lect15 - Condence Interval for Interpreting Condence...

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Confidence Interval for μ Interpreting Confidence Intervals Outline Confidence Interval for μ Interpreting Confidence Intervals 1 / 13 ISOM 2500 Lect 15: CI for μ
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Confidence Interval for μ Interpreting Confidence Intervals Confidence Interval for μ The Central Limit Theorem (Lect 14 page 5) tells us that for a large population, when the sample size is large, ¯ x will be within 1 . 96 σ n of μ about 95% of the time. i.e., about 95% of all the ¯ x are such that ¯ x - 1 . 96 σ n μ ¯ x + 1 . 96 σ n , This expression, however, is not useful as given because in virtually every problem, σ is unknown. 2 / 13 ISOM 2500 Lect 15: CI for μ
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Confidence Interval for μ Interpreting Confidence Intervals Confidence Interval for μ , ctd Since s estimates σ , we can use s / n to estimate σ/ n . An estimate of the standard deviation of the sampling distribution of a sample statistic is called the standard error (se) of the statistic Standard errors are often reported with the statistic to indicate its precision s / n is the standard error of ¯ x p ˆ p ( 1 - ˆ p ) / n is the standard error of ˆ p Note again that we are estimating the amount of sampling variation of ¯ x from the information in just the one observed sample x 1 , x 2 , · · · , x n . An exact confidence interval needs a second adjustment that accounts for the use of s in place of σ ; for which we’ll use the Student’s t Distribution. 3 / 13 ISOM 2500 Lect 15: CI for μ
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Confidence Interval for μ Interpreting Confidence Intervals The Student’s t Distribution Provided that the population is normally distributed, let ¯ X and S be the mean and SD of a random sample of size n .
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