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Topic: Check again
Replies: 0

 Luis A. Afonso Posts: 4,758 From: LIsbon (Portugal) Registered: 2/16/05
Check again
Posted: May 1, 2014 4:19 PM

__1__
Data was simulated (Box-Muller) as X~N(0,1): 100, sample mean mx evaluated,
__2__
Test Statistics z= mx/sqrt(1/n) . The null: | z | < z0= 1.959964 the CI bonds relative to the Probability inside 95% , Normal Distribution.

Why I didn´t consider the Student´s paradigm for the means of Normal/Gaussian samples? Am I wrong? Not at all (instead I had yet found here criticizing people for similar situations). Simply z expression indicates clearly that I took into account that ´I know´ that the Population standard deviation is 1 . . . Are you skeptics? Do perform the routine with, say n=10, and you will find the same 95 (as I did).
Now a challenge can be posed . . .
What happened though I deliberate to ignore that the standard deviation is 1? Simply we change
__1) the test statistic, with ssd= sum of squares deviations,
t= mx /sqrt (stdev/n) , stdev= ssd/(n-1)
__2) we read on a table (or software) the bounds relative to the Student Distribution and modify accordingly the routine at this point: for n=100, and 95% probability, the CI turns out [-1.984217, 1.984217].
Luis A. Afonso

REM "ut"
CLS
PRINT : PRINT : PRINT
PRINT " <ut> "
DEFDBL A-Z
n = 100
trial = 0
cutt = 1.959964

FOR t = 1 TO 20
pi = 4 * ATN(1)
FOR rpt = 1 TO 200
LOCATE 5, 40
PRINT USING "######"; 200 - rpt
RANDOMIZE TIMER
outt = 0
FOR j = 1 TO 100
mx = 0
FOR ix = 1 TO n
1 ru = RND
IF ru < 1E-08 THEN GOTO 1
x = SQR(-2 * LOG(ru)) * COS(2 * pi * RND)
mx = mx + x / n
NEXT ix
z = mx / SQR(1 / n)
COLOR 7
IF ABS(z) >= cutt THEN GOTO 4
outt = outt + 1
4 NEXT j
mean = mean + outt / 200
trial = trial + 1
NEXT rpt
PRINT : PRINT
LOCATE 7 + 2 * t, 40
PRINT USING "### ###.## ###### "; t; mean / t; trial
NEXT t
END