Hi all I am struggling with CT3-11 page 14. In the middle of the page, in the Core Reading part (i.e. bold writing), it says: "Therefore the inequalities can be inverted as follows: "X<= h1(theta) therefore theta >= theta1(X) "X>= h2(theta) therefore theta <= theta2(X)." Where are the functions theta1 and theta2 defined? Are they simply the inverses of h1 and h2 respectively? How are theta1 and theta2 used to follow on to the rest of the argument of this page, especially the Note near the end of the page? Many thanks
There is no general formula to invert, but yes it's like that. Personally I would ignore them and think as follows. If I observe X=x from a Bin(n,θ) and I want a 95% confidence interval then I find the θ that solves P(X<=x) = 2.5% and the θ that solves P(X>=x) = 2.5%. These two θ's give the confidence interval.
