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vladimir:
BUT if your data set is too small to work on statistical significance testing... and if you're not allowed to highlight single data points (if you don't want to be accused of being up to no good) then what can you do with small sets figures?
Then you don't use statistical arguments.

Just a simple question. How many coin tosses do you need to reject the hypothesis that a coin is unbiased with 99% confidence? 95%? 90%? And if your coin is used fewer times than that and then is lost, how are you going to use statistics to argue it was biased?

Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith

by Migeru (migeru at eurotrib dot com) on Fri Mar 20th, 2009 at 04:48:46 AM EST
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The ideal result should be heads 50% of the time and tails 50% of the time.

99% confidence means you are accepting a 1% error on the ideal.

95% means that you are accepting a 5% error on the ideal.

It would be hard to get within 1% or 5% of the ideal with very few tosses. If you do a few tosses you may actually conclude that the coin is biased even if it's not. I suspect that within 10 or 20 tosses you should seriously approach your ideal 50-50.

by vladimir on Fri Mar 20th, 2009 at 07:39:29 AM EST
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The right answer is that with 3 coin tosses, even if all 3 are Heads, the chance of that happening is 12.5% so you cannot reject the hypothesis that the coin is fair at 90% confidence.

With 4 coin tosses, HHHH has a probability of 6.25% which allows you to reject at 90% but not 95%.

With 5 coin tosses, HHHHH has a probability of 3.125% which allows you to reject at 95% but not at 99%.

The point is that, with less than 4 coin tosses you cannot show bias, no matter what. Sometimes you simply don't have enough data to argue statistically.

And statistics can only suggest where to look for actual evidence, it can't prove (or disprove) bias all by itself.

For instance, the contingency table analysis I did yesterday suggests looking for actual (not statistical) evidence of bias in the duration or the trials, not in the result. JakeS posted a theory that indictments were issued in the hopes of gathering sufficient evidence by the time the cases came to trial, which in some cases hasn't happened, resulting in prolongued imprisonments without trial rather than dismissals for lack of evidence. But a theory consistent with statistical suggestions is not evidence.

Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith

by Migeru (migeru at eurotrib dot com) on Fri Mar 20th, 2009 at 07:55:14 AM EST
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vladimir:
The ideal result should be heads 50% of the time and tails 50% of the time.
You urgently need to go read the first chapter of Feller's An Introduction to Probability Theory and its Applications which covers coin-tossing.

Most economists teach a theoretical framework that has been shown to be fundamentally useless. -- James K. Galbraith
by Migeru (migeru at eurotrib dot com) on Fri Mar 20th, 2009 at 07:57:04 AM EST
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It's only been about 20 years since my last statistics course.
by vladimir on Fri Mar 20th, 2009 at 08:06:54 AM EST
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