Determine the hypothesis test outcome about the mean \(\mu\) for a Gaussian process; significance \(\alpha\).

(a) \(H_{1}: \quad \mu eq 25 . \quad \alpha=0.05\). Statistics: \(n=27, S_{x}=715.333 . \quad \sigma_{0}=3.73\) (known).

(b) \(H_{1}: \mu>80 . \alpha=0.01\). We have 200 observations with \(\bar{x}=80.6741\). \(\sigma_{0}=5.1\) (known).

(c) \(H_{1}: \mu eq 25 . \alpha=0.02\). Data: \(\{28.1,42.1,22.7,38.8,28.8,37.0,20.4,37.3\}\). \(\sigma\) unknown.

(d) \(H_{1}: \mu<100 . \alpha=0.005\). Statistics: \(n=500, S_{x}=49650, S S_{x}=53803.8\). \(\sigma\) unknown.