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4. (2 points) The model {P}e is specified as follows: • → = {0, 1}. Po = N(0, 1). • P₁ = a continuous distribution whose PDF is 1½ exp (-12). Let X be a sample of size 1, and X are the following: ~ Po* for an unknown 0* = O. The hypotheses of interest Ho: 0* = 0, VS. H₁ 0* = 1. The following information might be helpful: Let & ~ N(0, 1). • • The quantile z0.975 is defined as the positive number that satisfies P(§ < 20.975) P(1)≈0.3374 > 0.3. = 0.975. Question: Please show that the following test is the uniformly most powerful test with a size of 0.05 T = 1 (|X|> 20.975).

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