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Unformatted text preview: Confidence Interval for μ Interpreting Confidence Intervals Outline Confidence Interval for μ Interpreting Confidence Intervals 1 / 13 ISOM 2500 Lect 15: CI for μ 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 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 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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This note was uploaded on 12/20/2011 for the course ACCT/MGMT 2010 taught by Professor A during the Spring '11 term at HKUST.
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