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Let $X$ be the number of tails when throwing a coin ten times. $X$ has the binomial distribution with parameters $n$ and $p$. My nullhypothesis is $$H_0:p\leq 1/2$$ and the alternative $$A:p>1/2.$$ I don't see why I can't test the hypothesis $$H_0':p>1/2$$and the alternative $$A':p\leq1/2$$ instead. Why must I test the first one?

  • Can the person who posted an answer and then deleted it, repost it please? I thought it was a nice answer –  Oct 04 '15 at 18:16
  • I was revising it, sorry about that. – Ian Oct 04 '15 at 18:19
  • @Ian Got pretty worried for a moment there! :D –  Oct 04 '15 at 18:20
  • Feel free to ask about my answer. Your question is partly mathematical but also partly philosophical and partly applied, so a "correct" answer to it is not really the end of the story. – Ian Oct 04 '15 at 18:26
  • @Ian would you mind taking a look at this question: http://math.stackexchange.com/questions/1464523/is-the-p-value-the-probability-that-your-null-hypothesis-is-true –  Oct 04 '15 at 21:50

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As you are probably aware, there are two types of errors in hypothesis testing. One could accept the null hypothesis incorrectly or reject it incorrectly. Ideally both of these would be unlikely, but the two are somewhat at odds, so our life is simpler if we prioritize one over the other. Statisticians take the conservative approach, which is to focus on reducing the probability of an incorrect rejection.

Thus, you want to ensure that the probability that you incorrectly reject the null hypothesis is small. So in your first scenario, you assume $p \leq 1/2$ and get a measurement $x$. Then the probability of seeing a measurement $y \geq x$ is some number $q(p,x)$. For $p \in [0,1/2]$, this number is largest when $p=1/2$. So your upper bound on the probability of an incorrect rejection is $q(1/2,x)$.

This example illustrates that our null hypothesis should contain the worst case scenario for whatever we are trying to justify. So it would be valid for a null hypothesis to be $p \geq 1/2$ or $p \leq 1/2$ but not $p<1/2$ or $p>1/2$.

Ian
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  • I understand why we should take $p\leq 1/2$, but it's still not exactly clear to me why not $p>1/2$ –  Oct 04 '15 at 18:33
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    @TonyStrong If you have "big" data, so you want to claim that your coins are biased towards tails, then the worst case scenario for your argument is that $p=1/2$. If it were any smaller then your argument would just be stronger. So your null hypothesis should allow for this, meaning that it should have $p \leq 1/2$ instead of $p<1/2$. – Ian Oct 04 '15 at 18:37
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    @TonyStrong On the other hand, if you have "small" data, so you want to claim they're biased towards heads, then the worst case is still $p=1/2$, and if it were any bigger your argument would just be stronger. So you get a null hypothesis of $p \geq 1/2$. The endpoint is included either way. – Ian Oct 04 '15 at 18:38