So I do not understand one part of the answer to the following question, here it goes from the beginning :

A coin is thrown $100$ times and it is counted the number of times heads and tails come up. Check the assumption that the probability of the appearance of heads and tails are equal if in those 100 throws, 45 have been tails. Work with the "theshold of sustainability(im not sure if im translating this notion correctly) " of $\alpha=0.05$. Also, find the smallest number of tail appearances from which there is no need to throw away the assumption that the coin is homogeneous.

Let $S_{100}$ be the number of tail appearances, so $S_{100}:\mathcal B(100,p)$ $p$-unknown probability of appearance of tails. Lets take hypothesis $H_0$ for the "equality" of appearance of tails and heads and $H_1$ otherwise. Then it is natural to take the critical area of $C=\{(x_1,...,x_{100}):|S_{100}-50|>c\}$

$0.95=P_{\frac{1}{2}}\{|S_{100}-50|>c\}$ Why is it like this? Should it not be $0.05$ instead of $0.95$?? Because I had an earlier definition as $P_{H_0}\{(x_1,...,x_{100})\in C\}=\alpha.$ What am I not seeing,understanding , is this a mistake?



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