fi - Stat 5102 Final Exam May 15, 2003 Name Student ID The...

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Stat 5102 Final Exam May 15, 2003 Name Student ID The exam is open book and open handouts. You may also use three 8 1 2 × 11 sheets of paper with formulas, etc. You may use a calculator. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from. The points for the questions total to 200. There are 4 pages and 6 prob- lems. 1. [30 pts.] Suppose X is an Exponential( θ ) random variable where θ is an unknown parameter satisfying θ > 0. Suppose we wish to test the hypotheses H 0 : θ = 1 H 1 : θ > 1 The the sensible test statistic is just X , so we reject H 0 when X c for some critical value c . (a) Find the critical value c that gives a hypothesis test having signif- icance level α = 0 . 05. (b) Suppose we observe X = 0 . 45, what is the P -value of the test? 2. [30 pts.] Suppose X 1 , X 2 , ... , X n are independent and identically dis- tributed with distribution Normal(0 , 1 ), that is, normal with mean zero and variance 1 (as usual in Bayesian problems involving the nor- mal distribution, the so-called precision θ is a more sensible parameter than the variance), the prior distribution for θ is the Exp(1) distribu- tion, that is, exponential with rate parameter 1. Find the posterior distribution of θ given data
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fi - Stat 5102 Final Exam May 15, 2003 Name Student ID The...

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