You are comparing two \(\beta\) distributed variables \(\psi \sim \beta_{(k, l)}\) and \(\pi \sim \beta_{(m, n)}\) to determine the hypothesis \(H_{1}\) (which is either \(\psi>\pi\), or \(\psi<\pi\) ) with significance \(\alpha\). Do this both by exact calculation using Rule A.3.3, and by Normal approximation. (The exact calculation requires good calculation tools.)

(a) \(\psi \sim \beta_{(2,5)}\) and \(\pi \sim \beta_{(4,3)}, \alpha=0.1, H_{1}: \psi<\pi\).

(b) \(\psi \sim \beta_{(23,17)}\) and \(\pi \sim \beta_{(17,23)}, \alpha=0.1, H_{1}: \psi>\pi\).

(c) \(\psi \sim \beta_{(20,20)}\) and \(\pi \sim \beta_{(17,23)}, \alpha=0.05, H_{1}: \psi>\pi\).

(d) Let the parameters of the Problem 16.c be 10 times as large. This corresponds to ten times as many positive observations and ten times as many negative observations. Do you think there should be a difference? Think about it for a while, and then calculate to see whether it matters or not!