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final-soln.pdf - Statistics 24400 Autumn 2016 Final...

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Statistics 24400 - Autumn 2016 Final Examination Solution December 2 and 5, 2016 Name (print): On my honor, I will not discuss this exam with ANY PERSON before 15:30 December 5, 2016. Signature 1. Please print your name in the space provided. If you are taking this exam on the 2nd or the morning of the 5th, you must sign the “temporary nondisclosure” line and conduct yourself accordingly in order to get credit for this final exam. 2. Do not sit directly next to another student. 3. Do not turn the page until told to do so. 4. This is a closed book examination. You are allowed a single page of notes, written on both sides. Please write your name on your notes and turn it in with the exam. You are permitted to have a calculator. Devices capable of communication (laptops, tablets, phones) must be powered down. Tables of the cumulative Normal and χ 2 distributions are at the end of the exam. 5. Please provide the answers in the space and blank pages provided. If you do not have enough space, please use the back of a nearby page, clearly indicating the identity of the continued problem. 6. Be sure to show your calculations. In order to receive full credit for a problem, you must show your work and explain your reasoning. Good work can receive substantial partial credit even if the final answer is incorrect. 7. Read through the exam before answering any questions. Our scale of credit for questions may not correlate with the level of di ffi culty you experience—use your time wisely! Question Points Score Question 1 20 Question 2 30 Question 3 30 Question 4 20 TOTAL 100 1
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1. True or False? (20 pts) There is a “guessing penalty” penalty of four questions (8 pts). You’ll begin to accrue points with your fifth correct answer. (a) (2 pts) T F If random variables X and Y are uncorrelated (i.e. Cor( X, Y ) = 0), then X and Y must be independent. F (b) (2 pts) T F If P ( A ) < P ( B ) and P ( C ) > 0, then P ( A | C ) P ( B | C ). F (c) (2 pts) T F The calculation of p -value does not depend on the alternative hypothesis once we know the null hypothesis. T (d) (2 pts) T F The maximum likelihood estimator (MLE) is always unbiased. F (e) (2 pts) T F For any hypothesis testing procedure, it is always possible to increase the power = 1 - β while keeping the type 1 error the same. F (f) (2 pts) T F If X is a continuous random variable with cdf F ( X ), and Y is a random variable such that Y = F ( X ), then Y is distributed uniformly on [0 , 1]. T (g) (2 pts) T F Γ ( 3 2 ) < Γ ( 1 2 ) T (h) (2 pts) T F Fisher’s method of combination on a set of tests of the same hypothesis with p -values p 1 , p 2 , . . . p k means that the p -value of the combined tests is given by P = p 1 p 2 . . . p k . F (i) (2 pts) T F P defined as above. Then - 2 log P χ 2 2 k . T (j) (2 pts) T F Suppose that X i N ( μ, σ 2 ), with μ and σ unknown. Then the distribution of X 1 - μ does not depend on any unknown parameter. F (k) (2 pts) T F As above, suppose that X i N ( μ, σ 2 ) with μ and σ unknown, Then the distribution of X 1 - X 2 X 3 - X 4 does not depend on any unknown parameter. T (l) (2 pts) T F For a Poisson process, conditional on the number of events N (0 , 1] = n , the number
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