Confidence Intervals
 
  • if we can specify with a specific degree of certainty that a sample statistic falls within a given range around population value, then we know with same degree of certainty that population value falls within same range around sample statistic

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  • if 95% of sample means are w/in 2 SE of population mean, there is a 95% chance that population mean is within 2 SE of sample mean

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  • confidence interval (CI) = range of values computed from sample data that includes population value to a specified degree of certainty

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  • 95% CI = standard in most of social and behavioral sciences; standards vary in other sciences (99%, 99.9%, 99.99% CIs)

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  • margin of error = 2 SE

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    Some principles about CIs
     

  • increased sample size yields narrower CI

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  • smaller sample sd yields narrower CI for mean

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  • width of CI grows with increasing confidence level (from 90% to 95% to 99%)

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  • CI widths vary across samples, because sample statistic (e.g., sd, prop.) used as estimate of pop.  parameter, and sample statistic varies across samples

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    annual household electricity cost in 1% sample of 1990 CA census

    mean = $703.09  sd = 585.68                  n = 290,968

    95% CI = $700.91-705.27

    SE =
     
     

    CI for difference between two means

    1) compute difference between means

    2) compute SE for each group

    3) compute SE for difference between means by squaring SE for each group, and then taking the square root of the sum of these squared SEs

    4) use this resulting SE for difference between means to construct CI
     

    reported number of children 1996 GSS

    n:           men = 1277     women = 1612

    mean:    men = 1.68      women = 1.95

    sd:         men = 1.66    women = 1.69

    can the difference between men and women be explained by sampling error?

    difference in means = 0.27

    SE (men) = 1.66 / SQRT 1277 = .05

    SE (women) = 1.69 / SQRT 1612 = .04

    SE (difference in means) = SQRT (.052 + .052) = .06

    95% CI = 0.27 +/- 2(.06) = .15 - .39

    conclusion: women reported more children than men, and this cannot be accounted for by sampling error
     

    Confidence interval for a proportion

    95% CI = sample proportion +/- 2 SE of proportion

    2000 Presidential election results - national popular vote

    total votes received (11/17/00):

    Gore 49,921,267 (48.6%)
    Bush 49,658,276 (48.3%)

    total = 102,780,000

    national exit poll (n = 13,130)

    48.2% for Gore
    47.8% for Bush

    SE for Gore =                          SQRT((.482 x .518)/13130) = .004

    95% CI for Gore = 47.4% - 49.0%

    Gallup telephone poll during the 2 days before the election

    n = 2,350 likely voters across U.S.

    46% Gore
    48% Bush

    margin of error = 2%

    95% CI for Gore = 44%-48%
     

    proportion of CA men who have ever served in military (1% sample of CA 1990 Census)

    proportion = .304 of all adult men had ever been in the military

    n = 108,867

    SE = SQRT ((.304 x .696)/ 108867) = .001

    95% CI = .302-.306
     

    1997-8 survey by National Sleep Foundation of 1,027 persons in U.S.

    23% reported falling asleep at the wheel in the last year

    SE = SQRT ((.23 x .77) / 1027) = .013

    90% CI = +/- 1.64 SE = .209-.251

    95% CI = +/- 2 SE = .204-.256

    99% CI = +/- 2.56 SE = .197-.263
     
     

  • rules for constructing CIs vary for different statistics, but interpretation remains the same

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  • some % chance that the true population value will fall in the reported range
  • standard errors often shown in graphs as "error bars"

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  • inferential statistics, like CIs, generally only appropriate when applied to data from probability samples

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