You are given a (posterior) distribution for \(\pi \sim \beta_{(a, b)}\), a significance \(\alpha\), and \(H_{1}\). Test the following competing hypotheses, to decide between them, both by direct calculation and by Normal approximation.

(a) Posterior: \(\pi \sim \beta_{(35,24)}, \alpha=0.1\), and \(H_{1}: \pi>0.5\).

(b) Posterior: \(\pi \sim \beta_{(78,21)}, \alpha=0.05\), and \(H_{1}: \pi<0.85\).

(c) Prior: \(\pi \sim \beta_{(1,1)}\). Observations: 22 positive outcomes and 51 negative. \(\alpha=0.02\), and \(H_{1}: \pi>0.2\).