Lecture10 - March 12th 1. Sample size estimation based on...

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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 (1 ) (1 ) , p p p p p Z p Z n n --- + 2 (1 ) lengh of your 100(1 )% CI 2 p p L Z n - =- = , , L p are given and we are interested in sample size n . Therefore, 2 2 2 2 2 2 2 2 2 1 1 4( ) 1 4( ) (1 ) ( ) 2 2 Z Z p p Z n L L L - - = = (When 1 2 p = , it has the maximum value.) Example. 0.02, 0.05, 0.54 ? L p n = = = = Example. 0.02, 0.05, 0.5 ? L p n = = = = 2. Sample size calculation for p based on the maximum error E . Definition. ( ) 1 P p p E - = - We want to estimate p within E with a probability of (1 ) - . Derive the formula for n 1 ( ) 1 ( ) 1 ( ) 1 and Z = ~ (0,1) (1 ) (1 ) (1 ) (1 ) : ( ) 1 (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 - = -- - = ---- = ------ = --- & 2 (1 ) E c Z p p n = =- 2 2 2 2 2 2 ( ) (1 ) ( ) 4 Z p p Z n E E - = Recall we also derived n based on L- the length of the 100(1 ) - % large sample confidence interval for p . Their relationship is 2 L E = ( ) 1 ( ) 1 ( ) 1 ( ) 1 P p p E P E p p E P E p p E p P p E p p E - = -- - = --- - - = -- + = - 2 The 100(1 ) - % confidence interval for p is [ ] , p E p E- + ....
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Lecture10 - March 12th 1. Sample size estimation based on...

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