Pengantar Metode Penarikan Contoh dan Sebaran Penarikan Contoh Metode Statistika Pengantar Metode Penarikan Contoh dan Sebaran Penarikan Contoh

Populasi vs Contoh population sample himpunan semua objek yang menjadi minat pengambilan kesimpulan himpunan bagian dari populasi melakukan pengamatan terhadap seluruh populasi seringkali tidak mungkin dilakukan ketika akan membuat kesimpulan, mengapa?

Mengapa harus dengan contoh? sumber daya terbatas 1 2 waktu yang tersedia terbatas pengamatan kadang bersifat merusah 3 4 mustahil mengamati seluruh anggota populasi bagaimana caranya dengan menggunakan data contoh kita dapat mengambil kesimpulan terhadap populasi? INI YANG KITA PELAJARI PADA MATA KULIAH INI

Ilustrasi Andaikan kita memiliki sepiring sambel buatan ibu kita.  Berapa banyak yang kita ambil untuk mencicipi rasa sambel tersebut ? Sebagian besar orang akan berpendapat bahwa seujung jari sudah cukup untuk mengetahui rasa sepiring sambel tersebut.  Tidak akan ada seorang pun yang menjawab bahwa kita harus merasakan setengah piring untuk menyatakan rasa sambel buatan ibu. Pengambilan contoh dari sebuah populasi bisa dianalogkan dengan mencicipi masakan seperti di atas.  Jika data masing-masing objek bermacam-macam, dengan kata lain karakteristik objeknya berbeda-beda, maka perlu diambil contoh yang banyak untuk mewakili setiap kelompok karakteristik.  Namun jika karakteristik objek pada populasi itu seragam, hampir sama, maka contoh yang sedikit sudah cukup.

Pengantar Metode Sampling Tujuan Utama: Mendapatkan sampel yang mencerminkan populasi  dapat digunakan untuk menduga populasi Metode Sampling  Probability vs Non Probability Sampling Masalah utama dalam sampling: Menentukan metode sampling yang sesuai Menentukan ukuran sampel yang mewakili populasi (dengan tingkat ketelitian yang diinginkan dan segala kendala yang ada)

Probability Sampling Metode Sampling yang berbasis pada pemilihan secara acak Acak  setiap unit memiliki peluang yang sama untuk terpilih  Butuh kerangka contoh (daftar seluruh unit atau anggota populasi) Beberapa definisi: N = banyaknya objek dalam kerangka contoh (sampling frame) n = banyaknya objek dalam contoh f = n/N = fraksi contoh

Beberapa Metode (Probability Sampling) Penarikan Contoh Acak Sederhana (Simple Random Sampling) Penarikan Contoh Acak Berlapis (Stratified Random Sampling) Penarikan Contoh Sistematis (Systematic Random Sampling) Penarikan Contoh Gerombol (Cluster Random Sampling) Penarikan Contoh Bertahap (Multi-Stage Sampling) Error  Sampling Error vs Non Sampling Error

Ukuran contoh optimum (n): Simple Random Sampling n = f(ragam, ukuran populasi, ketelitian yang diinginkan, biaya, waktu) Ukuran contoh yang diperlukan untuk menduga  dengan batas error pendugaan sebesar B adalah: Ukuran contoh yang diperlukan untuk menduga P dengan batas error pendugaan sebesar B adalah: Z=1.96 dengan SK 95%, V=Std relatif thd mean, =batas kesalahan yang diinginkan (% thd mean)

Contoh Penentuan ukuran contoh optimum (n) Tentukan ukuran contoh optimum untuk menduga rata-rata produksi petambak jika diketahui N=10000 dan range produksi petambak antara 10-20 ton, dan batas error yang diinginkan B=1 ton. Tentukan ukuran contoh optimum untuk menduga proporsi (p) indukan udang yang baik jika diketahui N=2000 dan diinginkan batas error B=0.05. Asumsikan proporsi awal tidak diketahui.

Non Probability Sampling Pemilihan tidak dilakukan secara acak Generalisasi terhadap populasi agak sulit dilakukan Sering digunakan dalam penelitian sosial, marketing research, dll., krn Probability Sampling tidak praktis atau bahkan tidak dapat diterapkan Accidental/Haphazard/Convenience vs Purposive Purposive  Model Instance Sampling, Expert Sampling, Quota Sampling, Heterogenety Sampling, Snowball Sampling

The Central Limit Theorem Section 7.5 Sampling Distributions And The Central Limit Theorem In general, a sampling distribution of a statistic is a probability distribution (such as the normal distribution) for all possible values of the statistic computed from a sample of size n. The sampling distribution of the sample mean is a probability distribution of all possible values of the random variable computed from a sample of size n from a population with mean  and standard deviation .

The idea behind obtaining the sampling distribution of the mean is as follows: Obtain a simple random sample of size n. Compute the sample mean. Assuming that we are sampling from a finite population, repeat steps 1 and 2 until all simple random samples of size n have been obtained. If population size N = 100 and sample size n = 5 What is the number of possible samples of size 5?

Since each sample of size n will have an observed value of and not all observed values will be exactly the same, is a random variable. Since is a random variable, we can ask the following questions: What is the E( ) ? What is the Var( ) ? What is the distribution of ?

The Mean and Standard Deviation of the Sampling Distribution of . Suppose that a simple random sample of size n is drawn from a population with mean  and standard deviation . The sampling distribution of will have a mean and standard deviation The standard deviation of the sampling distribution of , , is called the standard error of the mean. Now we have answered the questions: What is the E( )? What is the Var( )? Population mean Population variance / sample size

What about the distribution of ? If a random variable X is normally distributed with mean  and standard deviation , then the distribution of the sample mean, , is normally distributed with mean and standard deviation What happens if the distribution of X is not normal? .

The mean of the distribution is CENTRAL LIMIT THEOREM Suppose a random variable X has a population mean  and standard deviation  and that a random sample of size n is taken from this population. Then the sampling distribution of becomes approximately normal as the sample size n increases. The mean of the distribution is and standard deviation . Let us visualize this.

Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As the sample size gets large enough… n↑

If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency Sampling Distribution (becomes normal as n increases) Variation Larger sample size Smaller sample size

100 random draws n = 5 n = 100 n = 25

When is n large enough to assume normality? The size of n depends on how close to normal the original population is. If the population is normal, n = 1 is large enough As a rule of thumb, we will use n = 30 as “sufficiently large” Hence, when n  30 the sampling distribution of will be approximately normal.

Example Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2?

Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 25) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation

Example Solution (continued): (continued) Z X Population Distribution Sampling Distribution Standard Normal Distribution .1915 +.1915 ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? -0.5 0.5 7.8 8.2 Z X

Example: The length of human pregnancies is approximately normally distributed with a mean of 266 days and standard deviation of 16 days. What is the probability a randomly selected pregnancy lasts less than 260 days? What is the probability that a random sample of 20 pregnancies have a mean gestation period of 260 days or less? What is the probability that a random sample of 50 pregnancies have a mean gestation period of 260 days or less? What might you conclude if a random sample of 50 pregnancies resulted in a mean gestation period of 260 days or less?

What is the probability a randomly selected pregnancy lasts less than 260 days? What is the probability that a random sample of 20 pregnancies have a mean gestation period of 260 days or less?

3. What is the probability that a random sample of 50 pregnancies have a mean gestation period of 260 days or less? 4. What might you conclude if a random sample of 50 pregnancies resulted in a mean gestation period of 260 days or less? We might conclude that the population from which the 50 pregnancies were drawn from has a mean gestation period less than 266 days. (Only 0.4% chance)

Population Proportions, P P = the proportion of the population having some characteristic Sample proportion ( ) provides an estimate of P: 0 ≤ ≤ 1 has a binomial distribution, but can be approximated by a normal distribution when nP(1 – P) > 9

Sampling Distribution of P ^ Sampling Distribution of P Normal approximation: Properties: and Sampling Distribution .3 .2 .1 0 . 2 .4 .6 8 1 (where P = population proportion)

Z-Value for Proportions Standardize to a Z value with the formula:

Example If the true proportion of voters who support Proposition A is P = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45? i.e.: if P = .4 and n = 200, what is P(.40 ≤ ≤ .45) ?

Example if P = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? (continued) Find : Convert to standard normal:

Example if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? (continued) Use standard normal table: P(0 ≤ Z ≤ 1.44) = .4251 Standardized Normal Distribution Sampling Distribution .4251 Standardize .40 .45 1.44 Z