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Topic: No Tables and rigorously, Chi-square Goodness-of-fit Test
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Luis A. Afonso

Posts: 4,518
From: LIsbon (Portugal)
Registered: 2/16/05
No Tables and rigorously, Chi-square Goodness-of-fit Test
Posted: Oct 3, 2013 1:27 PM
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No Tables and rigorously, Chi-square Goodness-of-fit Test

It worth to emphasize that through the simulation of the interval contents we are able to calculate model rigorous p-values of the Chi-square Test, therefore avoiding both the rule that each interval must contain at least 5 (6?) observations and the use of the approximation the Chi.square Distribution does provide.

Suppose that 100 observations are distributed among the intervals as,
___14, 4, 11, 14, 14, 16, 21, 6_____
Uniformity as Null Hypotheses: Expected 12.5 by each interval.Test: 16.640, 7 df. P-value = 0.0199.


Method

By the statement INT(8*rnd)+1 the current value is directed to the memory 1, 2, ..., 8 and to the respective content a unit is added; performing 100 times the sample is simulated and the statistics Chi is calculated: if is larger than the sample one, 16.640 a significant value is found. Repeating a great number of times, say 40´000 the fraction of significant p-values is obtained, say 0.01961 (see below). I had performed 5 times this procedure.

Results

__0.01961 / 40´000 samples
__0.01983
__0.01976
__0.01956
__0.01980

It is shown that the experimental sample is no uniform at the ordinary 5% significance level.

Luis A. Afonso

REM "FREE"
CLS
COLOR 12
DEFDBL A-Z
DATA 14,4,11,14,14,16,21,6
FOR i = 1 TO 8: READ n(i)
s = s + n(i): NEXT i
E = s / 8
FOR i = 1 TO 8: chii = (n(i) - E) ^ 2 / E
chi = chi + chii: NEXT i
LOCATE 8, 50
PRINT USING " SAMPLE chi= ###.### "; chi
REM
REM
COLOR 7
LOCATE 11, 48: PRINT " p-value = "
RANDOMIZE TIMER
all = 40000
FOR rpt = 1 TO 5
FOR j = 1 TO all
LOCATE 10, 50: PRINT USING "########"; all - j
FOR kk = 1 TO 8: inn(kk) = 0: NEXT kk
REM
FOR nn = 1 TO 100
REM
kr = INT(8 * RND) + 1
inn(kr) = inn(kr) + 1
NEXT nn
REM
FOR i = 1 TO 8: c = (inn(i) - E) ^ 2 / E
chiR = chiR + c: NEXT i
IF chiR > chi THEN outt = outt + 1 / all
chiR = 0
NEXT j
LOCATE 11 + rpt, 50
PRINT USING "##.### "; outt
outt = 0
NEXT rpt
END



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