StatistikaTeknikUjiHipotesis

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MagisterPengelolaanAirdanAirLimbahUniversitasGadjahMada

UjiHipotesis• ModelMatematikavs Pengukuran• komparasigaristeoretik(prediksimenurutmodel)dandatapengukuran

• jikaprediksimodelsesuaidengandatapengukuran,makamodelditerima

• jikaprediksimodelmenyimpangdaridatapengukuran,makamodelditolak

• Dalamsejumlahkasus,yangterjadiadalah• hasilkomparasiprediksimodeldandatapengukurantidakcukupjelasuntukmenyatakanbahwamodelditerimaatauditolak

• ujihipotesissebagaialatanalisisdalamkomparasitersebut

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ProsedurUjiHipotesis• Rumuskanhipotesis• Rumuskanhipotesisalternatif• Tetapkanstatistikauji• Tetapkandistribusistatistikauji• Tentukannilaikritiksebagaibatasstatistikaujiharusditolak• Kumpulkandatauntukmenyusunstatistikauji• Kontrolposisistatistikaujiterhadapnilaikritis

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KemungkinanKesalahan

pilihankeadaannyata

hipotesisbenar hipotesissalah

menerima taksalah kesalahantipeII

menolak kesalahantipeI taksalah

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Notasi

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H0 =hipotesis(yangdiuji)

Ha =hipotesisalternatif

1−α =tingkatkeyakinan(confidencelevel)

UjiHipotesisNilaiRata-rata

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H0 : µ =µ1

Ha : µ =µ2

X <µ1− z1−α

σX

n⇒ Z < −z1−α

DistribusiNormalσX2 diketahui

Z =

nσX

X −µ1( )Statistikauji: berdistribusinormal

Jikaμ1 >μ2:H0 ditolakjika

Jikaμ1 <μ2:H0 ditolakjika X <µ1+ z1−α

σX

n⇒ Z > z1−α

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luas=α

z1−α

prob Z > z1−α( ) = α

UjiHipotesisNilaiRata-rata

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H0 : µ =µ1

Ha : µ =µ2

X <µ1− t1−α ,n−1

sX

n

DistribusiNormalσX2 tidakdiketahui

T =

nsX

X −µ1( )Statistikauji: berdistribusit

H0 ditolakjika: jikaμ1 >μ2

X >µ1+ t1−α ,n−1

sX

njikaμ1 <μ2

UjiHipotesisNilaiRata-rata

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H0 : µ =µ0

Ha : µ ≠µ0

Z =n

σX

X −µ0( ) > z1−α 2

DistribusiNormalσX2 diketahui

Z =

nσX

X −µ0( )Statistikauji: berdistribusinormal

H0 ditolakjika:

UjiHipotesisNilaiRata-rata

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H0 : µ =µ0

Ha : µ ≠µ0

t =n

sX

X −µ0( ) > t1−α 2,n−1

DistribusiNormalσX2 tidakdiketahui

T =

nsX

X −µ0( )Statistikauji: berdistribusit

H0 ditolakjika:

UjiHipotesisNilaiRata-rata• Hasilujihipotesisadalah• menolakH0,atau• tidakmenolakH0

• Artinya• H0:μ =μ0

• TidakmenolakH0 à “menerima”H0 berartibahwaμ tidakberbedasecarasignifikandenganμ0.

• Tetapitidakdikatakanbahwaμ benar-benarsamadenganμ0karenakitatidakmembuktikanbahwaμ =μ0.

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Ujihipotesisbedanilairata-rataduabuahdistribusinormal

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H0 : µ1−µ2 = δ

Ha : µ1−µ2 ≠ δ

z =n

σX

X −µ0( ) > z1−α 2

Z =X1−X2 −δ

σ12 n1+σ2

2 n2( )1 2Statistikauji: berdistribusinormal

H0 ditolakjika:

var X1( ) dan var X2( ) diketahui

Ujihipotesisbedanilairata-rataduabuahdistribusinormal

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H0 : µ1−µ2 = δ

Ha : µ1−µ2 ≠ δ

t =n

sX

X −µ0( ) > t1−α 2,n1+n2−2

T =X1−X2 −δ

n1+ n2( ) n1−1( ) s12 + n2 −1( ) s2

2#$

%&

n1n2 n1+ n2 −2( )#$ %&

'()

*)

+,)

-)

1 2Statistikauji:

berdistribusit dengan(n1+n2–2)degreesoffreedom

H0 ditolakjika:

var X1( ) dan var X2( ) tidak diketahui

UjiHipotesisNilaiVarian

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H0 : σ2 = σ02

Ha : σ2 ≠ σ02

χα 2,n−1

2 < χc2 < χ1−α 2,n−1

2

DistribusiNormal

χc

2 =Xi −X( )σ0

2i=1

n

∑Statistikauji: berdistribusichi-kuadrat

H0 diterima(tidakditolak)jika:

UjiHipotesisNilaiVarian

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H0 : σ12 = σ2

2

Ha : σ12 ≠ σ2

2

Fc > F1−α ,n1−1,n2−1

2DistribusiNormal

Fc =

s12

s22Statistikauji: berdistribusiF dengan

H0 ditolakjika:

n1−1( ) dan n2 −1( ) degrees of freedom

s12 > s2

2

UjiHipotesisNilaiVarian

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H0 : σ12 = σ2

2 = ... = σ k2

Ha : σ12 ≠ σ2

2 ≠ ... ≠ σ k2

Qh> χ1−α ,k−1

2

DistribusiNormal

Qh

Statistikauji: berdistribusichi-kuadratdengan(k – 1)degreesoffreedom

H0 ditolakjika:

Q = n−1( ) lnni −1( ) si

2

N − ki=1

k

∑#

$%%

&

'((i=1

k

∑ − n−1( ) ln si2

i=1

k

∑

h =1+1

3 k −1( )1

ni −1−

1N − k

)

*+

,

-.

i=1

k

∑

N = nii=1

k

∑

UjiHipotesis• Latihan• LihatkembalidatadebitpuncaktahunanSungaiXYZ.

• Ujihipotesisyangmenyatakanbahwadebitpuncaktahunanrerataadalah650m3/sdanvariansadalah45.000m6/s2.

• Contohujihipotesis.pdf• Exercisesonhypothesisthesis.pdf

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CDFPLOTONPROBABILITYPAPERGoodnessofFitTest

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TestingTheGoodnessofFitofDatatoProbabilityDistributions• Graphical(andvisual)methodstojudgewhetherornotaparticulardistributionadequatelydescribesasetofobservations:• plotandcomparetheobservedrelativefrequencycurvewiththetheoreticalrelativefrequencycurve

• plottheobserveddataonappropriateprobabilitypaperandjudgeastowhetherornottheresultingplotisastraightline

• Statisticaltests:• chi-squaregoodnessoffittest• theKolmogorov-Smirnovtest

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AnnualPeakDischargeofXYZRiver

0.00

0.05

0.10

0.15

0.20

Relativ

efreq

uency

Discharge(m3/s)

observeddatatheoreticaldistribution

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markers: observed dataline: theoretical distribution

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NormalDistributionPaper

Chi-squareGoodnessofFitTest

• Methodoftest• Comparisonbetweentheactualnumberofobservationsandtheexpectednumberofobservations(expectedaccordingtothedistributionundertest)thatfallintheclassintervals.

• Theexpectednumbersarecalculatedbymultiplyingtheexpectedrelativefrequencybythetotalnumberofobservations.

• Theteststatisticiscalculatedfromthefollowingrelationship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑

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Chi-squareGoodnessofFitTest˃ Theteststatisticiscalculatedfromthefollowingrelationship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑where:k isthenumberofclassintervalsOi isthenumberofobservationsintheithclassintervalEi istheexpectednumberofobservationsintheithclassinterval

accordingtothedistributionbeingtestedχc2 hasadistributionofchi-squarewith(k – p – 1)degreesoffreedom,

wherep isthenumberofparametersestimatedfromthedata

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Chi-squareGoodnessofFitTest˃ Theteststatisticiscalculatedfromthefollowingrelationship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑

˃ Thehypothesisthatthedataarefromthespecifieddistributionis rejected if:

χc2 > χ1−α ,k−p−1

2

1−α α

χ1−α ,k−p−12

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TheKolmogorov-SmirnovTest

• StepsintheKolmogorov-Smirnovtest:• LetPX(x)bethecompletelyspecifiedtheoreticalcumulativedistributionfunctionunderthenullhypothesis.

• LetSn(x)bethesamplecomulativedensityfunctionbasedonnobservations.Foranyobservedx,Sn(x)=k/n wherek isthenumberofobservationslessthanorequaltox.

• Determinethemaximumdeviation,D,definedby:D =max|PX(x)– Sn(x)|

• If,forthechosensignificancelevel,theobservedvalueofD isgreaterthanorequaltothecriticaltabulatedoftheKolmogorov-Smirnovstatistic,thehypothesisisrejected.TableofKolmogorov-Smirnovteststatisticisavailableinmanybooksonstatistics.

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TheKolmogorov-SmirnovTest

• NotesontheKolmogorov-Smirnovtest:• ThetestcanbeconductedbycalculatingthequantitiesPX(x)andSn(x)ateachobservedpointor

• Byplottingthedataontheprobabilitypaperandandselectingthegreatestdeviationontheprobabilityscaleofapointfromthetheoreticalline.• Thedatashouldnotbegroupedforthistest,i.e.ploteachpointofthedataontheprobabilitypaper.

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Chi-squareGoodnessofFitTestandTheKolmogorov-SmirnovTest• Exercise• Dothechi-squaregoodnessoffittestandtheKolmogorov-SmirnovtesttotheannualpeakdischargeofXYZRiveragainstnormaldistribution.

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Chi-squareGoodnessofFitTestandTheKolmogorov-SmirnovTest• Notesonbothtestswhentestinghydrologicfrequencydistributions.• Bothtestsareinsensitiveinthetailsofthedistributions.• Ontheotherhand,thetailsareimportantinhydrologicfrequencydistributions.

• Toincreasesensitivityofchi-squaretest• Theexpectednumberofobservationsinaclassshallnotbelessthan3(or5).

• Definetheclassintervalsothatunderthehypothesisbeingtested,theexpectednumberofobservationsineachclassintervalisthesame.• Theclassintervalswillbeofunequalwidth.• Theintervalwidthswillbeafunctionofthedistributionbeingtested.

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Chi-squareGoodnessofFitTestandTheKolmogorov-SmirnovTest• Exercise• Redothechi-squaregoodnessoffittestandtheKolmogorov-SmirnovtesttotheannualpeakdischargeofXYZRiveragainstnormaldistribution.• Definetheclassintervalssothattheexpectednumberofobservationsineachclassintervalisthesame.

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