Suppose Y is a random sample of size 1 from a population with density function f (y

Chapter 10, Problem 10.97

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Suppose Y is a random sample of size 1 from a population with density function f (y | ) = y1, 0 y 1, 0, elsewhere, where > 0. a Sketch the power function of the test with rejection region: Y > .5. b Based on the single observation Y , find a uniformly most powerful test of size for testing H0 : = 1 versus Ha : > 1. a Derive the likelihood function L() as a function of N1, N2, and N3. b Find the most powerful test for testing H0 : = 0 versus Ha : = a , where a > 0. Show that your test specifies that H0 be rejected for certain values of 2N1 + N2. c How do you determine the value of k so that the test has nominal level ? You need not do the actual computation. A clear description of how to determine k is adequate. d Is the test derived in parts (a)(c) uniformly most powerful for testing H0 : = 0 versus Ha : >0? Why or why not?