Teknik Pengolahan Data

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1 Universitas Gadjah Mada Jurusan Teknik Sipil dan Lingkungan Prodi Magister Teknik Pengelolaan Bencana Alam Teknik Pengolahan Data Uji Hipotesis (Hypothesis Tes/ng) 1

2 Uji Hipotesis Model Matema/ka vs Pengukuran 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

3 Prosedur Uji Hipotesis 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 3

4 Kemungkinan Kesalahan keadaan nyata pilihan hipotesis benar hipotesis salah menerima tak salah kesalahan /pe II menolak kesalahan type I tak salah 4

5 Notasi H 0 = hipotesis (yang diuji) H a = hipotesis alterna/f 1 α = /ngkat keyakinan (confidence level) 5

6 Uji Hipotesis Nilai Rata- rata H 0 : µ = µ 1 H a : µ = µ Distribusi Normal σ X diketahui Sta/s/ka uji: Z = n σ X X µ 1 ( ) Jika μ 1 > μ : H 0 ditolak jika berdistribusi normal X < µ 1 z 1 α σ X n Z < z 1 α Jika μ 1 < μ : H 0 ditolak jika X < µ 1 + z 1 α σ X n Z > z 1 α 6

7 z 1 α luas = α prob( Z > z ) 1 α = α 7

8 Uji Hipotesis Nilai Rata- rata H 0 : µ = µ 1 H a : µ = µ Distribusi Normal σ X /dak diketahui Sta/s/ka uji: T = n s X X µ 1 ( ) berdistribusi t s H X < µ 1 t X 0 ditolak jika: 1 α,n 1 jika μ n 1 > μ X > µ 1 + t 1 α,n 1 s X n jika μ 1 < μ 8

9 Uji Hipotesis Nilai Rata- rata H 0 : µ = µ 0 H a : µ µ 0 Distribusi Normal σ X diketahui Sta/s/ka uji: H 0 ditolak jika: Z = Z = n σ X X µ 0 ( ) n ( X µ σ 0 ) > z 1 α X berdistribusi normal 9

10 Uji Hipotesis Nilai Rata- rata H 0 : µ = µ 0 H a : µ µ 0 Distribusi Normal σ X /dak diketahui Sta/s/ka uji: H 0 ditolak jika: T = t = n s X X µ 0 ( ) n ( X µ s 0 ) > t 1 α,n 1 X berdistribusi t 10

11 Uji Hipotesis Nilai Rata- rata Hasil uji hipotesis adalah menolak H 0, atau /dak menolak H 0 Ar/nya H 0 : μ = μ 0 Tidak menolak H 0 à menerima H 0 berar/ bahwa μ /dak berbeda secara signifikan dengan μ 0. Tetapi /dak dikatakan bahwa μ benar- benar sama dengan μ 0 karena kita /dak membuk/kan bahwa μ = μ 0. 11

12 Uji hipotesis beda nilai rata- rata dua buah distribusi normal H 0 : H a : µ 1 µ = δ µ 1 µ δ var( X 1 ) dan var( X ) diketahui Sta/s/ka uji: H 0 ditolak jika: Z = z = X 1 X δ ( σ 1 n 1 + σ n ) 1 n ( X µ σ 0 ) > z 1 α X berdistribusi normal 1

13 Uji hipotesis beda nilai rata- rata dua buah distribusi normal H 0 : H a : µ 1 µ = δ µ 1 µ δ var( X 1 ) dan var( X ) tidak diketahui Sta/s/ka uji: T = ' ) ( *) X 1 X δ n 1 + n # ( )s 1 + ( n 1)s % + $ &), # $ n 1 n ( n 1 + n ) % & -) ( ) n 1 1 berdistribusi t dengan (n 1 +n ) degrees of freedom 1 H 0 ditolak jika: t = n s X X µ 0 ( ) > t 1 α,n1 +n 13

14 Uji Hipotesis Nilai Varian H 0 : σ = σ 0 H a : σ σ 0 Distribusi Normal Sta/s/ka uji: χ c = n i=1 ( X i X) σ 0 H 0 diterima (/dak ditolak) jika: berdistribusi chi- kuadrat χ α,n 1 < χ c < χ 1 α,n 1 14

15 Uji Hipotesis Nilai Varian H 0 : σ 1 = σ H a : σ 1 σ Distribusi Normal Sta/s/ka uji: F c = s 1 s s 1 > s F c F 1 α,n1 1,n 1 berdistribusi F dengan ( n 1 1) dan ( n 1) degrees of freedom H 0 ditolak jika: 15

16 Uji Hipotesis Nilai Varian H 0 : σ 1 = σ =... = σ k H a : σ 1 σ... σ k Distribusi Normal Sta/s/ka uji: H 0 ditolak jika: Q h berdistribusi chi- kuadrat dengan (k 1) degrees of freedom k # k ( n Q = ( n 1) ln i 1)s & k % i ( n 1 i=1 $ % i=1 N k '( i=1 k ) 1 h =1+ n i 1 1, +. * N k - N = k i=1 1 3 k 1 ( ) n i Q h > χ 1 α,k 1 i=1 ( ) lns i 16

17 Uji Hipotesis La/han Lihat kembali data debit puncak tahunan Sungai XYZ. Uji hipotesis yang menyatakan bahwa debit puncak tahunan rerata adalah 650 m 3 /s dan varians adalah m 6 /s. Contoh uji hipotesis.pdf Exercises on hypothesis thesis.pdf 17

18 Goodness of Fit Test CDF PLOT ON PROBABILITY PAPER 18

19 Testing The Goodness of Fit of Data to Probability Distributions Graphical (and visual) methods to judge whether or not a par/cular distribu/on adequately describes a set of observa/ons: 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 19

20 Annual Peak Discharge of XYZ River 0.0 theore/cal distribu/on observed data Rela%ve frequency Discharge (m 3 /s) 0

21 h8p://is/arto.staff.ugm.ac.id markers: observed data line: theoretical distribution 1

22 Normal Distribution Paper

23 Chi- square Goodness of Fit Test Method of test 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: χ c = k ( O i E ) i i=1 E i 3

24 Chi- square Goodness of Fit Test The test sta/s/c is calculated from the following rela/onship: χ c = k ( O i E ) i i=1 E i where: k is the number of class intervals O i is the number of observa/ons in the ith class interval E i is the expected number of observa/ons in the ith class interval according to the distribu/on being tested χ c 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 4

25 Chi- square Goodness of Fit Test The test sta/s/c is calculated from the following rela/onship: χ c = k ( O i E ) i i=1 E i The hypothesis that the data are from the specified distribu/on is rejected if: χ c > χ 1 α,k p 1 1 α α χ 1 α,k p 1 5

26 The Kolmogorov- Smirnov Test Steps in the Kolmogorov- Smirnov test: Let P X (x) be the completely specified theore/cal cumula/ve distribu/on func/on under the null hypothesis. Let S n (x) be the sample comula/ve density func/on based on n observa/ons. For any observed x, S n (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 P X (x) S n (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. 6

27 The Kolmogorov- Smirnov Test Notes on the Kolmogorov- Smirnov test: The test can be conducted by calcula/ng the quan//es P X (x) and S n (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. 7

28 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 distribu/on. 8

29 Chi- square Goodness of Fit Test and The Kolmogorov- Smirnov Test Notes on both tests when tes/ng hydrologic frequency distribu/ons. 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. 9

30 Chi- square Goodness of Fit Test and The Kolmogorov- Smirnov Test Exercise 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. 30

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