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# pracmid2 - X 1,X 2,X n following N μ,σ 2 with μ 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 ( μ,σ 2 ) with μ known but σ 2 unknown. In this case, ∑ n i =1 ( X i-μ ) 2 /σ 2 is a valid pivot and can be used to ﬁnd a conﬁdence interval for σ 2 . (2 × 5 = 10 points). (2) Consider the model Y i = β X i + ² i for i = 1 , 2 ,...,n . Here the X i ’s are ﬁxed constants and the ² ’s are i.i.d N (0 ,σ 2 ) random variables. (a) Are the Y i ’s independent and identically distributed in this case? Explain. (b) Find the M.L.E’s of ( β,σ ) ( 7 points). (3) Let X 1 and X 2 be two i.i.d. observations from an Exponential( λ ) distribution. Suppose X 1 = 1 . 8 and X 2 = 2 . 6. Find a conﬁdence interval for λ based on the above data. (6 points) Note: Tables of quantiles of standard distributions are available in the Appendix Section of Rice’s book. 1...
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