R studio homework help

1.Asampleof12radondetectorsofacertaintypewasselected,andeachwasexposedto100 pCi/L ofradon.Theresultingreadingswereasfollows:105.6,90.9,91.2,96.9,96.5,91.3,

100.1,105.0,99.6,107.7,103.3,and92.4.

(a)    Dothedatasuggest that thepopulationmeanreadingundertheseconditionsdiffers from100?Stateandtestappropriatehypothesesusingα=0.05.Clearlystateyour assumptions.

H₀: μ = 100 (The population mean reading under these conditions is 100)

H_a:μ ≠ 100(The population mean reading under these conditions differ from 100)

            α=0.05        n=12         s=6.1167      mean=98.375     

(b)   Constructa95%confidenceintervalforthe populationmeanreading.

(c)    Supposethat priortotheexperiment,avalueofσ=9.0hadbeenassumed.What samplesizewouldhavebeenappropriate toobtainβ=0.10forthealternativeµ=95?

2.SupposewearetestingH0:µ=5againstH1:µ<5foranormalpopulationwithσ=1.A

randomsampleofsizen=9isavailable from thispopulation.Theztestisusedwithα=0.05,

the rejection region for this test is or, equivalently, .

(a)Onthesamegraph,useRtoplotthesamplingdistribution oftheteststatisticwhen

µ=5andwhenµ=4.2.

(b) Onyourgraph,shadeandlabeltheareathatrepresentstheprobabilityoftypeIerror

[eitherusingRorbyhand].

(c)Onyourgraph,shadeandlabeltheareathatrepresentstheprobabilityof typeIIerror

[eitherusingRorbyhand].

(d) ComputetheprobabilityoftypeIIerrorwhenµ=4.2.ProvidetheappropriateRcodes.

3.Astudywasconductedtoassessthedietoftwo sizemorphsoftheeasternhornedlizard Phrynosomadouglassibrevirostreinwhichthestomachcontentsof45lizardscollectedover timewheredetermined.Theresearch seekstodetermineifthereisanyevidencethatthe meancontentsofmilligramsofdrybiomassofColeoptrainthestomachsofadultfemalesis largerthanthatofadultmalesandyearlingfemales.Foradultmalesandyearlingfemales thevaluesobtainedfrom 24lizardswere256,209,0,0,0,44,49,117,6,0,0,75,34,13,0,90,

0,32,0,205,332,0,31,and0.Foradultfemalesthevaluesobtainedfrom21lizardswere0,

89,0,0,0,163,286,3,843,0,158,443,311,232,179,179,19,142,100,0,and432.Assume thatthesedataareobtained asindependentrandomsamplesfromtherespectivepopulations.

(a)    Statethenullandalternativeshypothesesclearly.Carefullydefineallquantities.

(b)   Conductatestofhypothesesusingthetwo-samplepooledttest,youmaytakeα=0.05. Inreportingtheresultsyoushouldclearlystatethenecessaryassumptionsforthetest, showthecalculationoftheobservedvalueoftheteststatistic, definetherejection region,presenttheP-value,andinterprettheconclusionsofthetestinthecontextof thisexample.AttachanyRcodesthatyouuse.

(c)    Repeatpart(b)forthetwo-sampleunpooledttest.

(d)   Repeatpart(b)fortheWilcoxonranksumtest. ComputetheP-valuebysimulating from thepermutationdistribution.

(e)    Whichofthethreetestsabovewouldyourecommend? Useappropriategraphs/summary statisticstojustifyyouranswerbasedontheassumptionsofeachtest.

4.      SupposewetakearandomsamplefromaN(µ,σ2)population.Considerthefollowingthree estimatorsofσ2:

,          ,            

(a)ConductasimulationstudyinR tocompletethefollowingtablewhenµ=0andσ2=1.

PresentyourRcode.

1      2              1      2              1      2

(b) Comparethebiasofthesethreeestimators.

(c)Comparethevarianceofthesethreeestimators.

(d) Comparethemeansquarederrorofthesethreeestimators.

(e)Basedonyourresults,whichestimatorwouldyourecommend?

5.Suppose thatwewishtoestimatetheaverage sulfateconcentration(ppm)observedinground- wateratabackgroundwell.Supposethatitisreasonabletoassumethatthepopulationof sulfateconcentrationmeasurementsingroundwateratthebackgroundwellisdescribedbya normaldistribution. Thereforea100(1−α) %confidenceintervalformeansulfateconcentra- tioncanbeobtainedusingtheusualonesampletinterval.

WhenthesamplestandarddeviationSiscomputedfromaaniidsamplefromaN(µ,σ2)

population,itcanbeshownthat

E(S) =