410Hw09 -...

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STAT 410 Homework #9 Spring 2009 (due Monday, July 27, by 4:00 p.m.) 1. 4.3.16 Hint: n n e t n t n t e z n t 6 2 1 3 2 + + + = for some z between 0 and n t . 2. 4.3.17 Hint: Use Theorem 4.3.9. 3. 4.3.19 4. 5.1.3 “Hint”: Table II ( p. 673 ) gives quantiles (percentiles) of χ 2 distribution.
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5. 5.4.13 6. a) 7.2.4 b) 7.2.7 7. a) 7.2.6 b) 7.2.8 8. Let X 1 , X 2 , … , X n be a random sample of size n from a N ( 0 , σ 2 ) distribution. a) Find the sufficient statistic Y for σ 2 . ( See 7.2.1 . ) b) Show that the maximum likelihood estimator for σ 2 is a function of Y . c) Is the maximum likelihood estimator for σ 2 unbiased?
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9. Let X 1 , X 2 , … , X n be a random sample of size n from a distribution with p.d.f. f ( x ; θ ) = θ x θ – 1 , 0 < x < 1, where θ > 0. a) Find the sufficient statistic Y for θ . b) Show that the maximum likelihood estimator for θ is a function of Y . _________________________________________________________________________
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Unformatted text preview: _________________________________________________________________________ If you are registered for 4 credit hours: ( to be handed in separately ) 10. Let X have a Binomial distribution with parameters n and p . Recall that ( 29 1 X p p n p n--has an approximate Standard Normal N ( 0, 1 ) distribution, provided that n is large enough, and ( 29 - <--<- 1 1 X P 2 2 z p p n p n z . Show that an approximate 100 ( 1 ) % confidence interval for p is ( 29 n z n z n p p z n z p 2 2 2 2 2 2 2 2 1 4 1 2 + +- + , where n p X = . This interval is called the Wilson interval. Note that for large n , this interval is approximately equal to ( 29 n p p z p 2 1- ....
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This note was uploaded on 10/29/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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410Hw09 -...

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