Suppose that a random sample of size \(n\) is drawn from a normal population distribution for which \(\mu\) is assumed to be known and equal to the given value \(\mu_{*}\). Define UMPU level \(\alpha\) tests of the following null hypotheses:

(a) \(H_{0}: \sigma^{2} \leq \sigma_{0}{ }^{2}\) versus \(H_{a}: \sigma^{2}>\sigma_{0}{ }^{2}\)

(b) \(H_{0}: \sigma^{2} \geq \sigma_{0}{ }^{2}\) versus \(H_{a}: \sigma^{2}<\sigma_{0}{ }^{2}\)

(c) \(H_{0}: \sigma^{2}=\sigma_{0}{ }^{2}\) versus \(H_{a}: \sigma^{2} eq \sigma_{0}{ }^{2}\)