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# lec28 - Examples for CI of proportion Example 1 Suppose we...

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Examples for CI of proportion Example 1: Suppose we want to estimate the fraction of records in the 2000 IRS data base that have a taxable income over 35K. Question: We want to get a 98% confidence interval and wish to estimate the quantity to within 0.01. How many samples we need? The size of CI is actually 0 . 02 to satisfy the desire condition (W.L.O.G., we choose a conservative confidence interval for easy computation ): Using the second definition: recall that P ( | ˆ θ θ | < e ) 1 α is the second definition, we have 2 e 0 . 02 ⇐⇒ z α/ 2 2 n 0 . 01 ⇐⇒ n 2 . 33 2 · 0 . 01 = 116 . 5 so that n 13573 . n 0 . 25 ( z α/ 2 ) 2 , is the half of the desired size of confidence interval. 1

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Example 2: Suppose that we are interested in the large time probability p that a server is available. Doing 100 simulations has shown, that in 60 of them a server was available at time t = 1000 hrs. What is a 95% confidence interval for this probability? If 60 out of 100 simulations showed a free server, we can use ˆ p = 60 / 100 = 0 . 6 as an estimate for p , or, use the conservative one ˆ p = 0 . 5 . For a 95% confidence interval, z α/ 2 = z (1 - 0 . 95) / 2 = Φ - 1 (0 . 975) = 1 . 96 . The conservative confidence interval is: ˆ p ± z α/ 2 1 2 n = 0 . 6 ± 1 . 96 1 2 · 100 = 0 . 6 ± 0 . 098 . For the confidence interval using substitution we get: ˆ p ± z α/ 2 ˆ p (1 ˆ p ) n = 0 . 6 ± 1 . 96 0 . 6 · 0 . 4 100 = 0 . 6 ± 0 . 096 . 2
Related C.I. methods Target: We want to study the confidence interval for mean difference μ 1 μ 2 , or, the difference between two proportion, p 1 p 2 . ¯ X 1 ¯ X 2 and ˆ p 1 ˆ p 2 are the unbiased estimators for μ 1 μ 2 and p 1 p 2 Confidence intervals are summarized as: large n confidence interval for μ 1 μ 2 (based on independent ¯ X 1 and ¯ X 2 ) large n confidence interval for p 1 p 2 (based on independent ˆ p 1 and ˆ p 2 ) ¯ X 1 ¯ X 2 ± z α/ 2 s 2 1 n 1 + s 2 2 n 2 ˆ p 1 ˆ p 2 ± z α/ 2 2 1 n 1 + 1 n 2 ( conservative ) or ˆ p 1 ˆ p 2 ± z α/ 2 ˆ p 1 (1 -

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lec28 - Examples for CI of proportion Example 1 Suppose we...

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