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Ch.1the Field of Organizational Behavior

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DISTRIBUSI PROBABILITAS
Amiyella Endista
Email : amiyella.endista@yahoo.com Website : www.berandakami.wordpress.com

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Distribusi Probabilitas




Kunci aplikasi probabilitas dalam statistik adalah memperkirakan terjadinya peluang/probabilitas yang dihubungkan dengan terjadinya peristiwa tersebut dalam beberapa keadaan. Jika kita mengetahui keseluruhan probabilitas dari kemungkinan outcome yang terjadi, seluruh probabilitas kejadian tersebut akan membentuk suatu distribusi probabilitas.

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Macam Distribusi Probabilitas
1.
2. 3.

Distribusi Binomial (Bernaulli) Distribusi Poisson Distribusi Normal (Gauss)

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1. Distribusi Binomial
Penemu Distribusi Binomial adalah James Bernaulli sehingga dikenal sebagai Distribusi Bernaulli.  Menggambarkan fenomena dengan dua hasil atau outcome. Contoh: peluang sukses dan gagal,sehat dan sakit, dsb.


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Syarat Distribusi Binomial
1.

2.

Jumlah trial merupakan bilangan bulat. Contoh melambungkan coin 2 kali, tidak mungkin 2 ½ kali. Setiap eksperiman mempunyai dua outcome (hasil). Contoh: sukses/gagal,laki/perempuan, sehat/sakit,setuju/tidak setuju.
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Syarat Distribusi Binomial
3. Peluang sukses sama setiap eksperimen. Contoh: Jika pada lambungan pertama peluang keluar mata H/sukses adalah ½, pada lambungan seterusnya juga ½. Jika sebuah dadu, yang diharapkan adalah keluar mata lima, maka dikatakan peluang sukses adalah 1/6, sedangkan peluang gagal adalah 5/6.Untuk itu peluang sukses dilambangkan p, sedangkan peluang gagal adalah (1-p) atau biasa juga dilambangkan q, di mana q = 1-p.

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Contoh Distribusi Binomial




Simbol peristiwa Binomial  b (x,n,p) b=binomial x=banyaknya sukses yang diinginkan (bilangan random) n= Jumlah trial p= peluang sukses dalam satu kali trial. Dadu dilemparkan 5 kali, diharapkan keluar mata 6 dua kali, maka kejadian ini dapat ditulis b(2,5,1/6)  x=2, n=5, p=1/6

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Contoh soal


Probabilitas seorang bayi tidak di imunisasi polio adalah 0,2 (p). Pada suatu hari di Puskesmas “X” ada 4 orang bayi. Hitunglah peluang dari bayi tersebut 2 orang belum imunisasi polio. Jadi, di dalam kejadian binomial ini dikatakan b (x=2, n=4, p=0,2)  b (2, 4, 0,2)

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Penyelesaian soal




Katakanlah bayi tersebut A,B,C,D. Dua orang tidak diimunisasi mungkin adalah A&B, A&C, A&D, B&C, B&D, C&D. Rumus untuk b (x,n,p) adalah: P (x)= n! Px(1-p)n-x x! (n-x)! = 4! 0,22 (1-0,2)4-2 2! (4-2)! = 4.3.2.1 0,22 x 0,82 = 0,1536 = 0,154 2.1 (2.1)

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Penyelesaian


Disamping memakai rumus binomial, permasalahan ini juga dapat dikerjakan dengan memakai tabel binomial, caranya adalah dengan menentukan n.misalnya dari contoh soal adalah 4, dilihat pada kolom pertama kolom kedua adalah kemungkinan x, dalam permasalahan ini adalah x=2. p dilihat pada baris paling atas dalam hal ini p=0,2, ditarik garis dari p= 0,2 sampai ke n = 4dan x = 2, ditabel didapatkan 0,973. Ini adalah peluang kumulatif dari p (x=0) + p (x=1) + p (x=2). Jadi kalau mau mendapatkan p(x=2) saja, maka 0,973-0,819 = 0,154

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2. Distribusi Poisson




Dalam mempelajari distribusi Binomial kita dihadapkan pada probabilitas variabel random diskrit (bilangan bulat) yang jumlah trial nya kecil (daftar binomial), sedangkan jika dihadapkan pada suatu kejadian dengan p > maka digunakan distribusi Poisson. Distribusi Poisson  dipakai untuk menentukan peluang suatu kejadian yang jarang terjadi, tetapi mengenai populasi yang luas atau area yang luas dan juga berhubungan dengan waktu.
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Contoh Distribusi Poisson
1. Disuatu gerbang tol yang dilewati ribuan mobil dalam suatu hari akan terjadi kecelakaan dari sekian banyak mobil yang lewat. 2. Dikatakan bahwa kejadian seseorang akan meninggal karena shock pada waktu disuntik dengan vaksin meningitis 0,0005. Padahal, vaksinasi tersebut selalu diberikan kalau seseorang ingin pergi haji.

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Rumus Distribusi Poisson p (x) = µx e-µ = λx e-λ x! x! µ = λ = n.p = E(x)  Nilai rata-rata e = konstanta = 2,71828 x = variabel random diskrtit (1,2,3, ….,x)

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Contoh:




Diketahui probabilitas untuk terjadi shock pada saat imunisasi dengan vaksinasi meningitis adalah 0,0005. Kalau di suatu kota jumlah orang yang dilakukan vaksinasi sebanyak 4000. Hitunglah peluang tepat tiga orang akan terjadi shock! Penyelesaian: µ = λ = n.p = 4000 x 0,0005 = 2 p(x=3) = 23 x 2,71828-2 = 0,1804 3 x 2x 1
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Penyelesaian dengan tabel Distribusi Poisson



Baris = µ = λ

Kolom = x  P (x=3) = 0,857 - 0,677 = 0,180

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3. Distribusi Normal (Gauss)




Pada kasus di mana n cukup besar dan p tidak terlalu kecil (tidak mendekati 0,….,1 dilakukan pendekatan memakai distribusi Normal (Gauss) Ditemukan pertama kali oleh matematikawan asal Prancis, Abraham D (1733), diaplikasikan lebih baik lagi oleh astronom asal  Distribusi Normal = Distribusi Jerman,Friedrich Gauss Gauss

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Rumus Eksposensial untuk Distribusi Normal
_ 1 (x- µ)2

∫ (x) =

1 e √2πσ2 -≈≈ -≈≈

2σ2

σ2 = 0 π = 3,14

e = 2,71828

Agar lebih praktis, telah ada tabel kurva normal di mana tabel ini menunjukkan luas kurva normal dari suatu nilai yang dibatasi nilai tertentu.

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Ciri Khas Distribusi Normal

0
   

Simetris Seperti lonceng Titik belok µ ± σ Luas di bawah kurva = probability = 1
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Kurva Normal Umum




-

Untuk dapat menentukan probabilitas di dalam kurva normal umum (untuk suatu sampel yang cukup besar, terutama untuk gejala alam seperti berat badan dan tinggi badan), nilai yang akan dicari ditransformasikan dulu ke nilai kurva normal standar melalui transformasi Z (deviasi relatif). Rumus: Z = x - µ Z=X–X σ S Kurva normal standar  N (µ = 0, σ = 1) Kurva normal umum  N (µ, σ)
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Contoh


Dari penelitian terhadap 150 orang laki-laki yang berumur 40 – 60 tahun didapatkan rata-rata kadar kolesterol mereka 215 mg % dan simpangan baku Sd = 45 mg %. Hitunglah peluang kita mendapatkan seorang yang kadar kolesterolnya: a. > 250 mg % b. < 200 mg % c. antara 200 – 275 mg %

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Penyelesaian


Nilai x ditransformasikan ke nilai z. Di dalam tabel nilai z berada pada kolom paling kiri dan baris paling atas. Ambillah nilai 2 ini tiga digit saja. Nanti 2 digit ada di kolom dan digit ketiga ada di baris.

a. Z = 250 -215 = 0,76 45

0,76 = 0,7 + 0.06  (Lihat tabel) = 0,7 dilihat pada kolom ; 0,06 pada baris  lihat lampiran tabel III didapat nilai 0,2764, ini adalah luas area antara 215 s.d 250.  yang ditanyakan adalah p (x > 250 mg%), jadi untuk mendpatkan area > 250 mg% adalah 0,5 – 0,2764 = 0,2236 BioStatistik

Penyelesaian
b. P (x < 200 mg%) Z = 200 -215 = 0,33  Tabel 0,1297 45 jadi P (x < 200 mg%) = 0,5 – 0,1297 = 0,3703

c. P (200 mg% < x < 275 mg%) pada soal b. sudah didapatkan area antara 215 mg% s.d 200 mg% = 0,1297  z = 275 – 215 = 1,33  Tabel 0,4082 45
Jadi P (200 mg% < x < 275 mg%) = 0,1297 + 0,4082 =0,5379
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Latihan
1. Peluang mahasiswa merokok 0,2. Bila diambil sampel sebanyak 8 mahasiswa. Hitung peluang : a. Tidak ada mahasiswa yang merokok b. 2 mahasiswa merokok c. Kurang dari 3 mahasiswa merokok d. Paling banyak 3 mahasiswa merokok 2. Diketahui peluang balita menderita gizi buruk di Depok 2 % (0,02). Bila diambil sampel 150 balita. Hitung peluang: a. Ada 3 balita menderita gizi buruk b. Ada < 6 balita menderita gizi buruk c. Ada > 4 balita menderita gizi buruk d. Paling banyak 5 balita menderita gizi buruk.
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Latihan
3. Diketahui rata-rata kadar ibu hamil 10 gr% dengan Sd 2 gr%. Bila data kadar Hb berdistribusi normal. Hitung pelung: a. Kadar Hb nya lebih dari 11 gr% b. kadar Hb nya kurang dari 9,5 gr% c. kadar Hb nya lebih dari 8 gr% d. Kadar Hb nya antara 11,5 s.d 14 gr%

BioStatistik…...

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