MATH 110 CHAPTER 7.pdf

MATH 110 CHAPTER 7, 8, 9 EXAM
QUESTIONS WITH CORRECT
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ProbabilitysDensitysCurves-
ssCORRECTsANSWERSthescurvesusedstosdescribesthesdistributionsofsascontinuoussrandomsvariable




NormalsCurves-ssCORRECTsANSWERSthesbell-
shapedscurvesthatsdescribessthesdistributionsofsmanysphysicalsandspsychologicalsattributes.sMostsscoressfal
lsnearsthesaverage,sandsfewersandsfewersscoressliesnearsthesextremes



Z-Scores-
ssCORRECTsANSWERSasmeasuresofshowsmanysstandardsdeviationssyousaresawaysfromsthesnorms(averagesor

smean)




Example:sAswomanswhosesheightsissx=67sinchessfromsasnormalspopulationswithsmeansμ=64sandsσ=3sinches
.s



WhatsissthesZ-score?s-ssCORRECTsANSWERSThesZ-scoresisszs=s(xs-sμ)/σs=s(67-64)/3s=s1



zsscoresformulas-ssCORRECTsANSWERSzs=s(xs-sμ)/σ



findingsthesareasundersthesnormalscurves-ssCORRECTsANSWERSnormalcdf(lower,supper,sμ,sσ)



Tosgetsit:s2nd+Vars->s2:snorm



Example:sFindsthesareastosthesleftsofszs=s1.26sinsasnormalscurves-ssCORRECTsANSWERSnormalcdf(-
1E99,s1.26,s0,s1)

,Example:sAsstudysreportedsthatstheslengthsofspregnancysfromsconceptionstosbirthsissapproximatelysnormall
ysdistributedswithsmeansμs=s272sdayssandsstandardsdeviationsσs=s9sdays.sWhatsproportionsofspregnanciessla
stslesssthans259sdays?s-ssCORRECTsANSWERSSteps1:sFindsthesZ-Score.s(259-272)/9s=s-1.444



Steps2:sUsesthesZ-ScorestosfindsareasleftsofsZs=s-1.444



normalcdf(-1E99,s-1.444,s0,s1)



FindingsXsfromsz-scores-ssCORRECTsANSWERSinvNorm(Areastosthesleft,sμ,sσ)



Tosgetstosit:s2nd+Varss->s3:invNorm



MensasissansorganizationswhosesmembersspossesssIQssinsthestops2%sofsthespopulation.sIfsthesIQssaresnorm
allysdistributedswithsasmeansofs100sandsasstandardsdeviationsofs15,swhatsissthesminimumsIQsnecessarysforsa
dmission?s-ssCORRECTsANSWERSFirst:sFindsthesz-scoresthatscorrespondsstosthesgivensareasusingsinvNorm.



zs=sinvNorm(0.98,s0,s1)



xs=sμs+szs*sσs=s100s+s2.0537(15)



xs=s130.8



xs≈s131



meansofssamplingsdistributionsofsx̄s-ssCORRECTsANSWERSμx̄s=sμ



Thesmeansofsthessamplingsdistributionsissdenotedsbysμx̄sandsissequalstosthesmeansofsthespopulationsμ.



standardsdeviationsofssamplingsdistributionsofsx̄s-ssCORRECTsANSWERSσx̄s=sσ/√n

, Thesstandardsdeviationsofsthessamplingsdistributions(StandardsError)sissdenotedsbysσx̄sandsequalssthesstand
ardsdeviationsofsthespopulationsdividedsbysthessquaresrootsofsthessamplessize.



Samplingsdistributionsofsasskewedspopulations-
ssCORRECTsANSWERSifsaspopulationsissskewed,saslargerssamplessizesissnecessarysforsthessamplingsdistributi

onsofsx̄stosbesapproximatelysnormal.



ThesCentralsLimitsTheorems-
ssCORRECTsANSWERSThestheorysthat,sasssamplessizesincreases,sthesdistributionsofssamplesmeanssofssizesn,sr

andomlysselected,sapproachessasnormalsdistribution.



Inspractice,sassamplessizesofsn>30sissenough.



Example:sBasedsonsdatasfromsthesU.S.sCensus,sthesmeansagesofscollegesstudentssins2011swassμs=s25syears,s
withsasstandardsdeviationsofsσs=s9.5syears.sAssimplesrandomssamplesofs125sstudentssissdrawn.sWhatsissthesp
robabilitysthatsthessamplesmeansagesofsthesstudentssissgreatersthans26syears?s-
ssCORRECTsANSWERSSteps1:sConfirmsthesassumptions.s



n=135>30

Thus,swesmaysusesthesnormalscurve.



Steps2:sThen,scomputesμx̄sandsσx̄.



μx̄s=sμs=s25

σx̄s=s9.5/√125s=s0.85



normalcdf(lower,supper,sμx̄,sσx̄)

normalcdf(26,s1E99,s25,s0.85)s=s0.1197



Meansofsassamplingsdistributionsofsp̂ s-ssCORRECTsANSWERSμp̂ s=sp̂



Thesmeansofsthessamplingsdistributionsofsp̂ sissequalstosthespopulationsproportion