Author: Uri Blass
Date: 15:58:36 02/03/01
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On February 03, 2001 at 17:12:10, Walter Koroljow wrote: >On February 03, 2001 at 12:16:20, Uri Blass wrote: > >>On February 03, 2001 at 11:09:02, Walter Koroljow wrote: >> ><snip> > >I was glad to see your reply. We agree on many things. > >>>Let We = win expectancy of player A. Then the simplest hypotheses >>>to use are: >>> >>>H0: We >= 0.5 >>>H1: We < 0.5. >>> >>>If we reject H0, we necessarily accept H1. B is the better player at the >>>confidence level of our test. >> >>I know statistics but the probability of 97% (when there is a 97% confidence) is >>the probability to accept H0 when We=0.5 >> >>3% is the probability for error when We=0.5. >>If We>0.5 the probability for error is even smaller so 3% is an upper bound for >>the probability of an error but it is an upper bound when H0 is correct and this >>model does not happen because often H1 is correct. > >Absolutely right, except for one thing -- when H1 is correct, it is correct to >reject H0 (with H0 and H1 defined as they are above). In fact, if we knew that >H1 occurred 50% of the time, we could say the error rate bound on false >acceptance of H1 (false rejection of H0) was 1.5%, since H1 is automatically >true half the time. > >> >>I think the interesting question is what is the probability that H1 is correct >>after knowing the result and the level of confidence does not give an answer for >>it. > >Yes, this is true. But a bound is a useful result. I think Amir's calculation >is arithmetically correct and very useful. He will not lose much money doing >such calculations! > >> >>You can say that you do not do the error of rejecting H0 when H0 is correct in >>at least 97% of the cases. > >True, but I would say (see the argument above) that we do not need the condition >that H0 is correct to make that statement. > >> >>If you reject H0 often then it means that you are right in rejecting H0 in most >>of the cases that you decide to reject H0. >> >>For example if you reject H0 in 50% of the cases that you test hypothesis then >>you can say that the probability that H0 is wrong when you reject H0 is at least >>94%. >> >>If you do not reject H0 often and Reject it only in 3% of the cases then it is >>possible that you always make the wrong decision when you decide that H0 is >>right when always We=0.5 <snipped> >I could not understand the last two paragraphs. How can it be a wrong decision >to accept H0 when We = 0.5? I suspect there is a language problem here. You are right. I should say you decide that H0 is wrong instead of you decide that H0 is right. If always We=0.5 you make errors everytime that you reject H0. I could say to make it clear: Assume level of confidence 97% in all of your tests. If you reject H0 in 3% of the times then it is possible that you are always wrong when you reject H0(for example when always We=0.5). If you reject H0 in 50% of the times then you are wrong only in at most 6% of the cases that you reject H0(to be more exact I need to say if the probability to reject H0 is 50% but the probability is something that you do not know and the % of the cases that you rejected H0 practically is something that you know). Uri
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