I have problems to understanding the following discussion.
The questions are:
1)In "Some computation shows that this rule had probability 0.083,..." how $0.083$ calculated?(In a different version wrote $.074$)
2)Why "The stopping rule doesn’t affect the posterior distribution"? and i do not understand why it told ? So what?
3)In the discussion, a stopping rule changed, so likelihood inference changed , but Bayesian not. Is it really a better property? What does that mean? More generally how can i challenge this discussion? Can i find a stopping time rule such that Frequentist Inference has a good property but bayesian Inference does not(opposite of the discussion )? (If it is so finding a rule that bayesian approach has a good property but Frequentist Inference does not, means nothing).
For question (1)
$$P_{H_0}(Z_{30}>1.645 \ or \ Z_{20}>1.645)= P(Z_{30}>1.645)+\mathbb P(Z_{20}>1.645)-P(Z_{30}>1.645 \ and \ Z_{20}>1.645)= 0.1-P(Z_{30}>1.645 \ and \ Z_{20}>1.645)$$ $$P(Z_{30}>1.645 \ and \ Z_{20}>1.645)=P(\frac{\sum_{j=1}^{20}X_j +\sum_{j=21}^{30}X_j}{\sqrt{30}}>1.645 \ and \ Z_{20}>1.645)$$
$$=P(\sqrt{20} Z_{20} +\sum_{j=21}^{30}X_j>1.645\sqrt{30} \ and \ Z_{20}>1.645)$$ But I am stuck here.
Thanks in advance for any help you are able to provide.
Source:casi.pdf, 3.3 Flaws in Frequentist Inference (Page 31).
Source: Another version casi2.pdf
0.074
where your's says0.083
. The book title states "Corrected November 10, 2017." $\endgroup$