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Unformatted text preview: Corrections and comments to discussion on 11/01/2011
1. Let X1, X2,…, Xn be a random sample from the distribution with p.d.f
() , We wish to test H0: θ ≥ 2 vs. H1: θ < 2.
a) If n=5, find a uniformly most powerful rejection region with the significance
level α=0.1 that is based on the statistic ∑
. That is, for which values of
∑
should H0 be rejected?
∏ () ∏
Since θ < 2 then Λ increases as ∑
most powerful rejection region is: ( )∑ decreases so we have that our uniformly ∑ ≥ The distribution of ∑
we have found a few times before and you can do for
example using the CDF method: () ( ) ( ) ( ) ( (
( ∑
( α
(∑ ( 
≥  )
) , x,θ )
) ,
(∑ (∑ ) ≥
≥  )
) ( ( )≥ ) θ = “λ” = 2
“θ” = 1/θ = 1/2 )
∑ ≥ b) Find the power of the test in part (a) if θ
( 
( ),
, ) (∑
(λ ≥  ) )
θ = “λ” = 1/2
“θ” = 1/θ = 2 8.11 No errors. A For problem A where we tested if the distribution was a Poisson(λ ) the test
statistic was wrong. I gave you 7.16189 and it should have been 4.53416. My
analysis of the results was based on the former value so technically I wrongfully
rejected the hypothesis. However I intended to design the problem so we would
reject to point out the fact that if we reject H0 there is no alternative hypothesis
that we look to. B No errors. ...
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This note was uploaded on 11/15/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.
 Fall '11
 STEPHANOV

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