) = 37982/40000 = 0.950,
1- alpha(D= 0.5) = 37903/40000 = 0.948
very close to 0.95, as it was demanded.

X=N(0,1):100 Y=N(0,1):100

__________D= -0.5___D=0_____D= 0.5__

__0________________1912_____37903__
__1________________1804_______716__
__2________________2008_______228__
__3________________1978_______120__
__4________________1892________58__
__5________________1981________33__
__6________________2050_____
__7________________2113_____
__8________________2091_____
__9________________2023_____
_10________________2084_____
_11________________2104_____
_12________________1974_____
_13________________1992_____
_14________34______2025_____
_15________64______1977_____
_16_______131______1858_____
_17_______240______2017_____
_18_______699______2108_____
_19_____37982______3009_____

Total____39209_____40000______39175
_____>1: 791_____________<0 : 825

( my first trial: May 30, 2010 12:32 AM)

Luis A. Afonso

REM "WATCH"
CLS
DEFDBL A-Z
PRINT : PRINT : PRINT "___WATCH____"
INPUT " Two N(0,1) samples , size "; size
INPUT " D= "; mu
INPUT " How many "; repeat
pi = 4 * ATN(1): cx = 1 / SQR(2 * pi)
DIM kk(20)
DEF fng (x, k) =
-.5 * x ^ 2 * (2 * k + 1) / ((k + 1) * (2 * k + 3))
FOR j = 1 TO repeat: RANDOMIZE TIMER
LOCATE 5, 50
PRINT USING "#########"; repeat - j
PRINT : PRINT
m1 = 0: m2 = 0: ss1 = 0: ss2 = 0
FOR i = 1 TO size: r1 = RND: r2 = RND
aa = SQR(-2 * LOG(r1))
x = aa * COS(2 * pi * r2)
y = aa * SIN(2 * pi * r2)
m1 = m1 + x / size: ss1 = ss1 + x * x
m2 = m2 + y / size: ss2 = ss2 + y * y
NEXT i
ssd1 = ss1 - size * m1 * m1
ssd2 = ss2 - size * m2 * m2
sz = 1 / size + 1 / size
t = (m1 - m2) / SQR((ssd1 + ssd2) / (size + size - 2) * sz)
x = t
REM
x = (m1 - m2 - mu) / SQR(2 / size)
REM ___ x^-1___
zu = ABS(x): s = cx * zu: ante = cx * zu
FOR k = 0 TO 7777777
xx = ante * fng(zu, k)
s = s + xx
IF ABS(xx) < .0000005 THEN GOTO 10
ante = xx
NEXT k
10 REM
IF x < 0 THEN np = .5 - s
IF x >= 0 THEN np = .5 + s
IF np < 0 THEN lft = lft + 1
IF np > 1 THEN rgt = rgt + 1
REM by classes
FOR c = 0 TO 19
t0 = c / 20: t1 = (c + 1) / 20
IF np > t0 AND np < t1 THEN kk(c) = kk(c) + 1
NEXT c
PRINT "______________wait please "
NEXT j
FOR y = 0 TO 19: sum = sum + kk(y)
PRINT USING " ###) ###### "; y; kk(y);
NEXT y
PRINT sum
PRINT USING " ###### "; lft; rgt
END