Pertemuan 7

Dasar Probabilita

Definitions

Probability: value between 0 and 1, describing relative

possibility an event will occur

Experiment/trial: A process that leads only one of

possible outcomes

1 kali lempar dadu atau 1kali lempar koin = 1 trial

Outcome: a result of a trial

Event: kumpulan 1 atau beberapa trial

Approaches to Probability

Classical Empirical

ObjectiveSubjective

Tanpa data sebelumnya

Probability Distribution

Probability Distribution

Definisi: listing of all outcomes of a trial and the

probability of each outcomes

Normal Probability Distribution

Normal Probability Distribution

Normal Probability Distribution

Fungsi:

memberikan visualisasi besarnya penyebaran probabilitas dari event, atau probabilitas dari range suatu event

Dari kiri sampai kanan, jumlah semua probabilitas = 1

Sifat:

Numerous continuous variables common in business have distributions that closely resemble the normal distribution.

The normal distribution can be used to approximate various discrete probability distributions.

The normal distribution provides the basis for classical statistical inference

Bentuk Lain: mean sama, stdev beda

Bentuk Lain: mean beda, stdev sama

Standard Normal Probability Distribution

Definisi: distribusi probabilitas normal yang

distandarkan mean menjadi 0 dan stdev jadi 1

Fungsi: Menstandarkan dan menyederhanakan

perhitungan probabiltas dari berbagai distribusi yang

beragam mean dan stdev

Rumus Asli dari Normal Distribution

Rumus Standard Normal Probability Distribution

Z = nilai normal standar

X = nilai dari variabel yang ingin dicari

Hubungan Z, mean dan stdev

68.26%

95.44%

99.74%

Penerapan 1: menentukan probabilitas

range event

Contoh: dari sampel yang diambil, diketahui bahwa

pengeluaran untuk kopi per rumah tangga tersebar

normal dengan mean $45.16 dan stdev $10

Berapakah probabilita rumah tangga mempunyai

pengeluaran untuk kopi antara $45.16 dan $50?

=

=

50 45.16

10= 0.484

Jenis 1 dari Tabel Standard Normal Probability

Distribution: cumulative from mean (0 to Z)

Z = 0.484 = 0.4 + 0.08

Prob pengeluaran kopi antara $45.16 dan $50 adalah 0.18

Atau sekitar 18% populasi mempunyai pengeluaran untuk kopi antara $45.16 dan $50

Jenis 2 dari Tabel Standard Normal Probability

Distribution: cumulative from to mean

Kalau menggunakan tabel jenis 2

(cumulative from to mean)

0.6844

0.5

0.6844-0.5 = 0.1844

Penerapan 2: menentukan nilai X

Contoh: dari sampel yang diambil, diketahui bahwa

pengeluaran untuk kopi per rumah tangga tersebar

normal dengan mean $45.16 dan stdev $10

Sebuah perusahaan ingin menarget 60% teratas dari

segmen pasar kopi tersebut. Berapakah pengeluaran

terkecil untuk kopi?

0.60

0.40

Dengan tabel tipe 2 (cumulative from to mean) cari nilai

probabilita yang Z-nya mendekati 0.4

Probabilita yang mendekati 40% adalah 0.4207, berarti nilai Z-nya adalah -0.200

Z=-0.200, mean=$45.16 dan stdev $10

=

; = +

= + = 0.2 45.16 + 10 = $0.968

Latihan: bagaimana cara menentukan nilai X dengan menggunakan tabel tipe 1?

Sampling Error & Central Limit Theorem

Sampling Error

Error di sini bukan kesalahan pengambilan sampel

Definisi: perbedaan antara mean sampel dan mean

populasinya

Latihan

Dengan data di file employee data.xls (anggap

datanya adalah POPULASI.)

Dengan variabel salary, ambil 5 kali sampel, di mana 1

sampel ada 3 responden

Hitung mean kelima sampel tersebut

Apakah mean kelima sampel tersebut (yang cuma ada

3 orang) beda dengan mean salary yang ada 474 orang?

Sampling Distribution of the Sample Mean

Definisi: distribusi probabilitas dari beberapa mean

sampel.

Latihan: dengan contoh tadi, kelima sampel tadi dibuat

tabel probabilitasnya

Hitung rata2 mean dari semua sample

Central Limit Theorem

Definisi: jika semua sampel diambil dari sebuah populasi,

penyebaran mean-nya (Sampling Distribution of the Sample Mean) akan mendekati distribusi normal dan sama dengan mean populasi

Makin besar n, makin mendekati distribusi normal

Kalau n>= 30, distribusinya sudah mendekati normal

Jika tidak semua sampel diambil, setidaknya Sampling Distribution of the Sample Mean mendekati mean populasi

Stedev dari Sampling Distribution of the Sample Mean sebesar

Manfaat Central Limit Theorem

Kita punya informasi atau target tentang populasi

(mean dan stdev)

Kita ambil sampel dari populasi tersebut

Mean sampel kemungkinan akan berbeda dari

populasi (atau dari target yang kita inginkan)

Apakah mean sampel memang beda atau kebetulan

beda dari mean populasi?

Kondisi Central Limit Theorem

Jika populasinya diketahui normal

Minimum jumlah sampel tidak masalah

Jika populasinya diketahui tidak normal, atau sama

sekali tidak diketahui

Minimum jumlah sampel harus sebesar 30

Latihan

Dari sensus perumahan, diketahui mean harga rumah

sebesar $224,200 dengan stdev $9,000.

Jika kita ambil sampel dan kemudian setelah dihitung

mean sampel adalah $230,000, berapa probabilita

sample tersebut menyimpang dari mean populasi?

=

Stdev populasi diketahui

=

Stdev populasi tidak diketahui

Latihan

Sebuah pabrik minuman mineral men-setting

mesinnya untuk mengisi air mineral sebanyak 1 liter ke

dalam botol, dengan deviasi sebesar 0.01 liter

Sebuah sampel sebanyak 30 botol lalu ditemukan

bahwa mean sampel sebesar 1.05 liter. Berapa besar

probabilita dari mean sampel ini berbeda dari 1 liter?

Confidence Interval Estimation

Manfaat

Statistical process control

Mengira-ngira rentang nilai rata2 atau proporsi populasi, dengan tingkat keyakinan tertentu

Menentukan jumlah responden/sampel

Mengira-ngira Rata2 Populasi: Rumus

Z atau t?

Populasi terdistribusi normal

Stdev populasi diketahui = gunakan tabel Z

Stdev populasi tidak diketahui = gunakan tabel t

Populasi tidak normal/tidak diketahui distribusinya

Jumlah sampelnya >= 30 : gunakan tabel Z

Jumlah sampelnya < 30: gunakan non parametrik

Catatan: sampel besar berarti >= 30 sampel

Rumus Z dan t untuk Estimasi Rata2

Populasi

X = rata2 sampel

Z = nilai z (sesuai confidence level)

t = nilai t (sesuai confidence level)

= stdev populasi

s = stdev sampel

n = jumlah sampel

Tabel t

Distribusi t

Estimasi Proporsi Populasi

p = proporsi sampel

Z = nilai z (sesuai confidence level)

n = jumlah sampel

Latihan

Sebuah pabrik minuman mineral men-setting

mesinnya untuk mengisi air mineral sebanyak 1 liter ke

dalam botol dan stdev 0.01 liter.

Sebuah batch sebanyak 5000 botol diproduksi, lalu

sebuah sampel 30 botol diambil secara acak dengan

mean 1.05 liter.

Sebesar 95% kemungkinan, berapa nilai maksimum

dan minimum dari rata2 populasinya (dari 5000 botol)?

Apakah rata2 populasi (1 liter) masih berada di

rentang minimum-maksimum di atas?

Latihan

A major department store chain is interested in

estimating the average amount its credit card customers

spent on their first visit to the chain's new store in the

mall. Fifteen credit card accounts were randomly

sampled and analyzed with the following results: =

$50.50 and S2 = 400. Construct a 95% confidence

interval for the average amount its credit card

customers spent on their first visit to the chain's new

store in the mall assuming that the amount spent

follows a normal distribution.

Latihan

An economist is interested in studying the incomes of

consumers in a particular region. The population

standard deviation is known to be $1,000. A random

sample of 50 individuals resulted in an average income

of $15,000. What is the width of the 90% confidence

interval?

Latihan

Private colleges and universities rely on money contributed by

individuals and corporations for their operating expenses.

Much of this money is put into a fund called an endowment,

and the college spends only the interest earned by the fund. A

recent survey of 8 private colleges in the United States revealed

the following endowments (in millions of dollars): 60.2, 47.0,

235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics

yield Xbar = 180.975 and S = 143.042. Calculate a 95%

confidence interval for the mean endowment of all the private

colleges in the United States assuming a normal distribution for

the endowments.

Latihan

A major department store chain is interested in

estimating the average amount its credit card customers

spent on their first visit to the chain's new store in the

mall. Fifteen credit card accounts were randomly

sampled and analyzed with the following results: =

$50.50 and S2 = 400. Construct a 95% confidence

interval for the average amount its credit card

customers spent on their first visit to the chain's new

store in the mall assuming that the amount spent

follows a normal distribution.

Latihan

As an aid to the establishment of personnel

requirements, the director of a hospital wishes to

estimate the mean number of people who are admitted

to the emergency room during a 24-hour period. The

director randomly selects 64 different 24-hour periods

and determines the number of admissions for each. For

this sample, = 19.8 and S2 = 25. Estimate the mean

number of admissions per 24-hour period with a 95%

confidence interval.

Latihan

True or false? A race car driver tested his car for time

from 0 to 60 mph, and in 20 tests obtained an average

of 4.85 seconds with a standard deviation of 1.47

seconds. A 95% confidence interval for the 0 to 60 time

is 4.52 seconds to 5.18 seconds.

Latihan

A university dean is interested in determining the

proportion of students who receive some sort of

financial aid. Rather than examine the records for all

students, the dean randomly selects 200 students and

finds that 118 of them are receiving financial aid. Use a

90% confidence interval to estimate the true proportion

of students who receive financial aid.

Latihan

True or False? A sample of 100 fuses from a very

large shipment is found to have 10 that are defective.

The 95% confidence interval would indicate that, for

this shipment, the proportion of defective fuses is

between 0 and 0.28.

Catatan untuk Populasi Finite

Jika jumlah populasi diketahui

Dan jumlah sampel >5% dari jumlah sampel

1

1

Menentukan Jumlah Sampel

n = jumlah sampel

Z = nilai Z sesuai confidence level

= standar deviasi populasi

e = nilai error (penyimpangan dari

rata2 populasi) yg diinginkan

Cara Mencari Stdev Populasi

Melihat penelitian sebelumnya yang mirip

respondennya atau topiknya

Melakukan pilot study dengan sampel kecil

Menggunakan range

(nilai max nilai min)/6

Latihan

Suppose a department store wants to estimate the

average age of the customers of its contemporary

apparel department, correct to within 2 years, with

level of confidence equal to 95%. Management

believes that the standard deviation is 8 years. What is

the sample size they should take?