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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 16: Confidence Intervals One Sample Mean 1 Say we have a population of interest and we want to determine what its mean is. = ? Population Introduction 2 We know the procedure is to generate a random sample X 1 X 2 ,..., X n and form the estimate X n X i i n = = 1 1 It would be grossly misleading to claim that is precisely equal to the observed X How Wrong Are We ? To detail our uncertainty about our estimate for , we can construct a confidence interval or interval estimate for . Example: what is the average weight of graduate students at Harvard ? point estimates interval estimates Gender Sample Mean Std. Err. DF L. Limit U. Limit 168.52577 2.663393 96 163.23898 173.81256 1 126.34821 3.4025612 55 119.52933 133.1671 3 The same idea works for proportions. What proportion of students are right handed ? Gender Count Total Sample Prop. Std. Err. L. Limit U. Limit 90 98 0.9183673 0.027658405 0.86415786 0.9725768 1 56 58 0.9655172 0.023958908 0.91855866 1.0124758 illy?) Confidence intervals are a vital aspect to statistics since an estimate is useless without some concept of its precisionthat is exactly what a confidence interval tells us; how good is our estimate . (silly?) 4 Point and Interval Estimates s A point estimate is a single number, s a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval 5 Critique go4ivy.com 6 This is what they send you 7 The General Estimation Process (mean, , is nknown) Population Random Sample Mean x = 50 I am 95% confident that is between 40 & 60. unknown) Sample 8 If X N then Z X N ~ ( , ) ~ ( , ) 2 01 = Review standardization rule 9 = = = ~ ( 0 , 1 ) , ( 1 1 ) 0 . 6 8 ( 1 . 9 6 1 . 9 6 ) 0 . 9 5 ( 3 3 ) 0 . 9 9 7 F o r Z N P Z P Z P Z We construct the interval estimate using the CLT. Recall that the CLT says that X N n ~ ( , ) 2 10 Then by the Standardization Rule : X n N / ~ ( , ) 0 1 We have X n N / ~ ( , ) 0 1 Then from what we know about the standard normal distribution: 11 P X n ( . / . ) < < = 196 196 95% 321 1 2 3 95% of the area is between 1.96 and 1.96 By doing some algebra , we may rearrange stuff so that We have P X n ( . / . ) < < = 196 196 95% 12 P X n X n ( . . ) % < < + = 196 196 95 truth lower bound upper bound ( 1.96 , 1.96 ) x x n n  + We are thus 95% confident that the true mean is in the interval hat does it mean that we are 95% confident ?...
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen

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