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Topic: Binomial to Normal approach
Replies: 0

 Luis A. Afonso Posts: 4,758 From: LIsbon (Portugal) Registered: 2/16/05
Binomial to Normal approach
Posted: May 22, 2014 1:43 PM

Binomial to Normal approach

Though the sample size is large a 95% confidence Interval for the mean can be evaluated from a Binomial (a Bernoulli independent events sequence each one with the probability p to occur) sample from which it was obtained y successes out of n, Critical Interval approximate,
y/n +/- 1.960*sqrt [y/n*(1-y/n)/n]
To simulate exactly Binomial r.v. is straightforward: from n events do count a success every time, a RND is lesser (or equal) than p, otherwise do not. The number of successes follows exactly a Binomial Distribution with parameters p, n.
Project
Ã routine, <saphir-0> , in view to illustrate the accuracy we got relative to the expected difference between two proportions pX- pY. For the Normal Approach
Sigma= sqrt[pX*(1-pX)/nX + pY*(1-pY)/nY]____(a)
Results
Mean difference (out of 20 attempts)
Mean sigma
(a)

Experiment 1 ______(0.5, 40 ) - (0.2, 35)
0.2993 against 0.3000
0.1040 against (a)= 0.1040
Experiment 2 ______(0.8, 40 ) - (0.5, 30)
0.3001 against 0.3000
0.1114 against (a)= 0.1111
Experiment 2´
0.3005
0.1108
Experiment 2´´
0.3002
0.1110
Experiment 3 ______(0.8, 200 ) - (0.5, 150)
0.2998 against 0.3000
0.0496 against (a)= 0.0497
Experiment 3´
0.3002
0.0497

It worth to be noted that simulating 2000 samples size afresh we get a null bias compared with the classical Normal Approach even for 30 to 40 sizes and p so far as 0.2 (or 0.8) from ½.
To corroborate this result I made a new experiment, (0.90, 30), (0.85, 30), providing, in mean, a difference= 0.0500 and sigma = 0.0852 when (1) give 0.0851.

Luis A. Afonso

REM "SAPHIR-0"
CLS
DEFDBL A-Z
PRINT
PRINT " <SAPHIR-0> "
PRINT
PRINT " X --- > BIN(0.8, 200) ";
PRINT " Y --- > BIN(0.5, 150) "
PRINT : PRINT
INPUT " pX , nX "; pX, nX
INPUT " pY , nY "; pY, nY
PRINT : PRINT : PRINT : PRINT
FOR encore = 1 TO 20
RANDOMIZE TIMER
rrpt = 2000
FOR rpt = 1 TO rrpt
REM
xi = 0: yi = 0
FOR i = 1 TO nX
IF RND < pX THEN xi = xi + 1
NEXT i
FOR i = 1 TO nY
IF RND < pY THEN yi = yi + 1
NEXT i
diff = xi / nX - yi / nY
Ediff = Ediff + diff / rrpt
ssdiff = ssdiff + diff * diff
NEXT rpt
PRINT " Ediff, ssdiff ";
PRINT USING " ###.### "; Ediff; ssdiff;
meanediff = meanediff + Ediff
var = (ssdiff - rrpt * Ediff * Ediff) / (rrpt - 1)
PRINT USING " sigma ##.#### "; SQR(var)
Esigma = Esigma + SQR(var) / 20
Ediff = 0: ssdiff = 0
NEXT encore
PRINT
PRINT USING " Meanediff= ##.#### "; meanediff / 20
PRINT USING " Esigma= ##.#### "; Esigma
PRINT : COLOR 3
PRINT " Normal approach"
u = pX * (1 - pX) / nX + pY * (1 - pY) / nY
u = SQR(u)
PRINT USING " ##.#### "; u
END