I have spent time studying about the binomial test, and I have been stuck for a long time. I have a question concerning a null hypotethesis. If, for instance, we have a coin that is being thrown 1000 times, and you have noted there were 400 crown in total. Then, you are making a null hypothesis, which says
$H_0$: the coin is honest.
Let $X$ denote the stochastic variable which represents the number of crownns. My book says, "Under assumption of $H_0$, $X\sim bin(400,1/2)$..."
The question is: What does "under assumption of $H_0$" mean? Here are some possibilities, I thought it would mean
- You assume $H_0$ is true.
- You claim that $H_0$ holds, i.e. you claim the coin is honest, and you never know if it is true or false.
I believe it is the last one. If not, please explain the meaning of the quotation.
The statement means that if we assume the null hypothesis is true, then we can conclude that the distribution of $X$ is the given binomial distribution. Or in other words, the null hypothesis being true implies that $X$ has the given distribution.
The rest of the hypothesis test then checks whether this suggests an unlikely event has occurred (i.e. whether the p-value is small), and uses that to determine whether there is evidence we could use to reject the null hypothesis.
Compare it to a standard proof by contradiction:
Assume that a statement $P$ is true.
Demonstrate that this assumption leads to something impossible.
Conclude that therefore $P$ must be false.
This is similar, except that in step 2 we are trying to demonstrate that the assumption leads to something unlikely, so that in step 3 we can't definitively conclude that $P$ is false but it may push us towards it.