Let X1,..., Xn be a random sample from a n(θ, Ï2) population. Consider testing H0: θ θ0.

Let X1,..., Xn be a random sample from a n(θ, σ2) population. Consider testing
H0: θ θ0.
(a) If σ2 is known, show that the test that rejects Ho when

is a test of size a. Show that the test can be derived as an LRT.
(b) Show that the test in part (a) is a UMP test.
(c) If σ2 is unknown, show that the test that rejects H0 when

is a test of size α. Show that the test can be derived as an LRT.

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