PENGUJIAN DISTRIBUSI

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PENGUJIAN DISTRIBUSI

1. Data Distribusi Binomial

Adapun untuk langkah-langkah untuk membangkitkan data distribusi

binomial terdapat pada flowchart berikut. Adapun ketentuan-ketentuan yang

diberikan pada data distribusi binomial adalah:

1. Jumlah data (generate) = 150

2. Banyak percobaan (number of trials) = 30

3. Peluang kejadian (event probability) = 0,5

Mulai

1. Klik Calc padaMenu Bar

2. Pilih Random3. Klik Binomial

Input:Generate : 150

Number of Trials : 30Event Probability : 0.5

Proses Random

Print :150

Selesai

Gambar 1. Flowchart Pembangkitan Bilangan Random Distribusi Binomial



Tabel 1. Data Bilangan Random Distribusi Binomial

No Hasil No Hasil No Hasil No Hasil No Hasil1 6 16 11 31 13 46 14 61 15

2 9 17 11 32 13 47 14 62 15

3 9 18 11 33 13 48 14 63 15

4 10 19 12 34 13 49 14 64 15

5 10 20 12 35 13 50 14 65 15

6 10 21 12 36 13 51 14 66 15

7 10 22 12 37 13 52 14 67 15

8 10 23 12 38 13 53 14 68 15

9 10 24 12 39 13 54 14 69 1510 10 25 12 40 14 55 14 70 15

11 10 26 12 41 14 56 15 71 15

12 10 27 12 42 14 57 15 72 15

13 11 28 12 43 14 58 15 73 15

14 11 29 13 44 14 59 15 74 15

15 11 30 13 45 14 60 15 75 15

76 15 91 16 106 17 121 17 136 19

77 15 92 16 107 17 122 17 137 19

78 15 93 16 108 17 123 17 138 19

79 15 94 16 109 17 124 17 139 19

80 16 95 16 110 17 125 18 140 19

81 16 96 16 111 17 126 18 141 19

82 16 97 16 112 17 127 18 142 19

83 16 98 16 113 17 128 18 143 20

84 16 99 16 114 17 129 18 144 20

85 16 100 16 115 17 130 18 145 20

86 16 101 17 116 17 131 18 146 21

87 16 102 17 117 17 132 18 147 21

88 16 103 17 118 17 133 18 148 21

89 16 104 17 119 17 134 18 149 21

90 16 105 17 120 17 135 19 150 22



1.1. Pengujian Distribusi Binomial

Adapun data bilangan random data distribusi binomial dapat dilihat pada

Tabel 2.

Tabel 2. Bilangan Random Data Distribusi Binomial

Bilangan Random Data Distribusi Binomial

6 11 13 14 15 15 16 17 17 19

9 11 13 14 15 15 16 17 17 19

9 11 13 14 15 15 16 17 17 19

10 12 13 14 15 15 16 17 17 19

10 12 13 14 15 16 16 17 18 1910 12 13 14 15 16 16 17 18 19

10 12 13 14 15 16 16 17 18 19

10 12 13 14 15 16 16 17 18 20

10 12 13 14 15 16 16 17 18 20

10 12 14 14 15 16 16 17 18 20

10 12 14 15 15 16 17 17 18 21

10 12 14 15 15 16 17 17 18 21

11 12 14 15 15 16 17 17 18 21

11 13 14 15 15 16 17 17 18 21

Langkah-langkah pengujian yang dilakukan adalah sebagai berikut:

1. Rumusan Hipotesis

Ho : data berdistribusi binomial

Hi : data tidak berdistribusi binomial

2. Asumsikan α = 0,05

3. Tentukan nilai max, min, range, banyak kelas, dan panjang kelas

Data maksimum = 22

Data minimum = 6

Setelah didapat nilai dari Max dan Min maka data tersebut dirangkum di

dalam sebuah tabel frekuensi. Tabel frekuensi dapat dilhat seperti Tabel 3 berikut

:



Tabel 3. Tabel Frekuensi Data Pengujian Distribusi Binomial

No. Kelas Frekuensi (Oi) Xi Xi.Oi

1 6 1 6 6

2 9 2 9 18

3 10 9 10 90

4 11 6 11 66

5 12 10 12 120

6 13 11 13 143

7 14 16 14 224

8 15 24 15 360

9 16 21 16 336

10 17 24 17 408

11 18 10 18 180

12 19 8 19 152

13 20 3 20 60

14 21 4 21 84

15 22 1 22 22

Jumlah 150 2263

4. Menghitung peluang (P) dari masing-masing kelas dengan menggunakan

software Minitab agar lebih mudah, sehingga akan didapatlkan hasil seperti

Tabel 4.



Tabel 4. Perhitungan Frekuensi Peluang Data Distribusi Binomial

No X Frekuensi (Oi) Xi.Oi Luas Kurva

1 6 1 6 0.000552996

2 9 2 18 0.013324572

3 10 9 90 0.027981601

4 11 6 66 0.050875638

5 12 10 120 0.080553093

6 13 11 143 0.111535052

7 14 16 224 0.13543542

8 15 24 360 0.144464448

9 16 21 336 0.13543542

10 17 24 408 0.111535052

11 18 10 180 0.080553093

12 19 8 152 0.050875638

13 20 3 60 0.027981601

14 21 4 84 0.013324572

15 22 1 22 0.005450961

Jumlah 150 2269 1,0000

5. Hitung nilai ekspektasinya (ei) untuk setiap interval.

ei = luas ke-i x n

misalnya: e1 = 0,000552996 x 150 = 0,082949409

Sehingga akan didapatkan hasil dalam tabel berikut ini.



Tabel 5. Perhitungan Nilai Ekspektasi (ei) Binomial

No X Frekuensi (Oi) Xi.Oi Luas Kurva Ei

1 6 1 6 0.000552996 0.082949409

3 9 2 18 0.013324572 1.998685766

4 10 9 90 0.027981601 4.197240109

5 11 6 66 0.050875638 7.631345652

6 12 10 120 0.080553093 12.08296395

7 13 11 143 0.111535052 16.73025778

8 14 16 224 0.13543542 20.31531301

9 15 24 360 0.144464448 21.66966721

10 16 21 336 0.13543542 20.31531301

11 17 24 408 0.111535052 16.73025778

12 18 10 180 0.080553093 12.08296395

13 19 8 152 0.050875638 7.631345652

14 20 3 60 0.027981601 4.197240109

15 21 4 84 0.013324572 1.998685766

16 22 1 22 0.005450961 0.817644177

Jumlah 150 2269 1 150

6. Setelah didapatkan nilai ei dan Oi maka dapat dihitung nilai χ 2

hitung

χ 2

hitung

Nilai dari χ 2

hitung untuk masing-masing kelas dapat dilihat pada Tabel 6.



Tabel 6. Tabel Nilai χ 2

hitung Data Pengujian Distribusi Binomial

Kelas Frekuensi (Oi) Xi Xi.Oi Luas Kurva Ei χ 2

6 1 6 6 0.000552996 0.082949409 10.13849036

9 2 9 18 0.013324572 1.998685766 8.64173E-07

10 9 10 90 0.027981601 4.197240109 5.495635698

11 6 11 66 0.050875638 7.631345652 0.348731241

12 10 12 120 0.080553093 12.08296395 0.359079017

13 11 13 143 0.111535052 16.73025778 1.962662776

14 16 14 224 0.13543542 20.31531301 0.916644818

15 24 15 360 0.144464448 21.66966721 0.25060149

16 21 16 336 0.13543542 20.31531301 0.023076005

17 24 17 408 0.111535052 16.73025778 3.158896457

18 10 18 180 0.080553093 12.08296395 0.359079017

19 8 19 152 0.050875638 7.631345652 0.017808921

20 3 20 60 0.027981601 4.197240109 0.341506285

21 4 21 84 0.013324572 1.998685766 2.003946159

22 1 22 22 0.005450961 0.817644177 0.04067007

Jumlah 150 2269 1 150 25.41682918

7. Hitung χ 2

tabel

Jumlah kelas (k) = 15

V (derajat bebas) = 15 – 3 = 12

α = 0,05

Sehingga nilai χ 2

tabel yang diperoleh = 21.02607

8. Kesimpulan: Ho ditolak (χ 2

hitung = 25,416829 > χ 2

tabel = 21.02607) atau dapat

disimpulkan bahwa data tersebut tidak berdistribusi binomial.



Adapun gambar kurva distribusi binomial adalah sebagai berikut :

Gambar 2. Kurva Distribusi Binomial

9. Kurtosis dan Skewness

Adapun data yang digunakan dalam perhitungan Kurtosis dan skewness dapat

dilihat pada tabel berikut :

Tabel 7. Data Perhitungan Kurtosis dan Skewness

No Xi X Xi-X S ((Xi-X)/S)3

((Xi-X)/S)4

1 6 15.1267 -9.1267 2.8668 -32.264737 102.715861

2 9 15.1267 -6.1267 2.8668 -9.760326 20.858629

3 9 15.1267 -6.1267 2.8668 -9.760326 20.858629

4 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

5 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

6 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

7 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

8 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

9 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

F r e k u e n s i

Xi

Oi

ei



Tabel 7. Data Perhitungan Kurtosis dan Skewness (Lanjutan)

No Xi X Xi-X S ((Xi-X)/S)3 ((Xi-X)/S)4

10 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

11 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

12 10 15.1267 -5.1267 2.8668 -5.718693 10.226548

13 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

14 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

15 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

16 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

17 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

18 11 15.1267 -4.1267 2.8668 -2.982564 4.293251

19 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

20 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

21 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

22 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

23 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

24 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

25 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

26 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

27 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

28 12 15.1267 -3.1267 2.8668 -1.297288 1.414865

29 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

30 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

31 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

32 13 15.1267 -2.1267 2.8668 -0.408216 0.302821





33 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

34 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

35 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

36 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

37 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

38 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

39 13 15.1267 -2.1267 2.8668 -0.408216 0.302821

40 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

41 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

42 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

43 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

44 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

45 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

46 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

47 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

48 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

49 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

50 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

51 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

52 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

53 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

54 14 15.1267 -1.1267 2.8668 -0.060698 0.023855

55 14 15.1267 -1.1267 2.8668 -0.060698 0.023855





56 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

57 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

58 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

59 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

60 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

61 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

62 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

63 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

64 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

65 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

66 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

67 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

68 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

69 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

70 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

71 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

72 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

73 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

74 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

75 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

76 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

77 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

78 15 15.1267 -0.1267 2.8668 -0.000086 0.000004





79 15 15.1267 -0.1267 2.8668 -0.000086 0.000004

80 16 15.1267 0.8733 2.8668 0.028270 0.008612

81 16 15.1267 0.8733 2.8668 0.028270 0.008612

82 16 15.1267 0.8733 2.8668 0.028270 0.008612

83 16 15.1267 0.8733 2.8668 0.028270 0.008612

84 16 15.1267 0.8733 2.8668 0.028270 0.008612

85 16 15.1267 0.8733 2.8668 0.028270 0.008612

86 16 15.1267 0.8733 2.8668 0.028270 0.008612

87 16 15.1267 0.8733 2.8668 0.028270 0.008612

88 16 15.1267 0.8733 2.8668 0.028270 0.008612

89 16 15.1267 0.8733 2.8668 0.028270 0.008612

90 16 15.1267 0.8733 2.8668 0.028270 0.008612

91 16 15.1267 0.8733 2.8668 0.028270 0.008612

92 16 15.1267 0.8733 2.8668 0.028270 0.008612

93 16 15.1267 0.8733 2.8668 0.028270 0.008612

94 16 15.1267 0.8733 2.8668 0.028270 0.008612

95 16 15.1267 0.8733 2.8668 0.028270 0.008612

96 16 15.1267 0.8733 2.8668 0.028270 0.008612

97 16 15.1267 0.8733 2.8668 0.028270 0.008612

98 16 15.1267 0.8733 2.8668 0.028270 0.008612

99 16 15.1267 0.8733 2.8668 0.028270 0.008612

100 16 15.1267 0.8733 2.8668 0.028270 0.008612

101 17 15.1267 1.8733 2.8668 0.279021 0.182326





102 17 15.1267 1.8733 2.8668 0.279021 0.182326

103 17 15.1267 1.8733 2.8668 0.279021 0.182326

104 17 15.1267 1.8733 2.8668 0.279021 0.182326

105 17 15.1267 1.8733 2.8668 0.279021 0.182326

106 17 15.1267 1.8733 2.8668 0.279021 0.182326

107 17 15.1267 1.8733 2.8668 0.279021 0.182326

108 17 15.1267 1.8733 2.8668 0.279021 0.182326

109 17 15.1267 1.8733 2.8668 0.279021 0.182326

110 17 15.1267 1.8733 2.8668 0.279021 0.182326

111 17 15.1267 1.8733 2.8668 0.279021 0.182326

112 17 15.1267 1.8733 2.8668 0.279021 0.182326

113 17 15.1267 1.8733 2.8668 0.279021 0.182326

114 17 15.1267 1.8733 2.8668 0.279021 0.182326

115 17 15.1267 1.8733 2.8668 0.279021 0.182326

116 17 15.1267 1.8733 2.8668 0.279021 0.182326

117 17 15.1267 1.8733 2.8668 0.279021 0.182326

118 17 15.1267 1.8733 2.8668 0.279021 0.182326

119 17 15.1267 1.8733 2.8668 0.279021 0.182326

120 17 15.1267 1.8733 2.8668 0.279021 0.182326

121 17 15.1267 1.8733 2.8668 0.279021 0.182326

122 17 15.1267 1.8733 2.8668 0.279021 0.182326

123 17 15.1267 1.8733 2.8668 0.279021 0.182326

124 17 15.1267 1.8733 2.8668 0.279021 0.182326





125 18 15.1267 2.8733 2.8668 1.006815 1.009097

126 18 15.1267 2.8733 2.8668 1.006815 1.009097

127 18 15.1267 2.8733 2.8668 1.006815 1.009097

128 18 15.1267 2.8733 2.8668 1.006815 1.009097

129 18 15.1267 2.8733 2.8668 1.006815 1.009097

130 18 15.1267 2.8733 2.8668 1.006815 1.009097

131 18 15.1267 2.8733 2.8668 1.006815 1.009097

132 18 15.1267 2.8733 2.8668 1.006815 1.009097

133 18 15.1267 2.8733 2.8668 1.006815 1.009097

134 18 15.1267 2.8733 2.8668 1.006815 1.009097

135 19 15.1267 3.8733 2.8668 2.466302 3.332178

136 19 15.1267 3.8733 2.8668 2.466302 3.332178

137 19 15.1267 3.8733 2.8668 2.466302 3.332178

138 19 15.1267 3.8733 2.8668 2.466302 3.332178

139 19 15.1267 3.8733 2.8668 2.466302 3.332178

140 19 15.1267 3.8733 2.8668 2.466302 3.332178

141 19 15.1267 3.8733 2.8668 2.466302 3.332178

142 19 15.1267 3.8733 2.8668 2.466302 3.332178

143 20 15.1267 4.8733 2.8668 4.912131 8.350130

144 20 15.1267 4.8733 2.8668 4.912131 8.350130

145 20 15.1267 4.8733 2.8668 4.912131 8.350130

146 21 15.1267 5.8733 2.8668 8.598952 17.616813

147 21 15.1267 5.8733 2.8668 8.598952 17.616813





148 21 15.1267 5.8733 2.8668 8.598952 17.616813

149 21 15.1267 5.8733 2.8668 8.598952 17.616813

150 22 15.1267 6.8733 2.8668 13.781414 33.041395

Jumlah -39.583155 449.957129

)583155,39()148)(149(

150

)2)(1(

3

Skewness

s

X Xi

nn

nSkewness

= -0.269249

√

SES = 0.19803837

Rasio Skewness = iai

Rasio Skewness =0,0



Rasio Skewness = -1,359578

Karena nilai rasio skewness -1,359578 dan berada diantara -1.96 dan 1.96,

berarti kurva distribusi binomial tersebut normal berdasarkan kemencengan

kurvanya.

)147)(148(

)149(3

)957129,449()147)(148)(149(

)151(150

)3)(2(

)1(3

)3)(2)(1(

)1(

2

24

Kurtosis

nn

n

s

X Xi

nnn

nnKurtosis

= 0.082576

√

SEK = 0,393583

Rasio Kurtosis = iai

Rasio Kurtosis =0,5

Rasio Kurtosis = 0,2098057

Karena nilai rasio kurtosis 0,2098057 dan berada diantara -1.96 dan 1.96, berarti

kurva distribusi binomial tersebut normal berdasarkan kecondongan kurvanya.



Berdasarkan perhitungan dengan menggunakan software Minitab 16, maka

dapat dilihat gambar grafik beserta hasil perhitungannya di bawah ini:

2118151296

Median

Mean

16.0015.7515.5015.2515.0014.7514.50

1st Quartile 13.000

M edian 15.0003rd Quartile 17.000

Maximum 22.000

14.664 15.589

15.000 16.000

2.575 3.234

A -Squared 1.30

P -V alue < 0.005

Mean 15.127

StDev 2.867

V ariance 8.219

Skewness -0.269249

Kurtosis 0.082576

N 150

M inimum 6.000

A nderson-Darling N ormality Test

95% C onfidence Interv al for Mean

95% C onfidence Interval for Median

95% C onfidence Interv al for StDev95 % Confidence Intervals

Summary for Binomial

Gambar 3. Kurva Skewness dan Kurtosis Distribusi Binomial

Keterangan: karena nilai skewness dan kurtosis di antara ± 1.96, maka

bentuk grafik normalitasnya mengumpul di tengah (berbentuk seperti lonceng).

2. Distribusi Gamma

Adapun untuk langkah-langkah untuk membangkitkan data distribusi

gamma terdapat pada flowchart berikut. Ketentuan-ketentuan yang diberikan

pada data distribusi gamma adalah:

1. Jumlah data (generate) = 150

2. Shape parameter = 7

3. Scale parameter = 5

4. Threshold parameter = 0,0



Mulai

1. Klik Calc padaMenu Bar

2. Pilih Random3. Klik Gamma

Input:

Generate : 150 Shape Parameter: 7

Scale Parameter: 5 Threshold

Parameter : 0,0

Proses Random

Print :150

Selesai

Gambar 4. Flowchart Pembangkitan Data Distribusi Gamma

Tabel 8. Data Bilangan Random Distribusi Gamma

No Hasil No Hasil No Hasil No Hasil No Hasil

1 10.9062 16 19.2220 31 23.3011 46 27.4442 61 30.5599

2 11.9965 17 19.9879 32 23.5032 47 27.6178 62 30.9813

3 12.2602 18 20.3335 33 23.6701 48 27.7520 63 31.1297

4 12.6182 19 20.4310 34 23.6773 49 28.1644 64 31.2004

5 15.1994 20 20.6145 35 23.9103 50 28.4409 65 31.3167

6 15.9978 21 21.2667 36 24.3371 51 28.7590 66 31.4351

7 16.0645 22 21.9321 37 24.3732 52 28.9346 67 31.5616

8 16.9282 23 22.1904 38 24.8321 53 29.4972 68 31.7142

9 16.9821 24 22.6218 39 24.9279 54 29.5745 69 31.8096

10 17.2982 25 22.6561 40 26.1507 55 29.7250 70 31.9729

11 17.4798 26 22.7865 41 26.2277 56 29.7989 71 32.1008

12 17.5922 27 22.9607 42 26.3328 57 29.9159 72 32.1543

13 17.7966 28 22.9789 43 27.0954 58 30.1183 73 32.4062

14 17.9416 29 23.2518 44 27.2112 59 30.1642 74 32.4369

15 19.1903 30 23.2825 45 27.3194 60 30.4446 75 32.5840



Tabel 8. Data Bilangan Random Distribusi Gamma (Lanjutan)

No Hasil No Hasil No Hasil No Hasil No Hasil76 33.0821 91 35.8414 106 40.2899 121 45.4045 136 53.6507

77 33.1069 92 36.6770 107 40.4862 122 45.7913 137 54.0913

78 33.7099 93 37.0458 108 41.5977 123 45.9445 138 54.1004

79 33.8580 94 37.5532 109 42.0100 124 46.3361 139 54.4601

80 33.9151 95 37.8077 110 42.4898 125 46.5509 140 54.7571

81 34.2309 96 38.3755 111 42.8176 126 47.1999 141 56.7374

82 34.3364 97 38.3774 112 43.9733 127 47.3150 142 57.2407

83 34.4234 98 38.4222 113 44.0152 128 47.7343 143 57.5343

84 34.5519 99 38.6773 114 44.3554 129 48.2174 144 58.204685 34.8480 100 38.7437 115 44.3898 130 48.8687 145 59.0535

86 35.0600 101 38.8017 116 44.5588 131 50.2410 146 59.4449

87 35.0999 102 39.5681 117 44.6934 132 50.4061 147 63.3753

88 35.3198 103 39.5773 118 44.8671 133 51.0824 148 63.3902

89 35.8126 104 39.9896 119 45.0164 134 52.5999 149 67.1089

90 35.8377 105 40.0168 120 45.3419 135 53.2459 150 78.0758

2.1. Pengujian Distribusi Gamma

Adapun data bilangan bilangan random data distribusi gamma dapat

dilihat pada Tabel 9.

Tabel 9. Bilangan Random Data Distribusi Gamma

Data Distribusi Gamma

10.9062 19.2220 23.3011 27.4442 30.5599 33.0821 35.8414 40.2899 45.4045 53.6507

11.9965 19.9879 23.5032 27.6178 30.9813 33.1069 36.6770 40.4862 45.7913 54.0913

12.2602 20.3335 23.6701 27.7520 31.1297 33.7099 37.0458 41.5977 45.9445 54.1004

12.6182 20.4310 23.6773 28.1644 31.2004 33.8580 37.5532 42.0100 46.3361 54.4601

15.1994 20.6145 23.9103 28.4409 31.3167 33.9151 37.8077 42.4898 46.5509 54.7571

15.9978 21.2667 24.3371 28.7590 31.4351 34.2309 38.3755 42.8176 47.1999 56.7374

16.0645 21.9321 24.3732 28.9346 31.5616 34.3364 38.3774 43.9733 47.3150 57.2407

16.9282 22.1904 24.8321 29.4972 31.7142 34.4234 38.4222 44.0152 47.7343 57.5343

16.9821 22.6218 24.9279 29.5745 31.8096 34.5519 38.6773 44.3554 48.2174 58.2046



Tabel 9. Bilangan Random Data Distribusi Gamma (Lanjutan)

Data Distribusi Gamma

17.2982 22.6561 26.1507 29.7250 31.9729 34.8480 38.7437 44.3898 48.8687 59.0535

17.4798 22.7865 26.2277 29.7989 32.1008 35.0600 38.8017 44.5588 50.2410 59.4449

17.5922 22.9607 26.3328 29.9159 32.1543 35.0999 39.5681 44.6934 50.4061 63.3753

17.7966 22.9789 27.0954 30.1183 32.4062 35.3198 39.5773 44.8671 51.0824 63.3902

17.9416 23.2518 27.2112 30.1642 32.4369 35.8126 39.9896 45.0164 52.5999 67.1089

19.1903 23.2825 27.3194 30.4446 32.5840 35.8377 40.0168 45.3419 53.2459 78.0758

Langkah-langkah pengujian yang dilakukan adalah sebagai berikut:

1. Rumusan Normal

Ho : data berdistribusi gamma

Hi : data tidak berdistribusi gamma

2. Asumsikan α = 0,05

3. Tentukan nilai max, min, range, banyak kelas, dan panjang kelas

Data maksimum = 78,0758

Data minimum = 10,9062

R = Data max – Data min = 78,0758 – 10,9062 = 67,1696

K = 1 + 3,3 log (N)

K = 1 + 3,3 log 150 = 8,1811 9

2103,8

1811,8

0758,78

I

K

R I

Setelah didapat nilai dari Max, Min, Range, jumlah kelas dan panjang

kelas maka data tersebut dirangkum di dalam sebuah tabel frekuensi. Tabel

frekuensi dapat dilhat seperti Tabel 10.



Tabel 10. Tabel Frekuensi Data Pengujian Distribusi Gamma

Interval BKB BKA Xi Frekuensi (Oi) Xi.Oi

10.9062 - 19.1164 -∞ 19.11645 15.0113 14 210.1582

19.1165 - 27.3267 19.11645 27.32675 23.2216 31 719.8696

27.3268 - 35.5370 27.32675 35.53705 31.4319 43 1351.5717

35.5371 - 43.7473 35.53705 43.74735 39.6422 23 911.7706

43.7474 - 51.9576 43.74735 51.95765 47.8525 22 1052.7550

51.9577 - 60.1679 51.95765 60.16795 56.0628 13 728.8164

60.1680 - 68.3782 60.16795 68.37825 64.2731 3 192.8193

68.3783 - 76.5885 68.37825 76.58855 72.4834 0 0.0000

76.5886 - 84.7988 76.58855 ∞ 80.6937 1 80.6937

Jumlah 150 5248.4545

4. Hitung nilai rata-rata dari tabel distribusi frekuensi.

io

Xioi

.

98969,34

150

4545,5248

5. Menghitung peluang (P) dari masing-masing kelas dengan menggunakan

software Minitab agar lebih mudah, sehingga akan didapatlkan hasil seperti

Tabel 11.



Tabel 11. Perhitungan Frekuensi Peluang Data Distribusi Gamma

Interval BKB BKA Oi P(X<BKB) P(X<BKA)Luas

Kurva

10.9062 - 19.1165 -∞ 19.11645 14 0 0.093086 0.093086

19.1166 - 27.3268 19.11645 27.32675 31 0.093086 0.308529 0.215443

27.3269 - 35.5372 27.32675 35.53705 43 0.308529 0.566167 0.257639

35.5373 - 43.7475 35.53705 43.74735 23 0.566167 0.769437 0.203270

43.7476 - 51.9578 43.74735 51.95765 22 0.769437 0.892678 0.123240

51.9579 - 60.1682 51.95765 60.16795 13 0.892678 0.955026 0.062349

60.1683 - 68.3785 60.16795 68.37825 3 0.955026 0.982675 0.027649

68.3786 - 76.5888 68.37825 76.58855 0 0.982675 0.993767 0.011092

76.5889 - 84.7992 76.58855 ∞ 1 0.993767 1 0.006233

Jumlah 150 1.0000

6. Hitung nilai ekspektasinya (ei) untuk setiap interval.

ei = luas ke-i x n

misalnya: e1 = 0,093087 x 150 = 13,96310

Sehingga akan didapatkan hasil dalam tabel berikut ini.

Tabel 12. Perhitungan Nilai Ekspektasi (ei) Gamma

Interval Oi ei

10.9062 - 19.1164 14 13.96288

19.1165 - 27.3267 31 32.31642

27.3268 - 35.5370 43 38.64582

35.5371 - 43.7473 23 30.49049

43.7474 - 51.9576 22 18.48602



Tabel 12. Perhitungan Nilai Ekspektasi (ei) Gamma (Lanjutan)

Interval Oi ei

51.9577 - 60.1679 13 9.35233

60.1680 - 68.3782 3 4.14729

68.3783 - 76.5885 0 1.66385

76.5886 - 84.7988 1 0.93489

Jumlah 150 150

Karena terdapat nilai ei < 5 maka akan dilakukan penggabungan kelas ke

interval terdekat dengan nilai ei tersebut. Maka tabel nilai ei menjadi:

Tabel 13. Penggabungan Nilai Ekspektasi (ei) Gamma

Interval Oi ei

10.9062 - 19.1165 14 13.96288

19.1166 - 27.3268 31 32.31642

27.3269 - 35.5372 43 38.64582

35.5373 - 43.7475 23 30.49049

43.7476 - 51.9578 22 18.48602

51.9579 - 60.1682 13 9.35233

60.1683 - 68.3785 3 4.14729

68.3786 - 76.5888 0 1.66385

76.5889 - 84.7992 1 0.93489

7. Setelah didapatkan nilai ei dan Oi maka dapat dihitung nilai χ 2

hitung

χ 2

hitung

Nilai dari χ 2hitung untuk masing-masing kelas dapat dilihat pada Tabel 14.



Tabel 14. Tabel Nilai χ 2

hitungData Pengujian Distribusi Gamma

Interval Oi Ei χ

10.9062 - 19.1165 14 13.96288 0.000099

19.1166 - 27.3268 31 32.31642 0.053624721

27.3269 - 35.5372 43 38.64582 0.49058021

35.5373 - 43.7475 23 30.49049 1.840162271

43.7476 - 51.9578 22 18.48602 0.667966133

51.9579 - 60.1682 13 9.35233 1.422689643

60.1683 - 84.7992 4 6.74603 1.117796443

Jumlah 5.5929

8. Hitung χ 2

tabel

Jumlah kelas (k) = 7

V (derajat bebas) = 7 – 4 = 3

α = 0,05

Sehingga nilai χ 2

tabel yang diperoleh = 7,81473

9. Kesimpulan: Ho diterima (χ 2

hitung = 5,5929 < χ 2

tabel = 7,81473) atau dapat

disimpulkan bahwa data tersebut berdistribusi gamma.

Adapun gambar kurva distribusi gamma adalah sebagai berikut :



Gambar 5. Kurva Distribusi Gamma

10. Kurtosis dan Skewness

Adapun data yang digunakan dalam perhitungan Kurtosis dan skewness dapat

dilihat pada tabel berikut :

Tabel 4.15. Data Perhitungan Kurtosis dan Skewness

No Xi X Xi-X S ((Xi-X)/S)

((Xi-X)/S)

1 10.9062 34.753908 -23.8477 12.726871 -6.579196 12.328126

2 11.9965 34.753908 -22.7574 12.726871 -5.717448 10.223579

3 12.2602 34.753908 -22.4937 12.726871 -5.521036 9.758000

4 12.6182 34.753908 -22.1357 12.726871 -5.261563 9.151376

5 15.1994 34.753908 -19.5545 12.726871 -3.627231 5.573144

6 15.9978 34.753908 -18.7561 12.726871 -3.200830 4.717191

7 16.0645 34.753908 -18.6894 12.726871 -3.166829 4.650497

8 16.9282 34.753908 -17.8257 12.726871 -2.747745 3.848592

9 16.9821 34.753908 -17.7718 12.726871 -2.722897 3.802259

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7

F r e k u e n s i

Kelas

Oi

ei



Tabel 4.15. Data Perhitungan Kurtosis dan Skewness (Lanjutan)


10 17.2982 34.753908 -17.4557 12.726871 -2.580183 3.538892

11 17.4798 34.753908 -17.2741 12.726871 -2.500467 3.393866

12 17.5922 34.753908 -17.1617 12.726871 -2.451962 3.306371

13 17.7966 34.753908 -16.9573 12.726871 -2.365420 3.151697

14 17.9416 34.753908 -16.8123 12.726871 -2.305237 3.045235

15 19.1903 34.753908 -15.5636 12.726871 -1.828796 2.236422

16 19.2220 34.753908 -15.5319 12.726871 -1.817647 2.218261

17 19.9879 34.753908 -14.7660 12.726871 -1.561798 1.812034

18 20.3335 34.753908 -14.4205 12.726871 -1.454696 1.648274

19 20.4310 34.753908 -14.3229 12.726871 -1.425382 1.604138

20 20.6145 34.753908 -14.1394 12.726871 -1.371292 1.523491

21 21.2667 34.753908 -13.4872 12.726871 -1.190147 1.261248

22 21.9321 34.753908 -12.8218 12.726871 -1.022555 1.030186

23 22.1904 34.753908 -12.5635 12.726871 -0.961984 0.949637

24 22.6218 34.753908 -12.1321 12.726871 -0.866260 0.825780

25 22.6561 34.753908 -12.0978 12.726871 -0.858923 0.816467

26 22.7865 34.753908 -11.9674 12.726871 -0.831448 0.781832

27 22.9607 34.753908 -11.7932 12.726871 -0.795658 0.737285

28 22.9789 34.753908 -11.7750 12.726871 -0.791978 0.732742

29 23.2518 34.753908 -11.5021 12.726871 -0.738185 0.667145

30 23.2825 34.753908 -11.4714 12.726871 -0.732301 0.660064

31 23.3011 34.753908 -11.4528 12.726871 -0.728735 0.655783

32 23.5032 34.753908 -11.2507 12.726871 -0.690843 0.610716





33 23.6701 34.753908 -11.0838 12.726871 -0.660546 0.575269

34 23.6773 34.753908 -11.0766 12.726871 -0.659265 0.573782

35 23.9103 34.753908 -10.8436 12.726871 -0.618521 0.526995

36 24.3371 34.753908 -10.4168 12.726871 -0.548319 0.448791

37 24.3732 34.753908 -10.3807 12.726871 -0.542641 0.442605

38 24.8321 34.753908 -9.9218 12.726871 -0.473810 0.369379

39 24.9279 34.753908 -9.8260 12.726871 -0.460220 0.355321

40 26.1507 34.753908 -8.6032 12.726871 -0.308902 0.208815

41 26.2277 34.753908 -8.5262 12.726871 -0.300674 0.201432

42 26.3328 34.753908 -8.4211 12.726871 -0.289692 0.191682

43 27.0954 34.753908 -7.6585 12.726871 -0.217902 0.131124

44 27.2112 34.753908 -7.5427 12.726871 -0.208172 0.123376

45 27.3194 34.753908 -7.4345 12.726871 -0.199336 0.116443

46 27.4442 34.753908 -7.3097 12.726871 -0.189467 0.108820

47 27.6178 34.753908 -7.1361 12.726871 -0.176287 0.098846

48 27.7520 34.753908 -7.0019 12.726871 -0.166525 0.091616

49 28.1644 34.753908 -6.5896 12.726871 -0.138805 0.071868

50 28.4409 34.753908 -6.3130 12.726871 -0.122049 0.060540

51 28.7590 34.753908 -5.9949 12.726871 -0.104514 0.049230

52 28.9346 34.753908 -5.8194 12.726871 -0.095600 0.043713

53 29.4972 34.753908 -5.2567 12.726871 -0.070464 0.029104

54 29.5745 34.753908 -5.1794 12.726871 -0.067402 0.027430

55 29.7250 34.753908 -5.0289 12.726871 -0.061694 0.024378





56 29.7989 34.753908 -4.9550 12.726871 -0.059015 0.022976

57 29.9159 34.753908 -4.8380 12.726871 -0.054934 0.020883

58 30.1183 34.753908 -4.6356 12.726871 -0.048323 0.017601

59 30.1642 34.753908 -4.5897 12.726871 -0.046902 0.016914

60 30.4446 34.753908 -4.3094 12.726871 -0.038821 0.013145

61 30.5599 34.753908 -4.1941 12.726871 -0.035788 0.011794

62 30.9813 34.753908 -3.7726 12.726871 -0.026046 0.007721

63 31.1297 34.753908 -3.6242 12.726871 -0.023093 0.006576

64 31.2004 34.753908 -3.5535 12.726871 -0.021767 0.006077

65 31.3167 34.753908 -3.4372 12.726871 -0.019699 0.005320

66 31.4351 34.753908 -3.3188 12.726871 -0.017733 0.004624

67 31.5616 34.753908 -3.1923 12.726871 -0.015781 0.003958

68 31.7142 34.753908 -3.0397 12.726871 -0.013624 0.003254

69 31.8096 34.753908 -2.9443 12.726871 -0.012382 0.002864

70 31.9729 34.753908 -2.7810 12.726871 -0.010434 0.002280

71 32.1008 34.753908 -2.6531 12.726871 -0.009059 0.001888

72 32.1543 34.753908 -2.5997 12.726871 -0.008523 0.001741

73 32.4062 34.753908 -2.3477 12.726871 -0.006277 0.001158

74 32.4369 34.753908 -2.3170 12.726871 -0.006034 0.001099

75 32.5840 34.753908 -2.1699 12.726871 -0.004956 0.000845

76 33.0821 34.753908 -1.6718 12.726871 -0.002267 0.000298

77 33.1069 34.753908 -1.6470 12.726871 -0.002167 0.000280

78 33.7099 34.753908 -1.0440 12.726871 -0.000552 0.000045





79 33.8580 34.753908 -0.8959 12.726871 -0.000349 0.000025

80 33.9151 34.753908 -0.8388 12.726871 -0.000286 0.000019

81 34.2309 34.753908 -0.5230 12.726871 -0.000069 0.000003

82 34.3364 34.753908 -0.4175 12.726871 -0.000035 0.000001

83 34.4234 34.753908 -0.3305 12.726871 -0.000018 0.000000

84 34.5519 34.753908 -0.2020 12.726871 -0.000004 0.000000

85 34.8480 34.753908 0.0941 12.726871 0.000000 0.000000

86 35.0600 34.753908 0.3061 12.726871 0.000014 0.000000

87 35.0999 34.753908 0.3460 12.726871 0.000020 0.000001

88 35.3198 34.753908 0.5659 12.726871 0.000088 0.000004

89 35.8126 34.753908 1.0587 12.726871 0.000576 0.000048

90 35.8377 34.753908 1.0838 12.726871 0.000618 0.000053

91 35.8414 34.753908 1.0875 12.726871 0.000624 0.000053

92 36.6770 34.753908 1.9231 12.726871 0.003450 0.000521

93 37.0458 34.753908 2.2919 12.726871 0.005840 0.001052

94 37.5532 34.753908 2.7993 12.726871 0.010641 0.002341

95 37.8077 34.753908 3.0538 12.726871 0.013815 0.003315

96 38.3755 34.753908 3.6216 12.726871 0.023043 0.006557

97 38.3774 34.753908 3.6235 12.726871 0.023080 0.006571

98 38.4222 34.753908 3.6683 12.726871 0.023946 0.006902

99 38.6773 34.753908 3.9234 12.726871 0.029297 0.009032

100 38.7437 34.753908 3.9898 12.726871 0.030810 0.009659

101 38.8017 34.753908 4.0478 12.726871 0.032172 0.010232





102 39.5681 34.753908 4.8142 12.726871 0.054127 0.020475

103 39.5773 34.753908 4.8234 12.726871 0.054437 0.020631

104 39.9896 34.753908 5.2357 12.726871 0.069624 0.028643

105 40.0168 34.753908 5.2629 12.726871 0.070716 0.029243

106 40.2899 34.753908 5.5359 12.726871 0.082302 0.035800

107 40.4862 34.753908 5.7323 12.726871 0.091373 0.041155

108 41.5977 34.753908 6.8438 12.726871 0.155496 0.083617

109 42.0100 34.753908 7.2561 12.726871 0.185332 0.105666

110 42.4898 34.753908 7.7359 12.726871 0.224577 0.136507

111 42.8176 34.753908 8.0637 12.726871 0.254351 0.161155

112 43.9733 34.753908 9.2194 12.726871 0.380142 0.275377

113 44.0152 34.753908 9.2613 12.726871 0.385347 0.280416

114 44.3554 34.753908 9.6015 12.726871 0.429395 0.323948

115 44.3898 34.753908 9.6359 12.726871 0.434016 0.328605

116 44.5588 34.753908 9.8049 12.726871 0.457260 0.352277

117 44.6934 34.753908 9.9395 12.726871 0.476357 0.372029

118 44.8671 34.753908 10.1132 12.726871 0.501766 0.398720

119 45.0164 34.753908 10.2625 12.726871 0.524311 0.422784

120 45.3419 34.753908 10.5880 12.726871 0.575808 0.479038

121 45.4045 34.753908 10.6506 12.726871 0.586083 0.490470

122 45.7913 34.753908 11.0374 12.726871 0.652288 0.565700

123 45.9445 34.753908 11.1906 12.726871 0.679823 0.597761

124 46.3361 34.753908 11.5822 12.726871 0.753717 0.685927





125 46.5509 34.753908 11.7970 12.726871 0.796438 0.738249

126 47.1999 34.753908 12.4460 12.726871 0.935241 0.914601

127 47.3150 34.753908 12.5611 12.726871 0.961420 0.948894

128 47.7343 34.753908 12.9803 12.726871 1.060948 1.082079

129 48.2174 34.753908 13.4635 12.726871 1.183875 1.252394

130 48.8687 34.753908 14.1148 12.726871 1.364138 1.512903

131 50.2410 34.753908 15.4871 12.726871 1.801952 2.192759

132 50.4061 34.753908 15.6522 12.726871 1.860209 2.287787

133 51.0824 34.753908 16.3285 12.726871 2.111902 2.709558

134 52.5999 34.753908 17.8460 12.726871 2.757150 3.866165

135 53.2459 34.753908 18.4920 12.726871 3.067534 4.457102

136 53.6507 34.753908 18.8968 12.726871 3.273417 4.860360

137 54.0913 34.753908 19.3374 12.726871 3.507751 5.329728

138 54.1004 34.753908 19.3465 12.726871 3.512703 5.339761

139 54.4601 34.753908 19.7062 12.726871 3.712306 5.748109

140 54.7571 34.753908 20.0032 12.726871 3.882671 6.102496

141 56.7374 34.753908 21.9835 12.726871 5.153780 8.902279

142 57.2407 34.753908 22.4868 12.726871 5.515909 9.745918

143 57.5343 34.753908 22.7804 12.726871 5.734826 10.265032

144 58.2046 34.753908 23.4507 12.726871 6.256046 11.527449

145 59.0535 34.753908 24.2995 12.726871 6.960323 13.289416

146 59.4449 34.753908 24.6910 12.726871 7.302179 14.166742

147 63.3753 34.753908 28.6214 12.726871 11.373846 25.578582





148 63.3902 34.753908 28.6363 12.726871 11.391622 25.631896

149 67.1089 34.753908 32.3550 12.726871 16.430830 41.771429

150 78.0758 34.753908 43.3219 12.726871 39.441966 134.259413

Jumlah 82.0452 457.0577

)0452,82()148)(149(

150

)2)(1(

3

Skewness

s

X Xi

nn

nSkewness

= 0,558080

√

SES = 0.19803837

Rasio Skewness = iai

Rasio Skewness =0,0

Rasio Skewness = 2,8180413



Karena nilai rasio skewness 2,8180413 dan tidak berada diantara -1,96

dan 1,96, berarti kurva distribusi gamma tersebut tidak normal berdasarkan

kemencengan kurvanya.

)147)(148(

)149(3)0577,457(

)147)(148)(149(

)151(150

)3)(2(

)1(3

)3)(2)(1(

)1(

2

24

Kurtosis

nn

n

s

X Xi

nnn

nnKurtosis

= 0,132189

√

SEK = 0,393583

Rasio Kurtosis = iai

Rasio Kurtosis =0,5

Rasio Kurtosis = 0,3358601

Karena nilai rasio kurtosis 0,3358601 dan berada diantara -1,96 dan 1,96,

berarti kurva distribusi gamma tersebut normal berdasarkan kecondongan

kurvanya.

Berdasarkan perhitungan dengan menggunakan software Minitab 16, maka

dapat dilihat gambar grafik beserta hasil perhitungannya di bawah ini:



7560453015

Median

Mean

37363534333231

1st Quartile 24.717

M edian 32.833

3rd Q uartile 44.100

Maximum 78.076

32.701 36.807

31.168 35.201

11.431 14.356

A -Squared 0.88

P -V alue 0.024

Mean 34.754

StDev 12.727

V ariance 161.973

Skewness 0.558080

Kurtosis 0.132189

N 150

M inimum 10.906

A nderson-Darling N ormality Test

95% C onfidence Interval for Mean

95% C onfidence Interval for Median

95% C onfidence Interv al for StD ev95 % Confidence Intervals

Summary for Gamma

Gambar 6. Kurva Skewness dan Kurtosis Distribusi Gamma

Keterangan: karena nilai skewness dan kurtosis di antara ± 1,96, maka

bentuk grafik normalitasnya mengumpul di tengah.



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