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I don't understand what's wrong with this chain of logic.

  1. Null Hypothesis: A coin is fair
  2. Alternative Hypothesis: A coin is biased.

Suppose you flip a coin 900 times, and 450 times it comes up as heads. The probability of this is ~2%, so under a 5% significance level, we reject the null hypothesis.

More generally, I don't understand how hypothesis testing deals with events that are rare in general (e.g. getting exactly 450 heads). The null hypothesis seems to get rejected whenever the event is rare, even though the event can have maximum probability under the null hypothesis. In this case, the expected proportion of heads would be 0.5, which matches up with the null hypothesis.

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    $\begingroup$ With a $5\%$ significance level, you reject the null hypothesis if, assuming the null hypothesis is correct, the probability of seeing what you observed or something else as or more extreme is less than $5\%$. With a fair coin, every other potential result would have been more extreme so the probability would have been $100\%$, which is not less than $5\%$, so you would not reject the null hypothesis $\endgroup$
    – Henry
    Commented Aug 28, 2022 at 23:25

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Defining your hypothesis is one thing. Deciding to accept or reject your hypothesis is another thing. The decision is based on critical value. The critical value is based on Type1 error, Power and the assumed distribution under the null and alternate hypothesis. If your event is rare you have to use a different distribution and not normal.

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