You are given a probability distribution for \(\tau \sim \gamma_{(k, \lambda)}\), a significance \(\alpha\), and \(H_{1}\). Determine the hypothesis test, both by direct calculation and by using the Normal approximation.

(a) Posterior: \(\tau \sim \gamma_{(6,3)}, \alpha=0.04\), and \(H_{1}: \tau<4\).

(b) Posterior: \(\tau \sim \gamma_{(10,29.7)}, \alpha=0.05\), and \(H_{1}: \tau>0.2\).

(c) Posterior: \(\tau \sim \gamma_{(17.5,53.4)}, \alpha=0.1\), and \(H_{1}: \tau>0.5\).

(d) Posterior: \(\tau \sim \gamma_{(20 / 3, \sqrt{32})}, \alpha=0.001\), and \(H_{1}: \tau<3.2\).