For a more precise estimation a small margin of error

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Unformatted text preview: ily, CLT comes to rescue. For large enough n, x is approximately normally ¯ distributed, so we can build confidence intervals. CI for µ, Case 3: Any Population, σ unknown If population has unknown distribution and sample standard deviation is s , an approximate 100(1 − α)% confidence interval for µ can be constructed via: √ x ± zα/2 s / n ¯ only if n > 30 Utku Suleymanoglu (UMich) Interval Estimation 14 / 17 Interval Estimation of Population Mean Sample Size and Desired Margin of Error As we have seen, margin of error depends on sample size in all the cases. For a more precise estimation, a small margin of error is desired. Suppose you get to design your study and you want to keep the margin of error low. Say, at 0.5. You can do that, if you can collect enough data. Think about the case where σ is known. 2 zα/2 σ 2 σ ME = zα/2 √ → n = ME 2 n So for example, if σ = 2, α = 0.05, so that zα/2 = 1.96, to ensure ME = 0.5, we need at least 1.962 22 = 61.46 0.52 62 observations, (just to be on the safe side-with 62, ME will be slightly smaller than 0.5). n= Utku Suleymanoglu (UMich) Interval Estimation 15 / 17 Interval Estimation for Proportions Confidence Invervals for p Just like µ, you can build confidence intervals for any population parameters, as long as you know its sampling distribution. In the previous chapter, we saw that we can estimate proportions and the estimator has an approximate normal distribution. CI for p If p is the estimate for population proportion of p , we can build a confidence interval ¯ estimate p as: p ± zα/2 ¯ p (1 − p ) ¯ ¯ n Same spirit as before: example in the section. Utku Suleymanoglu (UMich) Interval Estimation 16 / 17 Interval Estimation for Proportions Additional Note for The Entire Chapter For practice: Think about what it means to have a narrower or a wider CI using the correct interpretation of CIs. Think about what happens to CIs when one these increase while keeping others constant, do they get narrower or wider? Think about the intiution, too. α n σ or s p (1 − p ) ¯ ¯ Utku Suleymanoglu (UMich) Interval Estimation 17 / 17...
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This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

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