Week 10 Wed Nov 3

# Week 10 Wed Nov 3 - Posted before class WEEK 10 Wed Nov 3...

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1 Posted before class WEEK 10 Wed, Nov 3 Correct Formulas for Sample-Size Determination The equations that should be solved when using simple random sampling from a population of size N are n N n N z B 1 2 / (*) n p p N n N z B ) 1 ( 1 2 / (**) with results 1 / / 2 2 2 2 / 2 2 2 2 / B z N B N z n c (6-10c) 1 / ) 1 ( / ) 1 ( 2 2 2 / 2 2 2 / B p p z N B N p p z n c (6-11c) where we have chosen to label the correct sample size as n c . There is a better way to state these results : To evaluate (6-10c) or (6-11c), first calculate n b using the book’s (6- 10) or (6-11) and then plug the result into (***). N n n n b b c ) 1 ( 1 . (***)

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2 Example . CI estimation of μ. Suppose the population size is N = 854, σ is unknown, desired margin of error is B = 2, and desired confidence level is 99%. Suppose σ = 10 is used for planning the sample size . We use (6-10c). We find n b = 165.89 and then by (***) we find n c = 139.1. This suggests that the sample size should be n = 140 for a simple random sample selected from the population of size N = 854.. Hypothetical Implementation. Take n = 140 and use the t-interval estimate of μ with 139 df once the data are in. Using the book’s Table 3 and the conservative approach to finding the t-multiplier t .005 , we would use the multiplier t .005 = 2.617 (120 df ). With some hypothetical data x = 76.4, s = 10.3, the t-interval estimate for the population mean μ is ) 140 / 3 . 10 ( 1 854 140 854 617 . 2 4 . 76 / 1 005 . n s N n N t x which evaluates to 1 . 2 4 . 76 . We can present the interval estimate as 74.3 < μ < 78.5 with 99% confidence. Note that the realized margin of error 2.1 is a little larger than the planned B = 2. Discuss. Example . CI estimation of p. Population size N = 253, desired margin of error B = .04, 95% confidence level, p = .5 for planning purposes. Then n b = 600.25 and n c = 178.2. Take n = 179.
3 Insurance Company (IC) Example (a real case) . The IC conducted an audit of N = 4,492 billings from a medical device supplier that it had received and paid over a period of time. The IC used a simple random sample of size n = 188 billings and audited the sample billings. Pages IC-1 and IC-2 to follow provide the information IC provided to attorneys when asked how it came to the sample size n = 188.

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