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# 410Hw11ans - STAT 410 Homework#11(due Friday November 13 by...

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STAT 410 Homework #11 Fall 2009 (due Friday, November 13, by 3:00 p.m.) 1. 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 α α α - < - - < - 1 1 X P 2 2 z p p n p n z . Show that an approximate 100 ( 1 – α ) % 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 ˆ ˆ ˆ α α α α + + - ± + , where n p X ˆ = . This interval is called the Wilson interval. Note that for large n , this interval is approximately equal to ( n p p z p 2 ˆ ˆ ˆ 1 - ± α . ( 29 2 1 X α z p p n p n < - - ( 29 ( 29 2 2 2 1 X α z p p n p n < - - X 2 2 n X p + n 2 p 2 < n 2 2 α z p n 2 2 α z p 2 p ˆ 2 2 p ˆ p + p 2 < n z 2 2 α p n z 2 2 α p 2 2 2 2 2 2 2 ˆ ˆ 2 1 p p n z p p n z + + - + α α < 0

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a p 2 + b p + c = 0 p 1, 2 = a c a b b 2 2 4 - ± - a p 2 + b p + c < 0 p 1 < p < p 2 ( a > 0 ) p 1, 2 = + + - + ± + 1 2 1 4 2 2 2 2 2 2 2 2 2 2 2 2 ˆ ˆ ˆ n z p n z n z p n z p α α α α = n z p n z n z p n z p 2 2 2 2 2 2 2 2 2 2 1 1 2 2 ˆ ˆ ˆ α α α α + + - + ± + = n z p n z p n z n z p p n z p 2 2 2 2 2 2 2 4 2 2 2 2 2 2 1 4 2 2 2 ˆ ˆ ˆ ˆ ˆ α α α α α + - - + + ± + = n z n z p n z n z p n z p 2 2 2 4 2 2 2 2 2 2 2 2 1 4 2 ˆ ˆ ˆ α α α α α + + - ± + = ( 29 n z n z n p p z n z p 2 2 2 2 2 2 2 2 1 4 1 2 ˆ ˆ ˆ α α α α + + - ± +
2. 5.1.3 “Hint”: Table II ( p. 673 ) gives quantiles (percentiles) of χ 2 distribution. a) E ( X ) = μ = - 0 θ 1 θ dx e x x = θ . b) M X 1 ( t ) = ( 1 – θ t ) – 1 , t < 1 / θ . X M ( t ) = X E t e = ( + + + n t n e 2 1 X ... X X E = n n t 1 X M = n n t - - 1 θ , t < n / θ . X has a Gamma distribution with α = n , β = θ / n . c) M Y ( t ) = X M ( 2 n t / θ ) = ( 29 n t - - 2 1 , t < 1 / 2 . Y has a χ 2 distribution with 2 n degrees of freedom. d) < < X 2 P θ d n c = 0.95 < < X 2 X 2 P θ c n d n = 0.95.

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410Hw11ans - STAT 410 Homework#11(due Friday November 13 by...

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