1. Show that the likelihood ratio principle leads to the same test when testing a simple…

1. Show that the likelihood ratio principle leads to the same test when testing a simple hypothesis H0 against an alternative simple hypothesis H1, as that given by the Neyman–Pearson theorem. Note that there are only two points in Ω

2. Let X1, X2,…,Xn be a random sample from the normal distribution N(θ, 1). Show that the likelihood ratio principle for testing H0 : θ = θ , where θ is specified, against H1 : θ = θ leads to the inequality |x − θ | ≥ c

(a) Is this a uniformly most powerful test of H0 against H1?

(b) Is this a uniformly most powerful unbiased test of H0 against H1?

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