UniversitasGadjahMada FakultasTeknik’ … Uji... · 2015-10-15 · Statistika Uji$Hipotesis$ $ $ 1 UniversitasGadjahMada FakultasTeknik’ Jurusan’Teknik’Sipil’dan’Lingkungan’

Statistika Uji Hipotesis

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arto.staff.ugm.ac.id

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Universitas Gadjah Mada Fakultas Teknik Jurusan Teknik Sipil dan Lingkungan Prodi S2 Teknik Sipil

Uji Hipotesis •  Model Matema4ka vs Pengukuran •  komparasi garis teore4k (prediksi menurut model) dan data pengukuran

•  jika prediksi model sesuai dengan data pengukuran, maka model diterima

•  jika prediksi model menyimpang dari data pengukuran, maka model ditolak

•  Dalam sejumlah kasus, yang terjadi adalah •  hasil komparasi prediksi model dan data pengukuran 4dak cukup jelas untuk menyatakan bahwa model diterima atau ditolak

•  uji hipotesis sebagai alat analisis dalam komparasi tersebut

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Prosedur Uji Hipotesis •  Rumuskan hipotesis •  Rumuskan hipotesis alterna4f •  Tetapkan sta4s4ka uji •  Tetapkan distribusi sta4s4ka uji •  Tentukan nilai kri4k sebagai batas sta4s4ka uji harus ditolak •  Kumpulkan data untuk menyusun sta4s4ka uji •  Kontrol posisi sta4s4ka uji terhadap nilai kri4s

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Kesalahan

pilihan situasi nyata

hipotesis benar hipotesis salah

menerima tak salah kesalahan 4pe II

menolak kesalahan type I tak salah

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Notasi

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H0 = hipotesis (yang diuji)

Ha = hipotesis alterna4f

1−α = 4ngkat keyakinan (confidence level)

Uji Hipotesis Nilai Rata-‐rata

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H0 : µ =µ1

Ha : µ =µ2

X <µ1− z1−α

σX

n⇒ Z < −z1−α

Distribusi Normal σX2 diketahui

Z =

nσX

X −µ1( )Sta4s4ka uji: berdistribusi normal

Jika μ1 > μ2: H0 ditolak jika

Jika μ1 < μ2: H0 ditolak jika X <µ1+ z1−α

σX

n⇒ Z > z1−α

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luas = α

z1−α

prob Z > z1−α( ) = α


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H0 : µ =µ1

Ha : µ =µ2

X <µ1− t1−α ,n−1

sX

n

Distribusi Normal σX2 4dak diketahui

T =

nsX

X −µ1( )Sta4s4ka uji: berdistribusi t

H0 ditolak jika: jika μ1 > μ2

X >µ1+ t1−α ,n−1

sX

njika μ1 < μ2


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H0 : µ =µ0

Ha : µ ≠µ0

Z =n

σX

X −µ0( ) > z1−α 2

Distribusi Normal σX2 diketahui

Z =

nσX

X −µ0( )Sta4s4ka uji: berdistribusi normal

H0 ditolak jika:


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H0 : µ =µ0

Ha : µ ≠µ0

t =n

sX

X −µ0( ) > t1−α 2,n−1

Distribusi Normal σX2 4dak diketahui

T =

nsX

X −µ0( )Sta4s4ka uji: berdistribusi t

H0 ditolak jika:

Uji Hipotesis Nilai Rata-‐rata •  Hasil uji hipotesis adalah •  menolak H0, atau •  4dak menolak H0

•  Ar4nya •  H0: μ = μ0

•  Tidak menolak H0 à “menerima” H0 berar4 bahwa μ 4dak berbeda secara signifikan dengan μ0.

•  Tetapi 4dak dikatakan bahwa μ benar-‐benar sama dengan μ0 karena kita 4dak membuk4kan bahwa μ = μ0.

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Uji hipotesis beda nilai rata-‐rata dua buah distribusi normal

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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 2Sta4s4ka uji: berdistribusi normal

H0 ditolak jika:

var X1( ) dan var X2( ) diketahui

Uji hipotesis beda nilai rata-‐rata dua buah distribusi normal

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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 2Sta4s4ka uji:

berdistribusi t dengan (n1+n2–2) degrees of freedom

H0 ditolak jika:

var X1( ) dan var X2( ) tidak diketahui

Uji Hipotesis Nilai Varian

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H0 : σ2 = σ02

Ha : σ2 ≠ σ02

χα 2,n−1

2 < χc2 < χ1−α 2,n−1

2

Distribusi Normal

χc

2 =Xi −X( )σ0

2i=1

n

∑Sta4s4ka uji: berdistribusi chi-‐kuadrat

H0 diterima (4dak ditolak) jika:


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H0 : σ12 = σ2

2

Ha : σ12 ≠ σ2

2

Fc > F1−α ,n1−1,n2−1

2 Distribusi Normal

Fc =

s12

s22Sta4s4ka uji: berdistribusi F dengan

H0 ditolak jika:

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

s12 > s2

2


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H0 : σ12 = σ2

2 = ... = σ k2

Ha : σ12 ≠ σ2

2 ≠ ... ≠ σ k2

Qh> χ1−α ,k−1

2

Distribusi Normal

Qh

Sta4s4ka uji: berdistribusi chi-‐kuadrat dengan (k – 1) degrees of freedom

H0 ditolak jika:

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

∑

Uji Hipotesis •  La4han •  Lihat kembali data debit puncak tahunan Sungai XYZ.

•  Uji hipotesis yang menyatakan bahwa debit puncak tahunan rerata adalah 650 m3/s dan varians adalah 45.000 m6/s2.

•  Contoh uji hipotesis.pdf •  Exercises on hypothesis thesis.pdf

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CDF PLOT ON PROBABILITY PAPER Goodness of Fit Test

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Testing The Goodness of Fit of Data to Probability Distributions •  Graphical (and visual) methods to judge whether or not a par4cular distribu4on adequately describes a set of observa4ons: •  plot and compare the observed rela4ve frequency curve with the theore4cal rela4ve frequency curve

•  plot the observed data on appropriate probability paper and judge as to whether or not the resul4ng plot is a straight line

•  Sta4s4cal tests: •  chi-‐square goodness of fit test •  the Kolmogorov-‐Smirnov test

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Annual Peak Discharge of XYZ River

0.00

0.05

0.10

0.15

0.20

Rela%v

e freq

uency

Discharge (m3/s)

observed data theore4cal distribu4on

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markers: observed data line: theoretical distribution

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Normal Distribution Paper

Chi-‐square Goodness of Fit Test

•  Method of test •  Comparison between the actual number of observa4ons and the expected number of observa4ons (expected according to the distribu4on under test) that fall in the class intervals.

•  The expected numbers are calculated by mul4plying the expected rela4ve frequency by the total number of observa4ons.

•  The test sta4s4c is calculated from the following rela4onship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑

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Chi-‐square Goodness of Fit Test ˃  The test sta4s4c is calculated from the following rela4onship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑where: k is the number of class intervals Oi is the number of observa4ons in the ith class interval Ei is the expected number of observa4ons in the ith class interval

according to the distribu4on being tested χc2 has a distribu4on of chi-‐square with (k – p – 1) degrees of freedom,

where p is the number of parameters es4mated from the data

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Chi-‐square Goodness of Fit Test ˃  The test sta4s4c is calculated from the following rela4onship:

χc

2 =Oi − Ei( )2

Eii=1

k

∑

˃  The hypothesis that the data are from the specified distribu4on is rejected if:

χc2 > χ1−α ,k−p−1

2

1−α α

χ1−α ,k−p−12

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The Kolmogorov-‐Smirnov Test

•  Steps in the Kolmogorov-‐Smirnov test: •  Let PX(x) be the completely specified theore4cal cumula4ve distribu4on func4on under the null hypothesis.

•  Let Sn(x) be the sample comula4ve density func4on based on n observa4ons. For any observed x, Sn(x) = k/n where k is the number of observa4ons less than or equal to x.

•  Determine the maximum devia4on, D, defined by: D = max |PX(x) – Sn(x)|

•  If, for the chosen significance level, the observed value of D is greater than or equal to the cri4cal tabulated of the Kolmogorov-‐Smirnov sta4s4c, the hypothesis is rejected. Table of Kolmogorov-‐Smirnov test sta4s4c is available in many books on sta4s4cs.

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The Kolmogorov-‐Smirnov Test

•  Notes on the Kolmogorov-‐Smirnov test: •  The test can be conducted by calcula4ng the quan44es PX(x) and Sn(x) at each observed point or

•  By plosng the data on the probability paper and and selec4ng the greatest devia4on on the probability scale of a point from the theore4cal line. •  The data should not be grouped for this test, i.e. plot each point of the data on the probability paper.

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Chi-‐square Goodness of Fit Test and The Kolmogorov-‐Smirnov Test

•  Exercise •  Do the chi-‐square goodness of fit test and the Kolmogorov-‐Smirnov test to the annual peak discharge of XYZ River against normal distribu4on.

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•  Notes on both tests when tes4ng hydrologic frequency distribu4ons. •  Both tests are insensi4ve in the tails of the distribu4ons. •  On the other hand, the tails are important in hydrologic frequency distribu4ons.

•  To increase sensi4vity of chi-‐square test •  The expected number of observa4ons in a class shall not be less than 3 (or 5).

•  Define the class interval so that under the hypothesis being tested, the expected number of observa4ons in each class interval is the same. •  The class intervals will be of unequal width. •  The interval widths will be a func4on of the distribu4on being tested.

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•  Exercise •  Redo the chi-‐square goodness of fit test and the Kolmogorov-‐Smirnov test to the annual peak discharge of XYZ River against normal distribu4on. •  Define the class intervals so that the expected number of observa4ons in each class interval is the same.

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UniversitasGadjahMada FakultasTeknik’ … Uji... · 2015-10-15 · Statistika Uji$Hipotesis$ $ $ 1 UniversitasGadjahMada FakultasTeknik’ Jurusan’Teknik’Sipil’dan’Lingkungan’

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