pracmid2 - X 1 ,X 2 ,...,X n following N ( , 2 ) with known...

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Midterm 2: Stat 426. Moulinath Banerjee University of Michigan December 3, 2004 Announcement: The total number of points is 23 but the maximum you can score is 20. (1) Decide on whether the following statements are TRUE or FALSE with a brief justification. (a) A confidence interval of level 1 - α for μ based on i.i.d. data X 1 ,X 2 ,...,X n following N ( μ,σ 2 ) will generally be larger than a confidence interval of level 1 - γ , based on the same data, if α > γ . (0 < α,γ < 1). (b) Constructing a confidence interval with confidence level 1 is the best thing to do, since we are then guaranteed to trap the true parameter value in our interval. (c) The MLE can always be obtained by differentiating the log likelihood function. (d) In a coin tossing experiment, the sample proportion of heads approaches the true underlying probability of heads as the number of tosses gets larger and larger (e) Consider i.i.d. data
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Unformatted text preview: X 1 ,X 2 ,...,X n following N ( , 2 ) with known but 2 unknown. In this case, n i =1 ( X i- ) 2 / 2 is a valid pivot and can be used to nd a condence interval for 2 . (2 5 = 10 points). (2) Consider the model Y i = X i + i for i = 1 , 2 ,...,n . Here the X i s are xed constants and the s are i.i.d N (0 , 2 ) random variables. (a) Are the Y i s independent and identically distributed in this case? Explain. (b) Find the M.L.Es of ( , ) ( 7 points). (3) Let X 1 and X 2 be two i.i.d. observations from an Exponential( ) distribution. Suppose X 1 = 1 . 8 and X 2 = 2 . 6. Find a condence interval for based on the above data. (6 points) Note: Tables of quantiles of standard distributions are available in the Appendix Section of Rices book. 1...
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This note was uploaded on 04/14/2010 for the course STATS 426 taught by Professor Moulib during the Winter '08 term at University of Michigan.

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