If I flip a coin 10 times, and it comes up heads 7 of them, do you believe it's fair? What about 70 out of 100? 700 out of 1,000? When specifically do you start doubting. What about if it was a binary event other than a coin, say the sex ratio of newborn sea turtles one year?
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You take a limit
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If we test against 5% level, 7/10 passes with 17%, but the others are extremely unlikely with 0.004% and 9*10^-36% chance that it is biased - that is assuming binomial distribution...
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The sex of a sea turtle is determined after fertilization and it's actually based on the temperature of the developing egg. There will be serious doubt getting 700/1000 heads because coins are fairly close to binary but getting 700/1000 female sea turtles should alarm us honestly
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You could use Bayesian model selection to estimate which model is more likely: that which posits the coin flip probability is 0.5, and the other that allows any value between 0 and 1 (where the parameter is integrated out). Ofc this also comes down to how you pick model priors
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I think one of the interesting concepts here is how to come up with a reasonable prior for some binary process of arbitrary and perhaps unknown complexity. [almost] cylinder rotation is physically simpler than "all of vertebrate biology," so can a [uniform] prior be stronger?
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I think it's a fair coin even if it was 100% one way. Sea turtles though. I don't know what percent of start worrying something was up.
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Do we ever have to doubt? Imagine we didn't know what a coin was. If heads came up 70%, and many others got the same result under diff conditions, then that's what it is. We develop a theory based on that result. 1/3
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If you want a model for belief in H0 (coin is fair) after gathering evidence, use the density of a Beta distribution. If you want to know specifically when you have enough evidence to dispel belief in H0, use SPRT https://en.wikipedia.org/wiki/Sequential_probability_ratio_test …
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Strong law of large numbers! https://en.m.wikipedia.org/wiki/Law_of_large_numbers …
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