SAMPLING PLAN Dasar - Dasar Penarikan Contoh (Sampling)
Apa Itu Sampling? • Pendugaan karakteristik suatu populasi berdasarkan contoh (sample) yang diambil dari populasi tersebut • pengukuran hanya dilakukan pada sebagian elemen dari populasi: tidak semua elemen dalam populasi diukur • Digunakan untuk memperoleh nilai dugaan dari populasi yang sedang dipelajari
Apa itu sampling…(2) • Perumpamaan: “seorang koki yang mencicipi satu sendok sup untuk mengatakan bahwa satu panci sup yang dimasaknya memang lezat.”
Bandingkan…! Adakah sampling untuk “yang satu ini…?”
Untuk diingat …!!!!!
Populasi & contoh…(2) • Ukuran populasi: – banyaknya unit populasi di dalam populasinya dinotasikan: N
• Ukuran contoh: – banyaknya unit populasi yang terambil sbg contoh dinotasikan: n
• Intensitas sampling (IS): – Proporsi ukuran contoh terhadap ukuran populasi:
Populasi & contoh…(3) • Parameter: nilai yang mencerminkan karakteristik populasi • Statistik: nilai mencerminkan karakteristik contoh Misalnya: – Rata-rata/nilai tengah (mean) – Ragam (variance)
Studi Kasus • Misalkan dalam kelas Anda ini: Apabila akan dipilih 10 orang wakil: – Apakah populasinya? – Berapakah ukuran populasinya? – Apakah wujud unit populasinya? – Apakah sampling frame-nya? – Berapakah ukuran contohnya? – Berapakah intensitas sampling-nya?
Mengapa Sampling? Keuntungan sampling: – – – –
Menghemat sumberdaya: biaya, waktu, tenaga Kecepatan mendapatkan informasi (up to date) Ruang lingkup (cakupan) lebih luas Data/informasi yang diperoleh lebih teliti dan mendalam – Pekerjaan lapangan lebih mudah dibanding cara sensus
Mengapa Sampling ? Sampling lebih disukai dibanding inspeksi 100% bilamana – inspeksi bersifat destruktif. – inspeksi butuh biaya yang mahal – inspeksi 100% tidak layak untuk dijalankan – Sarana inspeksi otomatis tak tersedia
Kesalahan dalam Sampling Jenis kesalahan: A. Kesalahan non-sampling: • Kesalahan yang bukan berasal dari pengambilan contoh, muncul karena: – kesalahan pengukuran (measurement error) – kesalahan alat (instrumental error) – Kesalahan karena faktor pengukur (human error) – Kesalahan karena faktor lingkungan (environmental error)
• Besarnya kesalahan jenis ini sulit dihitung secara pasti
Kesalahan dalam sampling…(2) B.Kesalahan sampling (sampling error, SE): – Kesalahan yang disebabkan oleh pengambilan contoh (sampling) yang dilakukan secara tidak tepat – Besarnya kesalahan dapat dihitung dengan formula berikut:
Ukuran contoh yang paling optimal adalah satu titik dimana banyaknya unit populasi yang terambil sebagai contoh akan menghasilkan total error yang paling minimal
Teliti, Akurat, Bias • Dalam sampling: Contoh (sample) digunakan untuk memperoleh nilai dugaan (estimate) yang akurat/tepat bagi parameter populasi
Teliti, Akurat, Bias Ingatlah selalu ilustrasi berikut:
Teliti, akurat, bias Ketelitian (precision): – Derajat kesesuaian (degree of agreement) dari suatu rangkaian pengukuran – Dalam sampling : Penyimpangan nilai-nilai pengukuran (dari contoh) thdp nilai rata-ratanya sendiri {Ditunjukan oleh nilai simpangan baku (s)}
Keakuratan/ketepatan (accuracy): – Derajat kedekatan suatu nilai pengukuran terhadap nilai sebenarnya – Dalam sampling : Besarnya penyimpangan nilai-nilai dugaan dari contoh thdp nilai parameter populasinya (Catatan: parameter populasi seringkali tdk diketahui)
Teliti, akurat, bias Bias kesalahan sistematis yg dapat disebabkan oleh : kesalahan dlm prosedur pengukuran, alat, prosedur sampling, perhitungan, pencatatan, …dsb…
Hubungan ketiganya:
(A=akurasi, B=bias, P=presisi) Akurasi = presisi, jika bias dapat dihilangkan
Dua Metoda Pengambilan Contoh 1. Pengambilan contoh secara acak (random sampling): Sampel diambil tanpa mengikuti satu pola tertentu, setiap unit populasi punya peluang yang sama untuk terambil dan menjadi bagian dari contoh
Dua Metoda Pengambilan Contoh 2. Pengambilan contoh secara sistematik (systematic sampling): • Unit contoh diambil dengan pola ttt, keteratutan ttt (sistematis) • Tidak memiliki penduga ragam yang sah: – Dalam penerapannya, sering dimodifikasi menjadi “systematic sampling with random start”
Acceptance Sampling
Acceptance Sampling The inspection and classification of a sample of nits selected at random from a larger batch or lot and ultimate decision about disposition of the lot – Lot Disposition or Lot Sentencing
• Another area of quality control and improvement • Closely connected with inspection and testing of product • Inspection can occur at many points in a process
Acceptance Sampling • Berhubungan dengan keputusan untuk menerima atau menolak lot produk • Acceptance sampling dari lot ke lot adalah suatu proses pengambilan keputusan • Suatu keputusan yang didasarkan pada sample yang diambil dari unit - unit yang terdapat dalam lot produk
Acceptance Sampling A batch (lot) of items has been produced. – Before shipment, the producer tests the lot for outgoing quality,
or – After receiving shipment, the consumer tests the lot for incoming quality.Acceptance Sampling
Acceptance Sampling
Persyaratan Suatu Lot Lots should be homogeneous: –
–
produced on the same machines, by same operators, from common raw materials, at approximately the same time period (come from a single source) units should be as similar to each other as possible.
Larger lots are more preferable than smaller lots
Dasar Penentuan Suatu Acceptance Sampling Plans 1. By Attributes – each sampled item is inspected and classified as conforming or nonconforming (defective). – Or, the number of defects or nonconformities is counted for each item. – Large numbers of nonconforming or nonconformities indicate poor quality
Dasar Penentuan Suatu Acceptance Sampling Plans 2. By Variables – some quantitative quality characteristic is measured for each sampled item. – the sample mean of the measurements is computed. – if the sample mean lies outside an acceptable range, the lot is rejected.
Beberapa Skema Acceptance Sampling Plan Sampling Plan for Attributes
Sampling Plan for Variables
• Single-Sampling Attributes plans • Dodge-Romig Plans • Double Sampling Plans • MIL-STD-105E • Sequential Sampling • Variables Sampling Plans
Single-Sampling Attributes Plans • Inspected units should be selected at random • each combination of n items has an equal chance of comprising the sample. • Inspected units should represent all items in the lot • eliminates bias
Single-Sampling Attributes Plans • •
•
A lot of size N has been submitted for inspection A simple random sample is selected from a lot for inspection. – n = sample size. – c = acceptance number. If there are more than c nonconforming items in the sample, then the lot is rejected. Example N = 10,000 n= 89 c=2
N = 10,000
n = 89
Contoh A manufacturer of silicon chips produces lots of 1000 chips for shipment. A single-sampling plan with n = 65 and c = 2 is used to test for bad outgoing lots. – In one sample, 4 defective chips are discovered. Since 4 > 2, the lot is rejected and not shipped. The source of the defective chips should be found and eliminated. – In a random sample of 65 chips taken from another lot, 2 defective chips are discovered. This lot is acceptable for shipment.
Fungsi Operating Characteristic (OC) Sebuah fungsi untuk mengukur performa design suatu sampling plan – Probabilitas diterimanya suatu lot dihitung berdasar fungsi OC berikut:
p = true proportion nonconforming.
The OC Curve
Plot of OC Function Curve plots the probability of accepting the lot (Pa) versus the lot fraction defective (p)
The OC Curve
Latihan 1. A single sampling plan has parameters n = 2 and c = 0. – What is the OC function for such a plan? – How much the probability of accepting a lot with 5% defective ? – How much the probability of acceptance of a lot with 20% defective ?
2. What is the OC function for a plan having n = 2 and c = 1? 3. What is the OC function for a plan with n = 10 and c = 2?
Latihan 4. An apple producer has 500 baskets of apples, containing 20 each. A buyer wants to inspect 10 of the apples before accepting the lot (if 2 or less are bruised). – How many n , N, and c ? – Suppose 20% of the apples in the lot are bruised, What is the probability of accepting such a lot? – If 50 % of the apples are bruised, then the probability of accepting such a lot is …………
DOUBLE SAMPLING PLAN
Contoh Suatu sampling plan dengan: n1 = 10 n2 = 20 ac1 = 0 ac2= 3 re1 = re2 = 3 N = 1000 D = 50
Example B: Consider a double sampling plan with n1 = 100, c1 = 2, r1 = 6, n2 = 200, c2 = 5. Find that the probability of acceptance If p = .01
Military Standard 105E
Military Standard 105E Procedure
Choose the AQL Choose the inspection level Determine the lot size Fine the appropriate sample size code letter (from Table 14-4) Determine the appropriate type of sampling plan to use (single) Enter the appropriate table to find the type of plan to be used Determine the corresponding normal and reduced inspection plans to be used when required
Acceptable quality level (AQL) •
The poorest quality level for the supplier’s process that a consumer would consider to be acceptable
•
A property of the supplier’s manufacturing process, not a property of the sampling plan
•
This is the maximum proportion of defectives allowed. The producer requires that the probability of acceptance at this level be fairly high. i.e. Pa(AQL) should be large, typically this should be near 0.99 or 0.95. Type I error probability: α = 1−Pa(AQL); typically .05 or .01
•
Lot tolerance percent defective (LTPD) • or consumer’s quality level • The protection obtained for individual lots of poor quality • This is a numerical definition of an unacceptable lot. The consumer requires that this kind of lot be rejected with high probability. In other words Pa(LTPD) should be small, typically this is chosen to be 0.1 • Pa(LTPD) is often referred to as , the probability of Type II error. • Also called rejectable quality level (RQL) and the limiting quality level (LQL)
Latihan 1. Suppose MIL STD 105E is used, lots of 15000 are to be examined, and AQL is .4 %. a. If double sampling is used. How many n and c we need b. Now, suppose single sampling is used. Calculate n and c we need
Latihan c.
Suppose that the following sequence of rejected (R) and accepted (A) lots with the number of defectives D occur:
Lot
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A/R
A
R
A
A
R
A
A
A
A
A
A
A
A
A
A
D
0
14
0
0
12
1
0
2
1
0
1
0
1
0
0
What mode of inspection will be applicated to that supplier?
MENDISAIN ACCEPTANCE SAMPLING PLAN
Mendesign Suatu Acceptance Sampling plan How can we decide 1. how many items n to sample, and 2. how many nonconforming items c in that sample are enough to convince us that the lot is unacceptable
Effects of c on OC curves
Optimal Plans • The ideal OC curve places high probability on accepting good lots (p close to 0), and low probability on accepting bad lots. • The producer wants all good lots to be accepted. (minimize type I error probability = producer’s risk). In other words, if p is small, then the producer wants Pa(p) to be close to 1 • The consumer wants all bad lots to be rejected. (minimize type II error probability = consumer’s risk). If p is large, then the consumer wants Pa(p) close to 0.
Finding a Plan, given α, β, LTPD and AQL To construct sampling plan such that
for n and c, where p1 = AQL and p2 = LTPD. To solve these equations, we could use a binomial nomograph OR Use Better alternative: normal approximation to binomial
Binomial Nomograph
Normal Approximation to Binomial
