Recalling
- From: "\"Luis A. Afonso\"" <licas_@xxxxxxxxxxx>
- Date: Sun, 16 Jul 2006 05:36:13 EDT
It is especially traumatic to those (like me) expecting to learn something new here to find an absolutely futile and wrong discussion on a so well-known and indisputable thing that is the meaning of Null Hypotheses.
The main question is:
When we must not reject the null hypotheses about the statement of equality between:
_______parameter W of the Population A =
_______parameter W of B ?
It is not because they are EQUAL (i.e. represented by the same number) but simply because the information brought by the data (usually a random sample from each population) does not allow us to state (at the chosen confidence level) that the parameters are different.
Self quoting from a May post:
*** If the Hypotheses Tests pioneers (K. Pearson, R. A. Fisher) could imagine that somebody, sometime, somewhere, could have even ascribe a literal meaning to the symbolic *equality* H0: m1=m2 (or m1-m2 =0) they, perhaps, had the care to substitute = by , I guess, n.s.d. (i.e. not statistically distinguishable from available data). But, how they could foresee, ever, the existence of someone so stupid as Bob and fellows? They thought that it was impossible, I’m sure. ***
Accordingly the only fact in Statistical Inference is that the information we get from sample data is always deficient in view to obtain definitely exact conclusions about Populations. They are relative, not absolute, and depending of the sampling incertitude.
A little exercise illustrates very well how the conclusions drawn from the Hypotheses Testing could give erroneous information.
Let be the normal Populations X1=N(0.05, 1) and X2=N(0, 1). They have different means, however, in a lot of times, we could not reject H0: m1=m0. This is the well-known Type II error.
Samples of equal size were simulating from the Populations.
Table of results
(beta, the prob. to accept H0 wrongly)
___________beta
___n=5_____0.43
_____6_____0.27
_____7_____0.15
_____8_____0.08
_____9_____0.03
____10_____0.01
Therefore we can state that sample sizes greater than 10 *assures us* that the difference 0.05 in the two Population means is sufficiently large to not accept unduly H0
The data above concerns the case that I know EXACTLY that the Populations s.d. are 1.
If it is not the case (and we have to estimate the variances from the samples) we must have larger sample sizes to get the same betas.
________licas (Luis A. Afonso)
.
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