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UJI NORMALITAS Kolmogorov-Smirnov & Chi-Square Oleh: Roni Saputra, M.Si.

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Presentasi berjudul: "UJI NORMALITAS Kolmogorov-Smirnov & Chi-Square Oleh: Roni Saputra, M.Si."— Transcript presentasi:

1 UJI NORMALITAS Kolmogorov-Smirnov & Chi-Square Oleh: Roni Saputra, M.Si

2 POPULASI SAMPLING SAMPEL KESIMPULAN PARAMETER :   statistik :  x sd

3 LANGKAH-LANGKAH MENARIK SIMPULAN UJI HIPOTESIS 1.Hipotesis 2.Tentukan  3.Tentukan rumus statistik penguji 4.Hitung rumus statistik penguji 5.Tentukan nilai df/db/dk 6.Lihat nilai tabel 7.Tentukan daerah penolakan 8.Simpulan

4 NOXiFTFT FSFS  F T - F S  1 2 3 4 5 dst Metode Kolmogorov- Smirnov

5 Keterangan : X i =Angka pada data Z=Transformasi dari angka ke notasi pada distribusi normal F T =Probabilitas komulatif normal ; komulatif proporsi luasan kurva normal berdasarkan notasi Zi, dihitung dari luasan kurva mulai ujung kiri kurva sampai dengan titik Z. F S =Probabilitas komulatif empiris (1/data ke n)

6 –Data berskala interval atau ratio (kuantitatif) –Data tunggal / belum dikelompokkan pada tabel distribusi frekuensi –Dapat untuk n besar maupun n kecil. Persyaratan

7 Signifikansi uji, nilai  FT - FS  terbesar dibandingkan dengan nilai tabel Kolmogorov Smirnov. –Jika nilai  FT - FS  terbesar < nilai tabel Kolmogorov Smirnov, maka Ho diterima ; Ha ditolak. –Jika nilai  FT - FS  terbesar ≥ nilai tabel Kolmogorov Smirnov, maka Ho ditolak ; Ha diterima. Siginifikansi

8 Contoh Suatu penelitian tentang berat badan peserta pelatihan kebugaran fisik/jasmani dengan sampel sebanyak 27 orang diambil secara random, didapatkan data sebagai berikut ; 78, 78, 95, 90, 78, 80, 82, 77, 72, 84, 68, 67, 87, 78, 77, 88, 97, 89, 97, 98, 70, 72, 70, 69, 67, 90, 97 kg. Selidikilah dengan  = 5%, apakah data tersebut di atas diambil dari populasi yang berdistribusi normal ?

9 Penyelesaian Hipotesis –Ho : tidak beda dengan populasi normal –Ha : Ada beda populasi normal Level signifikansi (  ) –Nilai  = 5% = 0,05 Rumus Statistik penguji

10 NOXi 167 2 368 469 570 6 772 8 977 1077 1178 1278 1378 1478 1580 1682 1784 1887 1988 2089 2190 2290 2395 2497 2597 2697 2798 Mean 81,2963 SD10,28372

11 NOXiFTFT FSFS  F T - F S  167 2 -1,39020,08230,07410,0082 368-1,29290,09850,11110,0126 469-1,19570,11510,14810,0330 570 6 -1,09850,13570,22220,0865 772 8 -0,90400,18410,29630,1122 977 1077-0,41780,33720,37040,0332 1178 1278 1378 1478-0,32050,37450,51850,1440 1580-0,12610,44830,55560,1073 16820,06840,52790,59260,0647 17840,26290,60260,62960,0270 18870,55460,70880,66670,0421 19880,65190,74220,70370,0385 20890,74910,77340,74070,0327 2190 22900,84640,80230,81480,0125 23951,33260,90820,85190,0563 2497 2597 26971,52700,93700,9630-0,0260 27981,62430,94741,0000-0,0526 Mean 81,2963 SD10,28372

12 Z0,000,010,020,030,040,050,060,070,080,09 0,00,50000,49600,49200,48800,48400,48010,47610,47210,46810,4641 0,10,46020,45620,45220,44830,44430,44040,43640,43250,42860,4247 0,20,42070,41680,41290,40900,40520,40130,39740,39360,38970,3859 0,30,38210,37830,37450,37070,36690,36320,35940,35570,35200,3483 0,40,34460,34090,33720,33360,33000,32640,32280,31920,31560,3121 0,50,30850,30500,30150,29810,29460,29120,28770,28430,28100,2776 0,60,27430,27090,26760,26430,26110,25780,25460,25140,24830,2451 0,70,24200,23890,23580,23270,22960,22660,22360,22060,21770,2148 0,80,21190,20900,20610,20330,20050,19770,19490,19220,18940,1867 0,90,18410,18140,17880,17620,17360,17110,16850,16600,16350,1611 1,00,15870,15620,15390,15150,14920,14690,14460,14230,14010,1379 1,10,13570,13350,13140,12920,12710,12510,12300,12100,11900,1170 1,20,11510,11310,11120,10930,10750,10560,10380,10200,10030,0985 1,30,09680,09510,09340,09180,09010,08850,08690,08530,08380,0823 1,40,08080,07930,07780,07640,07490,07350,07210,07080,06940,0681 1,50,06680,06550,06430,06300,06180,06060,05940,05820,05710,0559 1,60,05480,05370,05260,05160,05050,04950,04850,04750,04650,0455 1,70,04460,04360,04270,04180,04090,04010,03920,03840,03750,0367 1,80,03590,03510,03440,03360,03290,03220,03140,03070,03010,0294 1,90,02870,02810,02740,02680,02620,02560,02500,02440,02390,0233 2,00,02280,02220,02170,02120,02070,02020,01970,01920,01880,0183 2,10,01790,01740,01700,01660,01620,01580,01540,01500,01460,0143 2,20,01390,01360,01320,01290,01250,01220,01190,01160,01130,0110 2,30,01070,01040,01020,00990,00960,00940,00910,00890,00870,0084 2,40,00820,00800,00780,00750,00730,00710,00690,00680,00660,0064 2,50,00620,00600,00590,00570,00550,00540,00520,00510,00490,0048 2,60,00470,00450,00440,00430,00410,00400,00390,00380,00370,0036 2,70,00350,00340,00330,00320,00310,00300,00290,00280,00270,0026 2,80,00260,00250,00240,0023 0,00220,0021 0,00200,0019 2,90,00190,0018 0,00170,0016 0,0015 0,0014 3,00,0013 0,0012 0,0011 0,0010 3,10,00100,0009 0,0008 0,0007 3,20,0007 0,0006 0,0005 3,30,0005 0,0004 0,0003 3,40,0003 0,0002 3,50,0002 3,60,0002 0,0001 3,70,0001 3,80,0001

13 Df/db/dk –Df =  = tidak diperlukan Nilai tabel –Nilai Kuantil Penguji Kolmogorov,  = 0,05 ; N = 27 ;  0,254. Tabel Kolmogorov Smirnov Daerah penolakan –Menggunakan rumus –  0,1440  <  0,2540  ; berarti Ho diterima, Ha ditolak Kesimpulan –Sampel diambil dari populasi normal, pada  = 0,05.

14 Tingkat Signifikansi untuk tes satu sisi N0,1000,0750,0500,0250,010,005 Tingkat Signifikansi untuk tes dua sisi 0,2000,1500,1000,0500,0200,010 10,9000,9250,9500,9750,9900,995 20,6840,7260,7760,8420,9000,929 30,5650,5970,6420,7080,7850,828 40,4940,5250,5640,6240,6890,733 50,4460,4740,5100,5650,6270,669 60,4100,4360,47005210,5770,618 70,3810,4050,4380,4860,5380,577 80,3580,3810,4110,4570,5070,543 90,3390,3600,3880,4320,4800,514 100,3220,3420,3680,4100,4570,490 110,3070,3260,3520,391 0,437 0,468 120,2950,3130,3380,3750,4190,450 130,28403020,3250,3610,4040,433 140,2740,2920,3140,3490,3900,418 150,2660,2830,3040,3380,3770,404 160,2580,2740,2950,3280,3660,392 170,2500,2660,2860,3180,3550,381 180,2440,2590,2780,3090,3460,371 190,2370,2520,2720,3010,3370,363 200,2310,2460,2640,2940,3290,356 210,2260,2590,2870,3210,344 220,2210,2530,2810,3140,337 230,2160,2470,2750,3070,330 240,2120,2420,2690,3010,323 250,2080,220,2380,2640,2950,317 260,2040,2330,2590,2900,311 270,2000,2290,2540,2840,305 280,1970,2250,2500,2790,300 29 0,1930,2210,2460,2750,295 30 0,1900,200,2180,2420,2700,290 31 0,1870,2140,2380,2660,285 32 0,1840,2110,2340,2620,281 33 0,1820,2080,2310,2580,277 34 0,1790,2050,2270,2540,213 35 0,1710,190,2020,2240,2510,269 36 0,1740,1990,2210,2470,265 37 0,1720,1960,2180,2440,262 380,1700,1940,2150,2410,258 39 0,1680,1910,2130,2380,255 40 0,1650,1890,2100,2350,252 250,2080,2380,264 0,295 0,317 300,1900,2180,242 0,270 0,290 350,1770,2020,224 0,251 0,269 400,1650,1890,2100,2350,252 >40

15 Metode Chi-Square atau X 2 Uji Goodness of fit Distribusi Normal, menggunakan pendekatan penjumlahan penyimpangan data observasi tiap kelas dengan nilai yang diharapkan.

16 Rumus X 2 Keterangan : X 2 =Nilai X 2 O i =Nilai observasi E i =Nilai expected / harapan, luasan interval kelas berdasarkan tabel normal dikalikan N (total frekuensi)  p i x N N=Banyaknya angka pada data (total frekuensi)

17 N BATAS INTERVAL KELAS (batas tidak nyata)pi Oi Ei (pi x N) 1 2 3 ds t Keterangan : Xi=Batas tidak nyata interval kelas Z=Transformasi dari angka batas interval kelas ke notasi pada distribusi normal Pi=Luas proporsi kurva normal tiap interval kelas berdasar tabel normal Oi=Nilai observasi Ei=Nilai expected / harapan, luasan interval kelas berdasarkan tabel normal dikalikan N (total frekuensi)  pi x N

18 –Data tersusun berkelompok atau dikelompokkan dalam tabel distribusi frekuensi. –Cocok untuk data dengan banyaknya angka besar ( n > 30 ) –Setiap sel harus terisi, yang kurang dari 5 digabungkan. Persyaratan

19 –Signifikansi uji, nilai X 2 hitung dibandingkan dengan X 2 tabel (Chi-Square). –Jika nilai X 2 hitung < nilai X 2 tabel, maka Ho diterima ; Ha ditolak. –Jika nilai X 2 hitung ≥ nilai X 2 tabel, maka Ho ditolak ; Ha diterima. Signifikansi

20 TINGGI BADAN MASYARAKAT KALIMAS TAHUN 2006 NO.TINGGI BADANJUMLAH 1.140 – 1496 2.150 – 15922 3.160 – 16939 4.170 – 17925 5.180 – 1897 6.190 – 1991 JUMLAH100 Contoh Selidikilah dengan  = 5%, apakah data tersebut di atas berdistribusi normal ?

21 Penyelesaian : Hipotesis –Ho : tidak beda dengan populasi normal –Ha : Ada beda populasi normal Level signifikansi (  ) –Nilai  = = 5% = 0,05 Rumus Statistik penguji

22 N BATAS INTERVAL KELAS (batas tidak nyata)pi Oi Ei (pi x N) 1.139,5 – 149,5-2,49 – -1,530,0064 – 0,0630=0,056665,66 2.149,5 – 159,5-1,53 – -0,560,0630 – 0,2877=0,22472222,47 3.159,5 – 169,5-0,56 – 0,410,2877 – 0,6591=0,37143937,14 4.169,5 – 179,50,41 – 1,370,6591 – 0.9147=0,25562525,56 5.179,5 – 189,51,37 – 2,340,9147 – 0,9904=0,075777,57 6.189,5 – 199,52,34 – 3,300,9904 – 0,9995=0,009110,91 JUMLAH100 Telah dihitung Mean = 165,3 ; Standar deviasi = 10,36

23 Z0,000,010,020,030,040,050,060,070,080,09 0,00,50000,49600,49200,48800,48400,48010,47610,47210,46810,4641 0,10,46020,45620,45220,44830,44430,44040,43640,43250,42860,4247 0,20,42070,41680,41290,40900,40520,40130,39740,39360,38970,3859 0,30,38210,37830,37450,37070,36690,36320,35940,35570,35200,3483 0,40,34460,34090,33720,33360,33000,32640,32280,31920,31560,3121 0,50,30850,30500,30150,29810,29460,29120,28770,28430,28100,2776 0,60,27430,27090,26760,26430,26110,25780,25460,25140,24830,2451 0,70,24200,23890,23580,23270,22960,22660,22360,22060,21770,2148 0,80,21190,20900,20610,20330,20050,19770,19490,19220,18940,1867 0,90,18410,18140,17880,17620,17360,17110,16850,16600,16350,1611 1,00,15870,15620,15390,15150,14920,14690,14460,14230,14010,1379 1,10,13570,13350,13140,12920,12710,12510,12300,12100,11900,1170 1,20,11510,11310,11120,10930,10750,10560,10380,10200,10030,0985 1,30,09680,09510,09340,09180,09010,08850,08690,08530,08380,0823 1,40,08080,07930,07780,07640,07490,07350,07210,07080,06940,0681 1,50,06680,06550,06430,06300,06180,06060,05940,05820,05710,0559 1,60,05480,05370,05260,05160,05050,04950,04850,04750,04650,0455 1,70,04460,04360,04270,04180,04090,04010,03920,03840,03750,0367 1,80,03590,03510,03440,03360,03290,03220,03140,03070,03010,0294 1,90,02870,02810,02740,02680,02620,02560,02500,02440,02390,0233 2,00,02280,02220,02170,02120,02070,02020,01970,01920,01880,0183 2,10,01790,01740,01700,01660,01620,01580,01540,01500,01460,0143 2,20,01390,01360,01320,01290,01250,01220,01190,01160,01130,0110 2,30,01070,01040,01020,00990,00960,00940,00910,00890,00870,0084 2,40,00820,00800,00780,00750,00730,00710,00690,00680,00660,0064 2,50,00620,00600,00590,00570,00550,00540,00520,00510,00490,0048 2,60,00470,00450,00440,00430,00410,00400,00390,00380,00370,0036 2,70,00350,00340,00330,00320,00310,00300,00290,00280,00270,0026 2,80,00260,00250,00240,0023 0,00220,0021 0,00200,0019 2,90,00190,0018 0,00170,0016 0,0015 0,0014 3,00,0013 0,0012 0,0011 0,0010 3,10,00100,0009 0,0008 0,0007 3,20,0007 0,0006 0,0005 3,30,0005 0,0004 0,0003 3,40,0003 0,0002 3,50,0002 3,60,0002 0,0001 3,70,0001 3,80,0001

24

25

26 Df/db/dk Df = ( k – 3 ) = ( 5 – 3 ) = 2 Nilai tabel Nilai tabel X 2 ;  = 0,05 ; df = 2 ; = 5,991. Daerah penolakan Menggunakan gambar Menggunakan rumus  0,1628  <  5,991  ; berarti Ho diterima, Ha ditolak Kesimpulan Sampel diambil dari populasi normal, pada  = 0,05.

27 dfKemungkinan di bawah Ho bahwa X 2 Chi - Square 0,0050,0100,0250,0500,1000,200 17,8796,6355,0243,8412,7061,642 210,5979,2107,3785,9914,6053,219 312,83811,3419,3487,8156,2514,642 414,86013,27711,1439,4887,7795,989 516,75015,08612,83211,0709,2367,289 618,54816,81214,44912,59210,6458,558 720,27818,47516,01314,06712,0179,803 821,95520,09017,53515,50713,36211,030 923,58921,66019,02316,91914,68412,242 1025,18823,20920,48318,30715,98713,442 1126,75724,72521,92019,67517,27514,631 1228,30026,21723,33721,02618,54915,812 1329,81927,68824,73622,36219,81216,985 1431,31929,14126,11923,68521,06418,151 1532,80130,57827,48824,99622,30719,311 1634,26732,00028,84526,29623,54220,465 1735,71833,40930,19127,58724,76921,615 1837,15634,80531,52628,86925,98922,760 1938,58236,19132,85230,14427,20423,900 2039,99737,56634,17031,41028,41225,038 2141,40138,93235,47932,67129,61526,171 2242,79640,28936,78133,92430,81327,301 2344,18141,63838,07635,17232,00728,429 2445,55842,98039,36436,41533,19629,553 2546,92844,31440,64637,65234,38230,675 2648,29045,64241,92338,88535,56331,795 2749,64546,96343,19440,11336,74132,912 2850,99348,27844,46141,33737,91634,027 2952,33649,58845,72242,55739,08735,139 3053,67250,89246,97943,77340,25636,250

28 df 0,001 0,0050,0100,0200,0250,0500,1000,200 0,2500,300 1 10,83 7,8796,6355,415,0243,8412,7061,642 1,321,07 2 13,82 10,5979,2107,827,3785,9914,6053,219 2,772,41 3 16,27 12,83811,3419,849,3487,8156,2514,642 4,113,66 4 18,46 14,86013,27711,6711,1439,4887,7795,989 5,394,88 5 20,52 16,75015,08613,3912,83211,0709,2367,289 6,636,06 6 22,46 18,54816,81215,0314,44912,59210,6458,558 7,847,23 7 24,32 20,27818,47516,6216,01314,06712,0179,803 9,048,38 8 26,12 21,95520,09018,1717,53515,50713,36211,030 10,229,52 9 27,88 23,58921,66019,6819,02316,91914,68412,242 11,3910,66 10 29,59 25,18823,20921,1620,48318,30715,98713,442 12,5511,78 11 31,26 26,75724,72522,6221,92019,67517,27514,631 13,7012,90 12 32,91 28,30026,21724,0523,33721,02618,54915,812 14,8514,01 13 34,53 29,81927,68825,4724,73622,36219,81216,985 15,9815,12 14 36,12 31,31929,14126,8726,11923,68521,06418,151 17,1216,22 15 37,70 32,80130,57828,2627,48824,99622,30719,311 18,2517,32 16 39,29 34,26732,00029,6328,84526,29623,54220,465 19,3718,42 17 40,75 35,71833,40931,0030,19127,58724,76921,615 20,4919,51 18 42,31 37,15634,80532,2531,52628,86925,98922,760 21,6020,60 19 43,82 38,58236,19133,6932,85230,14427,20423,900 22,7221,69 20 45,32 39,99737,56635,0234,17031,41028,41225,038 23,8322,78 21 46,80 41,40138,93236,3435,47932,67129,61526,171 24,9323,86 22 48,27 42,79640,28937,6636,78133,92430,81327,301 26,0424,94 23 49,73 44,18141,63838,9738,07635,17232,00728,429 27,1426,02 24 51,18 45,55842,98040,2739,36436,41533,19629,553 28,2427,10 25 52,62 46,92844,31441,5740,64637,65234,38230,675 29,3428,17 26 54,05 48,29045,64242,8641,92338,88535,56331,795 30,4329,25 27 55,48 49,64546,96344,1443,19440,11336,74132,912 31,5330,32 28 56,89 50,99348,27845,4244,46141,33737,91634,027 32,6232,39 29 58,30 52,33649,58846,6945,72242,55739,08735,139 33,7132,46 30 59,70 53,67250,89247,9646,97943,77340,25636,250 34,8033,53 4066,7763,6959,3455,7651,80 45,62 5079,4976,1571,4267,5063,17 56,33 6091,9588,3883,3079,0874,40 66,98 70104,22100,4295,0290,5385,53 77,58 80116,32112,33106,63101,8896,58 88,13 90128,30124,12118,14113,14107,56 98,64 100140,17135,81129,56124,34118,50 10,9,14


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