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Unformatted text preview: March 12th 1. Sample size estimation based on the large sample C.I. for p From the interval 2 2 ˆˆˆˆ (1 ) (1 ) ˆˆ , p p p p p Z p Z n n α α  ⋅ + ⋅ 2 ˆˆ(1 ) lengh of your 100(1 )% CI 2 p p L Z n α α = = × ⋅ ˆ , , L p α are given and we are interested in sample size n . Therefore, 2 2 2 2 2 2 2 2 2 1 1 4( ) 1 ˆˆ 4( ) (1 ) ( ) 2 2 Z Z p p Z n L L L α α α   = ≤ = (When 1 ˆ 2 p = , it has the maximum value.) Example. ˆ 0.02, 0.05, 0.54 ? L p n α = = = ⇒ = Example. ˆ 0.02, 0.05, 0.5 ? L p n α = = = ⇒ = 2. Sample size calculation for p based on the maximum error E . Definition. ˆ ( ) 1 P p p E α ≤ =  We want to estimate p within E with a probability of (1 ) α . Derive the formula for n 1 ˆ ( ) 1 ˆ ( ) 1 ˆ ˆ ( ) 1 and Z = ~ (0,1) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (1 ) (1 ) (1 ) (1 ) : ( ) 1 ˆ ˆ ˆ ˆ (1 ) (1 ) P p p E P E p p E E p p E p p P N p p p p p p p p n n n n E E Thus P Z p p p p n n α α α α ≤ =  ≤ ≤ =  ≤ ≤ =  ≤ ≤ =  & 2 ˆ ˆ (1 ) E c Z p p n α = = 2 2 2 2 2 2 ˆ ˆ ( ) (1 ) ( ) 4 Z p p Z n E E α α ∴ = ≤ ⋅ Recall we also derived n based on L the length of the 100(1 ) α % large sample confidence interval for p . Their relationship is 2 L E = ⋅ ˆ ( ) 1 ˆ ( ) 1 ˆˆ ( ) 1 ˆˆ ( ) 1 P p p E P E p p E P E p p E p P p E p p E α α α α ≤ =  ≤ ≤ =  ≤  ≤ =  ≤ ≤ + =  2 ∴ The 100(1 ) α % confidence interval for p is [ ] ˆˆ , p E p E + ....
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Zhu,W

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