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Luis A. Afonso
Posts:
4,275
From:
LIsbon (Portugal)
Registered:
2/16/05
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You NEVER know if a coin is FAIR!
Posted:
Aug 6, 2011 1:23 PM
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You NEVER know if a coin is FAIR!
This is a rather trivial matter for a lot of people. However because statistical users largely exceed *abusers* I think the matter is not futile to present here.
Users, a large amount of them, were persuaded that Statistics, in particular Significance Tests, is able to decide though a Null Hypotheses is true. The *abusers*, those who had got the true problem?s realm, say peremptory: NO, NEVER.
To the *believers* I use to ask a simple question: How many finite flips should you perform in order that to be sure a coin is fair?. They are unable to get an answer: nobody does.
In fact I propose that a person should find n such that an interval [0.5-d , 0.5+d] which associated to the probability 1-x : d=0, x= 0: IMPOSSIBLE.
Values:
__________n____#heads interval___alpha_____C.V.__
_d=0.01__6765___[ 3315, 3450 ]____0.1_____1.645__
________ 9604___[ 4706, 4898 ]____0.05____1.960__
_______12544___ [ 6147, 6397 ]____0.025___2.240__
_______16589___ [ 8129, 8461 ]____0.01____2.576__
_______19698___ [ 9652, 10046]___ 0.005___2.807__
_______27077___ [13268,13809]___ 0.001___3.291__
_d=0.0001__
_______67650627____[ 33818548, 33832078 ]_____
_______96040009____[ 48010400, 48029608 ]_____
______125440007____[ 62707460, 62732548 ]_____
______165894405____[ 82930613, 82963792 ]_____
______196981224____[ 98470914, 98510310 ]_____
_____ 270767020___[ 135356434, 135410587 ]_____
___271 millions of trials, to achieve 0.5 +/- 0.001 with the probability 99.9% (!).
Luis A. Afonso
REM "YOUNEVER"
CLS
DEFDBL A-Z
COLOR 8
z(1) = 1.645: REM alpha/2:alpha = .1
z(2) = 1.96: REM alpha=0.05
z(3) = 2.24: REM =.025
z(4) = 2.576: REM =.010
z(5) = 2.807: REM =.005
z(6) = 3.291: REM =.001
PRINT : PRINT : PRINT
COLOR 14
delta = .01
PRINT USING " d= #.#### "; delta
COLOR 7
FOR zz = 1 TO 6
n = (.5 / delta) ^ 2 * z(zz) ^ 2
PRINT USING "############### "; n;
REM
u1 = (.5 - delta) * n
u2 = (.5 + delta) * n
PRINT USING " ############ "; u1; u2
NEXT zz
COLOR 14
delta = .001
PRINT USING " d= #.#### "; delta
COLOR 7
FOR zz = 1 TO 6
n = (.5 / delta) ^ 2 * z(zz) ^ 2
PRINT USING "###############"; n;
REM
u1 = (.5 - delta) * n
u2 = (.5 + delta) * n
PRINT USING " ############"; u1; u2
NEXT zz
COLOR 14
delta = .0001
PRINT USING " d= #.#### "; delta
COLOR 7
FOR zz = 1 TO 6
n = (.5 / delta) ^ 2 * z(zz) ^ 2
PRINT USING "###############"; n;
REM
u1 = (.5 - delta) * n
u2 = (.5 + delta) * n
PRINT USING " ############"; u1; u2
NEXT zz
LOCATE 15, 50
END
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