Why is type I error not affected by different sample size - hypothesis testing? I don't understand why the probability of getting a type I error when performing a hypothesis test, isn't affected. Increasing $n$ $\Rightarrow$ decreases standard deviation $\Rightarrow$ make the normal distribution spike more at the true $µ$, and the area for the critical boundary should be decreased, but why isn't that the case?
(Cross posted on Math Stack Exchange.)
 A: It seems that you're missing the main point that Type I error rate is also your criterion for cutoff. If your criterion for cutoff is not changing then alpha is not changing.
The $p$-value is the conditional probability of observing an effect as large or larger than the one you found if the null is true. If you select a cutoff $p$-value of 0.05 for deciding that the null is not true then that 0.05 probability that it was true turns into your Type I error.
As an aside, this highlights why you cannot take the same test and set a cutoff for $\beta$. $\beta$ can only exist if the null was not true whereas the test value calculated assumes it is.
Frank Harrell's point is excellent that it depends on your philosophy. Nevertheless, even under frequentist statistics you can choose a lower criterion in advance and thereby change the rate of Type I error.
A: If you're using standard hypothesis testing, then you are setting the confidence level $\alpha$ then comparing the test p-value to it. In this case the sample size will not impact the probability of type I error because your confidence level $\alpha$ is the probability of type I error, pretty much by defintition. In other words, you set the probability of Type I error by choosing the confidence level.
The probability of type I error is only impacted by your choice of the confidence level and nothing else. 
A: This is a question that is not asked often enough.  In frequentist statistics we tend to fix $\alpha$ by convention.  Then as $n\rightarrow\infty$ the type II error $\rightarrow 0$ (i.e., power $\rightarrow 1$) even though we also have the luxury for large $n$ of not allowing so many false positives had we chosen differently.  The result of this convention is that when $n$ is "large", one can detect trivial differences, and when there are many hypotheses there is a multiplicity problem.  By contrast, the likelihood school of inference tends to deal with the total of type I and type II errors, and lets type I error $\rightarrow 0$ as $n \rightarrow\infty$. This solves many of the problems of the frequentist paradigm.  Ironically, the frequentist performance characteristics of the likelihood method are also quite good.
See for example http://people.musc.edu/~elg26/SCT2011/SCT2011.Blume.pdf and http://onlinelibrary.wiley.com/doi/10.1002/sim.1216/abstract .
