I am trying to understand the level of a test better and was wondering if the level of a hypothesis is essentially equal to the probability that the null hypothesis is true. I have been trying to think of a counter example but haven't been able to confirm nor deny this association. Could anyone be kind enough to shed light on my understanding? Thank you!
I recently came across the paper . The author raised a few common misconceptions regarding hypothesis testing and p-values, about which I have two specific questions:   The p-value is technically $P({\rm observation}|H_{0})$, which, as pointed out by the paper, generally does not tell us anything about $P(H_{0}|{\rm observation})$, unless we happen to know the marginal distributions, which is rarely the case in "everyday" hypothesis testing. When we obtain a small p-value and "reject the null hypothesis," what exactly is the probabilistic statement that we are making, since we cannot say anything about $P(H_{0}|{\rm observation})$? The second question relates to a particular statement from page 6(652) of the paper:      Since the p-value, or range of p-values indicated by stars, is not set a priori, it is not the long-run probability of making a Type I error but is typically treated as such.   Can anyone help to explain what is meant by this statement?
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The code and image follow.  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{intersections}  \begin{document} \begin{tikzpicture}     \draw[name path=grid] [xstep=3,ystep=2] (0,0) grid (9,8);     \draw[-&gt;, name path=line] (2,1) -- (7,7);     \draw[name intersections={of=grid and line, sort by=line, name=i, total=\t}]         \foreach \s in {1,...,\t}{(i-\s) node {\s}}; \end{tikzpicture} \end{document}