The sampling distribution of the sample mean Xˉ is the distribution of the sample mean for all possible samples of size n from the population distribution of X.
- For any sample size n:
ar{X} ext{ follows } Nigg( rac{ar{oldsymbol{ ext{X}}}igg), rac{oldsymbol{ ext{σ}}}{ ext{√}n}igg)
- If XextfollowsN(extμ,extσ), then XˉextfollowsN(extμ,racextσext√n)</p></li></ul><h4id="aa8c71fb−92cd−454a−8c79−d9cf66e4a750"data−toc−id="aa8c71fb−92cd−454a−8c79−d9cf66e4a750"collapsed="false"seolevelmigrated="true">CentralLimitTheorem</h4><ul><li><p>TheCentralLimitTheoremstates:IfXisanyrandomvariablewithafinitemeanandvariance,thenasthesamplesizenincreases(wheren ext{ is typically } ext{≥ } 30),thedistributionofthesamplemeanar{X}approachesanormaldistribution.<br> −Irrespectiveoftheoriginalpopulationdistribution,thesamplingdistributionofthemeanwillbeapproximatelynormalforlargesamplesizes.<br> −Thus:<br> ar{X} ext{ follows } Nigg(oldsymbol{ ext{μ}}, rac{oldsymbol{ ext{σ}}}{ ext{√}n}igg)</p></li></ul><h4id="a8f71536−eb67−4750−bd4e−3e3a995cccd1"data−toc−id="a8f71536−eb67−4750−bd4e−3e3a995cccd1"collapsed="false"seolevelmigrated="true">StatisticalInference</h4><ul><li><p>Statisticalinferencereferstomethodsusedtodrawconclusionsaboutapopulationbasedonsampledata.<br> −Itallowsformakingpredictionsorgeneralizationsfromthesampledatatothebroaderpopulationofinterest.</p></li></ul><h5id="5dce9ee0−4682−44fc−8b08−6d9eec6e668b"data−toc−id="5dce9ee0−4682−44fc−8b08−6d9eec6e668b"collapsed="false"seolevelmigrated="true">QuantifyingUncertainty</h5><ul><li><p>Acknowledgesthatsampledatamaynotperfectlyrepresentthepopulation.</p></li><li><p>Usestheprinciplesofprobabilitytoquantifythisuncertainty:<br> −Dependencyontheprobabilityframeworkfacilitatesunderstandinglong−termpredictablebehaviorsinsample−basedconclusions.<br> −Goodsamples(e.g.,SimpleRandomSamples,SRS)helptoensurehigherconfidenceinconclusionsdrawn.</p></li></ul><h4id="2bd02a11−1526−43dc−a02d−73f67d0af3dd"data−toc−id="2bd02a11−1526−43dc−a02d−73f67d0af3dd"collapsed="false"seolevelmigrated="true">EvaluatingEstimatorPerformance</h4><ul><li><p>Thesamplemeanar{X}isusedasanestimatorforthepopulationmeanoldsymbol{ ext{μ}}.<br> −However,theprobabilitythatar{X} = oldsymbol{ ext{μ}}istheoreticallyzeroduetocontinuity.<br> −Reportingonlythesamplemeandoesnotprovideanyinformationregardingtheestimatedmean’saccuracy.</p></li></ul><h4id="e61bc487−a4a5−42df−8737−b2f167c97086"data−toc−id="e61bc487−a4a5−42df−8737−b2f167c97086"collapsed="false"seolevelmigrated="true">ConfidenceIntervals</h4><ul><li><p>Theaimistoconstructanintervalaroundthesamplemeanthatlikelycontainsthepopulationmeanoldsymbol{ ext{μ}}.<br> −Acknowledgesthatthesamplemeanvariesfromsampletosample,necessitatingarangeofvaluesinsteadofasingleestimate.</p></li><li><p>Awell−constructedconfidenceintervalgivesusahighdegreeofconfidence(typicallyabout95ar{X}.</p></li></ul><h5id="580dfc7b−2002−4daf−98c5−8bbadceaab7b"data−toc−id="580dfc7b−2002−4daf−98c5−8bbadceaab7b"collapsed="false"seolevelmigrated="true">95ar{X} ext{ } ± 2 rac{oldsymbol{ ext{σ}}}{ ext{√}n}</p></li><li><p>Thisfollowsfromthe68−95−99.7rule,whichindicatesthatforanormaldistribution:<br> −About95oldsymbol{ ext{σ}} = 0.40.</p></li><li><p>Calculatingfromasampleof25graduateswherethesamplemeanar{X} = 3.31:<br> −Thecalculationforthe95 ext{Margin of Error} = 2 rac{0.40}{ ext{√}25} = 0.16<br> −Therefore,theconfidenceintervalis:<br> −ar{X} ± 0.16 = (3.31 - 0.16, 3.31 + 0.16) = (3.15, 3.47)</p></li></ul><h5id="49b76c0e−d9c0−4f6c−9558−7e4777b12f3f"data−toc−id="49b76c0e−d9c0−4f6c−9558−7e4777b12f3f"collapsed="false"seolevelmigrated="true">InterpretationofConfidenceIntervals</h5><ul><li><p>A95oldsymbol{ ext{μ}}.</p></li><li><p>Confidenceintervalstaketheform:<br> − ext{estimate} ± ext{margin of error}<br> −Fora95ar{X}andthemarginoferroris2 rac{oldsymbol{ ext{σ}}}{ ext{√}n}</p></li></ul><h5id="e50d9b1d−b4ba−47df−93ba−9abb3c4315e4"data−toc−id="e50d9b1d−b4ba−47df−93ba−9abb3c4315e4"collapsed="false"seolevelmigrated="true">ConfidenceLevelC</h5><ul><li><p>EachconfidenceintervalhasanassociatedconfidencelevelC,indicatingtheprobabilitythattheintervalcapturesthetruepopulationmeanoldsymbol{ ext{μ}}.</p></li><li><p>Confidencelevelsarechosenbytheanalyst,buthigherconfidencelevels(C ext{ ≥ } 90 ext{%})arepreferredforgreateraccuracyinestimates.</p></li></ul><h5id="0fe5dc19−bd95−4a81−990f−2b03656e4827"data−toc−id="0fe5dc19−bd95−4a81−990f−2b03656e4827"collapsed="false"seolevelmigrated="true">VisualizationofConfidenceIntervals</h5><ul><li><p>Theconfidenceintervalcanberepresentedgraphicallyas:<br> −oldsymbol{ ext{μ}} - 2 rac{oldsymbol{ ext{σ}}}{ ext{√}n} < oldsymbol{ ext{μ}} < oldsymbol{ ext{μ}} + 2 rac{oldsymbol{ ext{σ}}}{ ext{√}n}</p></li></ul><h5id="ccbaee19−abb6−4d8c−a411−a3987a8c511c"data−toc−id="ccbaee19−abb6−4d8c−a411−a3987a8c511c"collapsed="false"seolevelmigrated="true">GeneralFormofConfidenceIntervalforPopulationMean</h5><ul><li><p>ThegeneralformulaforalevelCconfidenceintervalis:<br> −ar{X} ± z^{} rac{oldsymbol{ ext{σ}}}{ ext{√}n} −Wherez^{}isthecriticalvaluesuchthatP(-z^{} ext{ ≤ } Z ext{ ≤ } z^{}) = C.</p></li></ul><h5id="fcef6606−0634−45d6−a376−ce2107ade7b4"data−toc−id="fcef6606−0634−45d6−a376−ce2107ade7b4"collapsed="false"seolevelmigrated="true">FindingCriticalValues</h5><ul><li><p>Criticalvaluesz^{}canbefoundusingastandardnormaldistributiontable. −Foraconfidencelevelof90P(Z < z^{}) = 0.05 + 0.90 = 0.95,resultinginz^{} = 1.645. −Fora95z^{} ext{ is approximately } 1.96.</p></li></ul><h4id="59cac55b−4ca5−4c72−9e2c−aaef9b03c2a6"data−toc−id="59cac55b−4ca5−4c72−9e2c−aaef9b03c2a6"collapsed="false"seolevelmigrated="true">ApplicationExamples</h4><h5id="a045971c−bea8−4bbc−9f13−5df3fe7f07d1"data−toc−id="a045971c−bea8−4bbc−9f13−5df3fe7f07d1"collapsed="false"seolevelmigrated="true">SentenceTimesforCriminalsExample</h5><ul><li><p>Assumethestandarddeviationoldsymbol{ ext{σ}} = 28.7monthsforsentencetimesofacertaincrime.Asimplerandomsampleof10sentencesyields:<br> −(136,102,84,150,115,98,125,176,120,72)</p></li><li><p>Fora92 ar{X} = 117.8.<br> −Thecriticalvaluez^{}isdeterminedsuchthat: −P(Z < z^{}) = 0.04 + 0.92 = 0.96 o z^{*} = 1.75</p></li><li><p>The92ar{X} ± 1.75 rac{oldsymbol{ ext{σ}}}{ ext{√}10} = (101.9, 133.7),<br> −Meaningweare92n = 40: (18.26, 25.22)<br> −95n = 160: (20.00, 23.48)</p></li></ul><h5id="8ba602a2−9283−4fdc−a6f3−b1de60b5549a"data−toc−id="8ba602a2−9283−4fdc−a6f3−b1de60b5549a"collapsed="false"seolevelmigrated="true">Trade−OffbetweenConfidenceLevelandMarginofError</h5><ul><li><p>Astheconfidencelevelincreases,themarginoferroralsoincreases,leadingtowiderintervals.</p></li><li><p>Tomaintainahighconfidencelevelwhilealsoreducingthemarginoferror:<br> −Samplesizemustincreaseinaquadraticmannerrelativetothedesiredmarginoferrorchange.</p></li></ul><h4id="547793a7−ae55−41f7−99b6−4518d0883959"data−toc−id="547793a7−ae55−41f7−99b6−4518d0883959"collapsed="false"seolevelmigrated="true">MisinterpretationsofConfidenceIntervals</h4><h5id="0bf9d978−df9d−45fd−86b9−29c9eb1b8e57"data−toc−id="0bf9d978−df9d−45fd−86b9−29c9eb1b8e57"collapsed="false"seolevelmigrated="true">CommonMisinterpretations</h5><ol><li><p>MisleadingStatements:<br> −"Approximately9518.26and25.22perhour."<br> −<strong>Correction</strong>:Theintervalestimatesthemeanwage,notindividualwages.</p></li><li><p>IncorrectProbabilityConclusions:<br> −"Theprobabilitythatthecalculatedintervalcontainsthesamplemeanis9518.26and25.22perhour."<br> −<strong>Correction</strong>:Theintervalisforthepopulationmean,nototherpotentialsamples.</p></li><li><p>ParameterValuesMisinterpretation:<br> −"Theprobabilitythatoldsymbol{ ext{μ}}isbetween18.26and25.22is0.95."<br> −<strong>Correction</strong>:Thepopulationmeanisfixed;probabilitiesshouldbediscussedintermsoftheinterval,whichisrandom.</p></li></ol><h4id="1aca3868−7326−4125−9481−74011d01e026"data−toc−id="1aca3868−7326−4125−9481−74011d01e026"collapsed="false"seolevelmigrated="true">SampleSizeConsiderations</h4><h5id="8de46dd9−067f−4477−a1e0−00181d653256"data−toc−id="8de46dd9−067f−4477−a1e0−00181d653256"collapsed="false"seolevelmigrated="true">SampleSizevs.PopulationSize</h5><ul><li><p>Thelengthofaconfidenceintervalisdependentonthesamplesize,nottheoverallpopulationsize.<br> −Forfixedsamplesizes,theestimate’sprecisionremainsconsistentregardlessofthepopulationsize.</p></li><li><p>Forexample:<br> −Asampleof25studentsfromUofW(population10,000)vs.UofM(population30,000)yieldsthesamemarginoferror,assumingequalstandarddeviations.</p></li></ul><h5id="ceeab03c−11d2−439e−b522−2c218891506f"data−toc−id="ceeab03c−11d2−439e−b522−2c218891506f"collapsed="false"seolevelmigrated="true">DeterminingSampleSize</h5><ul><li><p>Importanttoconsiderthepurposeofdatacollectionanddesiredprecisionofestimates.<br> −Aimingforasmallmarginoferrorwithhighconfidenceleadstoappropriatesamplesizeselection.</p></li><li><p>Therelationshipbetweenmarginoferrormforrequiredsamplesizencanbedefinedas:<br> m = z^{*} rac{oldsymbol{ ext{σ}}}{ ext{√}n}</p></li></ul><h5id="9cc1bea8−01a4−4f45−a6fd−16d3b4403e8f"data−toc−id="9cc1bea8−01a4−4f45−a6fd−16d3b4403e8f"collapsed="false"seolevelmigrated="true">ExampleCalculationforSampleSize</h5><ul><li><p>Problem:Estimatetheaveragesleepdurationwithknownstandarddeviationoldsymbol{ ext{σ}} = 1.27hours,within0.5hoursata98z^{} = 2.326for98n = rac{oldsymbol{ ext{σ}}^{2}(z^{})^2}{m^{2}} = rac{(1.27)^{2}(2.326)^{2}}{(0.5)^{2}} = 34.935</p></li><li><p>Roundupforfinalsamplesize: n = 35.</p></li></ul><h5id="be2682ae−4e9a−4821−86cc−e78427b640a5"data−toc−id="be2682ae−4e9a−4821−86cc−e78427b640a5"collapsed="false"seolevelmigrated="true">EffectofMarginofErroronSampleSize</h5><ul><li><p>Reducingthemarginoferrornecessitatesincreasingsamplesize.<br> −Ifthemarginoferrorishalved,samplesizemustincreasetofourtimestheoriginalvalueduetothesquarerelation.</p></li></ul><h5id="5cb3f2f2−63f2−4400−8fe3−21dcdaa91cda"data−toc−id="5cb3f2f2−63f2−4400−8fe3−21dcdaa91cda"collapsed="false"seolevelmigrated="true">CautionsinConfidenceIntervalCalculations</h5><ol><li><p>Validitydependsonsamplingmethods;randomsamplingiscritical.Poorsampling(i.e.,haphazard)invalidatesstatisticalconclusions.</p></li><li><p>Stronginfluenceofoutliersonthesamplemeanandconfidenceintervals.</p></li><li><p>Theoreticaluseofpopulationstandarddeviationoldsymbol{ ext{σ}}$ is often impractical; sample deviations s may be more realistic in practical applications.