Teknik Pengolahan Data

Teknik Pengolahan Data Uji Hipotesis (Hypothesis Tes/ng)

15-‐Oct-‐15 h8p://is/arto.staff.ugm.ac.id

Universitas Gadjah Mada Jurusan Teknik Sipil dan Lingkungan Prodi Magister Teknik Pengelolaan Bencana Alam

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•  komparasi garis teore/k (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 /dak cukup jelas untuk menyatakan bahwa model diterima atau ditolak •  uji hipotesis sebagai alat analisis dalam komparasi tersebut

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•  Model Matema/ka vs Pengukuran

h8p://is/arto.staff.ugm.ac.id

Uji Hipotesis

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Rumuskan hipotesis Rumuskan hipotesis alterna/f Tetapkan sta/s/ka uji Tetapkan distribusi sta/s/ka uji Tentukan nilai kri/k sebagai batas sta/s/ka uji harus ditolak Kumpulkan data untuk menyusun sta/s/ka uji Kontrol posisi sta/s/ka uji terhadap nilai kri/s

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•  •  •  •  •  •  • 


Prosedur Uji Hipotesis

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Kemungkinan Kesalahan keadaan nyata hipotesis benar

hipotesis salah

menerima

tak salah

kesalahan /pe II

menolak

kesalahan type I

tak salah


pilihan

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H0 = hipotesis (yang diuji)

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Notasi

1−α = /ngkat keyakinan (confidence level)


Ha = hipotesis alterna/f

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µ = µ1

Ha :

µ = µ2

Distribusi Normal σX2 diketahui

Sta/s/ka uji: Z =

n σX

(X − µ ) 1

berdistribusi normal

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

σX

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

σX

n

n

⇒ Z < −z1−α

⇒ Z > z1−α

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H0 :


Uji Hipotesis Nilai Rata-‐rata

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z1−α

prob ( Z > z1−α ) = α


luas = α

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µ = µ1

Ha :

µ = µ2

Sta/s/ka uji:

Distribusi Normal σX2 /dak diketahui T=

n sX

berdistribusi t

(X − µ ) 1

H0 ditolak jika: X < µ1 − t1−α,n−1

X > µ1 + t1−α,n−1

sX n sX n

jika μ1 > μ2

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H0 :



jika μ1 < μ2 8

µ = µ0

Ha :

µ ≠ µ0

Sta/s/ka uji:

H0 ditolak jika:

Distribusi Normal σX2 diketahui Z=

n σX

Z=


(X − µ ) 0

n σX

(X − µ ) 0

> z1−α 2

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H0 :



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µ = µ0

Ha :

µ ≠ µ0

Sta/s/ka uji:

H0 ditolak jika:

Distribusi Normal σX2 /dak diketahui

T=

n sX

t =

berdistribusi t

(X − µ ) 0

n sX

(X − µ ) 0

> t1−α 2,n−1

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H0 :



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•  Hasil uji hipotesis adalah

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•  Ar/nya •  H0: μ = μ0 •  Tidak menolak H0 à “menerima” H0 berar/ bahwa μ /dak berbeda secara signifikan dengan μ0. •  Tetapi /dak dikatakan bahwa μ benar-‐benar sama dengan μ0 karena kita /dak membuk/kan bahwa μ = μ0.


•  menolak H0, atau •  /dak menolak H0

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µ1 − µ 2 = δ

Ha :

µ1 − µ 2 ≠ δ

Sta/s/ka uji:

H0 ditolak jika:

var (X1) dan var (X 2 ) diketahui

Z=

X1 − X 2 − δ

(

σ 12 n1 + σ 22 n2

z =

n σX

(X − µ ) 0

)

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> z1−α 2



H0 :

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

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µ1 − µ 2 = δ

Ha :

µ1 − µ 2 ≠ δ

Sta/s/ka uji:

T=

var (X1) dan var (X 2 ) tidak diketahui

X1 − X 2 − δ ') n + n # n −1 s 2 + n −1 s 2 % +) ( ) ( ) 1 2 ) $( 1 1 2 2 & ( , # % )* )$n1n2 ( n1 + n2 − 2)&

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berdistribusi t dengan (n1+n2–2) degrees of freedom H0 ditolak jika: t =

n sX

(X − µ ) 0

> t1−α 2,n1+n2 −2


H0 :

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

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Ha :

σ 2 = σ 02 2

σ ≠ σ0

Distribusi Normal

2

n

Sta/s/ka uji:

2 c

χ =∑ i=1

(X − X ) i

σ0

2

berdistribusi chi-‐kuadrat

2 2 2 χ < χ < χ H0 diterima (/dak ditolak) jika: α 2,n−1 c 1−α 2,n−1

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H0 :


Uji Hipotesis Nilai Varian

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Ha :

σ 12 = σ 22 2 1

σ ≠ σ2

Sta/s/ka uji:

2 Distribusi Normal

2

s12 Fc = 2 s2

(n −1) 1

berdistribusi F dengan

dan

(n

s12 > s22 Fc > F1−α,n1−1,n2 −1 H0 ditolak jika:

2

−1) degrees of freedom


H0 :

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Ha :

σ 12 = σ 22 = ... = σ k 2 2 1

2

σ ≠ σ 2 ≠ ... ≠ σ k

Sta/s/ka uji: Q h

2

Distribusi Normal

berdistribusi chi-‐kuadrat dengan (k – 1) degrees of freedom # k ( n −1) s 2 & k i ( − ∑ ( n −1) ln si 2 Q = ∑ ( n −1) ln %∑ i %$ i=1 N − k (' i=1 i=1 k ) 1 1 1 , h = 1+ − + . ∑ 3 ( k −1) i=1 * ni −1 N − k k

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H0 :



k

N = ∑ ni i=1

H0 ditolak jika:

Q 2 > χ1−α,k−1 h

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•  La/han

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Uji Hipotesis •  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


•  Lihat kembali data debit puncak tahunan Sungai XYZ.

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Goodness of Fit Test

CDF PLOT ON PROBABILITY PAPER

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•  plot and compare the observed rela/ve frequency curve with the theore/cal rela/ve frequency curve •  plot the observed data on appropriate probability paper and judge as to whether or not the resul/ng plot is a straight line

•  Sta/s/cal tests: •  chi-‐square goodness of fit test •  the Kolmogorov-‐Smirnov test

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•  Graphical (and visual) methods to judge whether or not a par/cular distribu/on adequately describes a set of observa/ons:


Testing The Goodness of Fit of Data to Probability Distributions

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0.20

observed data

0.15 h8p://is/arto.staff.ugm.ac.id

Rela%ve frequency

theore/cal distribu/on

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Annual Peak Discharge of XYZ River

0.10

0.05

0.00

Discharge (m3/s)

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markers: observed data line: theoretical distribution

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Normal Distribution Paper

•  Comparison between the actual number of observa/ons and the expected number of observa/ons (expected according to the distribu/on under test) that fall in the class intervals. •  The expected numbers are calculated by mul/plying the expected rela/ve frequency by the total number of observa/ons. •  The test sta/s/c is calculated from the following rela/onship: k

(O − E )

i=1

Ei

χ =∑ 2 c

2

i

i

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•  Method of test


Chi-‐square Goodness of Fit Test

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˃  The test sta/s/c is calculated from the following rela/onship: k

(O − E )

i=1

Ei

2

i

i

where: k is the number of class intervals Oi is the number of observa/ons in the ith class interval Ei is the expected number of observa/ons in the ith class interval according to the distribu/on being tested χc2 has a distribu/on of chi-‐square with (k – p – 1) degrees of freedom, where p is the number of parameters es/mated from the data


χ =∑ 2 c

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˃  The test sta/s/c is calculated from the following rela/onship: k

(O − E )

i=1

Ei

2

i

i

˃  The hypothesis that the data are from the specified distribu/on is rejected if: 2 χ2c > χ1−α,k−p−1

1−α


χ =∑ 2 c

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α 2 χ1−α,k−p−1

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•  Let PX(x) be the completely specified theore/cal cumula/ve distribu/on func/on under the null hypothesis. •  Let Sn(x) be the sample comula/ve density func/on based on n observa/ons. For any observed x, Sn(x) = k/n where k is the number of observa/ons less than or equal to x. •  Determine the maximum devia/on, 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 cri/cal tabulated of the Kolmogorov-‐ Smirnov sta/s/c, the hypothesis is rejected. Table of Kolmogorov-‐Smirnov test sta/s/c is available in many books on sta/s/cs.

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•  Steps in the Kolmogorov-‐Smirnov test:


The Kolmogorov-‐Smirnov Test

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•  The test can be conducted by calcula/ng the quan//es PX(x) and Sn(x) at each observed point or •  By plotng the data on the probability paper and and selec/ng the greatest devia/on on the probability scale of a point from the theore/cal 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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•  Notes on the Kolmogorov-‐Smirnov test:


The Kolmogorov-‐Smirnov Test

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

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•  Exercise


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

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•  Both tests are insensi/ve in the tails of the distribu/ons. •  On the other hand, the tails are important in hydrologic frequency distribu/ons.

•  To increase sensi/vity of chi-‐square test •  The expected number of observa/ons 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 observa/ons in each class interval is the same. •  The class intervals will be of unequal width. •  The interval widths will be a func/on of the distribu/on being tested.

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•  Notes on both tests when tes/ng hydrologic frequency distribu/ons.



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

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•  Exercise



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Teknik Pengolahan Data

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