Chapter 21. Recall from previous chapters: Statistical Thinking. Chapter What Is a Confidence Interval? Review: empirical rule

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1 Chapter 21 What Is a Cofidece Iterval? Chapter 21 1 Review: empirical rule Chapter 21 5 Recall from previous chapters: Parameter fixed, ukow umber that describes the populatio Statistic kow value calculated from a sample a statistic is used to estimate a parameter Samplig Variability differet samples from the same populatio may yield differet values of the sample statistic estimates from samples will be closer to the true values i the populatio if the samples are larger Chapter 21 6 Chapter 21 1

2 Recall from previous chapters: Samplig Distributio tells what values a statistic takes ad how ofte it takes those values i repeated samplig. sample proportios ( p ˆ s) from repeated samplig would have a ormal distributio with a mea equal to the populatio mea ad a certai stadard deviatio. sample meas ( x s) from repeated samplig have a ormal distributio with a mea equal to the populatio mea ad a certai stadard deviatio. Chapter % of all sample meas will be withi roughly 2 stadard deviatios (2*σ/ ) of the populatio parameter µ. " Because distaces are symmetrical, this implies that the populatio parameter µ must be withi roughly 2 stadard deviatios from the sample average x, i 95% of all samples. Red dot: mea value This reasoig is the essece of statistical iferece. of idividual sample Chapter Reworded " With 95% cofidece, we ca say that µ should be withi roughly 2 stadard deviatios (2*σ/ ) from our sample mea x bar. I 95% of all possible samples of this size, µ will ideed fall i our cofidece iterval. I oly 5% of samples would x be farther from µ. Chapter Chapter 21 2

3 The Rule for Sample Proportios If umerous simple radom samples of size are take from the same populatio, the sample proportios p ˆ from the various samples will have a approximately ormal distributio. The mea of the sample proportios will be p (the true populatio proportio). The stadard deviatio will be: p(1" p) Chapter Rule Coditios For a rule to be valid, it must have Radom sample Large sample size Chapter Derivatio of the Margi of Error Two formulas for a 95% cofidece iterval: sample proportio ±1/ (from Chapter 4) sample proportio ± 2(SD) Two formulas are equivalet whe the proportio used i the formula for SD is p=0.50. The 2(SD) is simply 1/ 2 p ˆ (1 " p ˆ ) 2 Chapter (1 " 0.5) = 1 Chapter 21 3

4 Cofidece Itervals from the Media Formula for a 95% cofidece iterval: sample proportio ± margi of error Chapter Figerprits A Questio Assume that the proportio of all me who have left asymmetry is 15%. Is it uusual to observe a sample of 66 me with a sample proportio ( p ˆ ) of 30% if the true populatio proportio (p) is 15%? Chapter Case Study: Figerprits Samplig Distributio p = 0.15 (= mea); = 66 p(1" p) 0.15(1" 0.15) = 66 = (= s.d.) Chapter Chapter 21 4

5 Case Study: Figerprits Aswer to Questio Where should about 95% of the sample proportios lie? mea plus or mius two stadard deviatios (0.044) = (0.044) = % should fall betwee [0.062, 0.238] Chapter Formula for a 95% Cofidece Iterval for the Populatio Proportio (Empirical Rule) ˆ p ± 2 p ˆ (1" p ˆ ) stadard error (estimated stadard deviatio of p ˆ ) Chapter Iferece for Populatio Meas Samplig Distributio, Cofidece Itervals makig coclusios about populatio meas rather tha populatio proportios icludes the rule for the samplig distributio of sample meas ( X ) icludes cofidece itervals for oe mea or a differece i two meas Chapter Chapter 21 5

6 The Rule for Sample Meas If umerous simple radom samples of size are take from the same populatio, the sample meas (X) from the various samples will have a approximately ormal distributio. The mea of the sample meas will be µ (the populatio mea). The stadard deviatio will be: " We do ot kow the value of σ! Chapter Stadard Error of the (Sample) Mea SEM = stadard error of the mea = s Chapter Results Exercise ad Pulse Rates mea std. dev. std. err. Exercisers Noexer Typical deviatio of a idividual pulse rate (for Exercisers) is s = 8.6 Typical deviatio of a mea pulse rate (for Exercisers) is s = 8.6 = Chapter Chapter 21 6

7 Cofidece Itervals Exercise ad Pulse Rates 95% C.I. for the populatio mea: sample mea ± 2 (stadard error) x ± 2" s Exercisers: 66 ± 2(1.6) = 66 ± 3.2 = (62.8, 69.2) Noexercisers: 75 ± 2(1.6) = 75 ± 3.2 = (71.8, 78.2) Do you thik the populatio meas are differet? Yes, because the itervals do ot overlap Chapter Cofidece Itervals: differece betwee meas Exercise ad Pulse Rates 95% C.I. for the differece i populatio meas (oexercisers mius exercisers): ± 2 (SE of the differece) X N " X E Differece i sample meas: = 9 SE of the differece = 2.26 (give) 95% cofidece iterval: (4.4, 13.6) iterval does ot iclude zero ( meas are differet) Chapter Careful Iterpretatio of a Cofidece Iterval We are 95% cofidet that the mea restig pulse rate for the populatio of all exercisers is betwee 62.8 ad 69.2 bpm. (We feel that plausible values for the populatio of exercisers mea restig pulse rate are betwee 62.8 ad 69.2.) ** This does ot mea that 95% of all people who exercise regularly will have restig pulse rates betwee 62.8 ad 69.2 bpm. ** Statistically: 95% of all samples of size 29 from the populatio of exercisers should yield a sample mea withi two stadard errors of the populatio mea; i.e., i repeated samples, 95% of the C.I.s should cotai the true populatio mea. Chapter Chapter 21 7

8 Uderstadig Cofidece Level For a cofidece level of 95%, we expect that about 95% of all such itervals will actually cover the true populatio value. The remaiig 5% will ot. Cofidece is i the procedure over the log ru. Chapter Levels of Cofidece 95% cofidece iterval: sample proportio ± 2(SD) I 95% of all samples, the true proportio will fall withi plus/mius 2 stadard deviatios of the sample proportio. (exact multipier= 1.96) 90% cofidece iterval: sample proportio ± 1.645(SD) 99% cofidece iterval: sample proportio ± 2.576(SD) More cofidece Wider Iterval Chapter Factors affectig the width of the cof iterval: The sample size, the larger is, the smaller the CI The stadard deviatio, s the smaller s is, the smaller the CI The cofidece level the smaller the level of cofidece, the smaller the CI Chapter Chapter 21 8

9 Which of the followig would cause a decrease i the width of a cofidece iterval? a) A icrease i b) A decrease i the SE c) A decrease i the cofidece level d) All of the above Chapter Summary of the Variety of Iformatio Give i Jourals Ca determie CIs for idividual meas or differece i two meas if you have: Direct cofidece itervals Meas ad stadard errors of the meas Meas, stadard deviatios, ad sample sizes Chapter Perceived racial/ethic harassmet ad tobacco use amog Africa America youg adults. We examied the associatio betwee perceived racial/ethic harassmet ad tobacco use i 2129 Africa America college studets i North Carolia. Age-adjusted ad multivariate aalyses evaluated the effect of harassmet o daily ad less-tha-daily tobacco use. Harassed participats were twice as likely to use tobacco daily (odds ratio = 2.01; 95% cofidece iterval=1.94, 2.08) compared with those with o reported harassmet experieces. Experieces of racial/ethic harassmet may cotribute to tobacco use behaviors amog some Africa America youg adults. Am J Public Health Feb;95(2): Chapter Chapter 21 9