Distribution of sampling means
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This document discusses the distribution of sample means and introduces three key principles: 1) There will usually be a difference between sample statistics and the true population mean due to random selection. 2) Larger sample sizes produce more accurate estimates of the population mean. 3) Greater variability in the population variable leads to greater differences between sample statistics and the population mean. It also discusses how the distribution of sample means approaches a normal distribution as sample size increases, and how this distribution can be used to determine probabilities regarding sample means.
Distribution of sampling means
- 1. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 1 Chapter 7 THE DISTRIBUTION OF SAMPLE MEANS
- 2. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS SAMPLING ?
- 3. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Prinsip-Prinsip Sampling Pada kebanyakan kasus dimana pengambilan sampel dilakukan terjadi perbedaan antara statistik sampel dan rata-rata populasi, yang dianggap disebabkan oleh pemilihan unit dalam sampel Contoh: Usia A = 18 tahun, B = 20 tahun, C = 23 tahun, D = 25 tahun. Usia rata-rata A, B, C, D adalah 21,5 tahun. 3
- 4. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Prinsip-Prinsip Sampling Jika kita ingin mengambil dua individu untuk memperkirakan usia rata-rata dari empat individu. 4 C2= 6 AB, AC, AD, BC, BD, CD 4 nCr = n! r! (n-r)!
- 5. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS n = 2 A = 18 B = 20 C = 23 D = 25 SAMPLE M μ M - μ AB 19 21,5 -2,5 AC 20,5 21,5 -1,5 AD 21,5 21,5 0 BC 21,5 21,5 0 BD 22,5 21,5 1,5 CD 24 21,5 2,5 5
- 6. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Prinsip-Prinsip Sampling Dari ke-enam kemungkinan kombinasi sampel, hanya dua yang tidak terdapat perbedaan antara statistik sampel dan rata- rata populasi. Perbedaan ini dianggap disebabkan sampel dan diketahui sebagai sampling error. 6
- 7. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Principle-ONE In majority of cases of sampling there will be a difference between the sample statistics and the true population mean, which is attributable to selection of the units in the sample 7
- 8. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Prinsip-Prinsip Sampling Jika kita ingin mengambil tiga individu untuk memperkirakan usia rata-rata dari empat individu. 4 C3= 4 ABC, ABD, ACD, BCD 8
- 9. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS n = 3 A = 18 B = 20 C = 23 D = 25 SAMPLE M μ M - μ ABC 20,33 21,5 -1,17 ACD 21 21,5 -0,5 ACD 22 21,5 -0,5 BCD 22,67 21,5 1,17 9
- 10. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Principle-TWO The greater sample size, the more accurate will be estimate of the true population mean 10
- 11. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Principle-THREE The greater difference in the variable under study in a population for a given sample size, the greater will be the difference between the sample statistics and the true population mean 11
- 12. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Perbedaan besar variabel yang diteliti pada populasi, besar pula perbedaan antara statistik sampel dan rata-rata populasi. Contoh: Usia A = 18 tahun, B = 26 tahun, C = 32 tahun dan D = 40 tahun. Dengan prosedur yang sama, diketahui rentang perbedaan jauh berbeda dengan contoh-contoh sebelumnya. Hal ini dianggap disebabkan perbedaan usia yang besar dalam populasi (heterogen) 12
- 13. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Faktor-faktor yang mempengaruhi kesimpulan yang ditarik dari sampel Prinsip-prinsip di atas menunjukkan terdapat dua faktor yang dapat mempengaruhi tingkat keyakinan tentang kesimpulan yang ditarik dari sampel. 1. Ukuran sampel Temuan yang didasarkan sampel yang besar lebih dapat diyakini dibandingkan dengan yang didasarkan dengan sampel yang lebih kecil. Sesuai prinsip, semakin besar ukuran sampel semakin akurat temuannya. 13
- 14. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS Faktor-faktor yang mempengaruhi kesimpulan yang ditarik dari sampel Prinsip-prinsip di atas menunjukkan terdapat dua faktor yang dapat mempengaruhi tingkat keyakinan tentang kesimpulan yang ditarik dari sampel. 2. Besarnya variasi populasi Variasi besar dalam karakteristik populasi, besar pula ketidakyakinannya (semakin besar standar deviasi, semakin tinggi standard error). 14
- 15. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 15 THE DISTRIBUTION OF SAMPLE MEANS Two separate samples probably will be different even though they are taken from the same population The sample will have different individual, different scores, different means, and so on The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population
- 16. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 16 COMBINATION Consider a population that consist of 5 scores: 3, 4, 5, 6, and 7 Mean population = ? Construct the distribution of sample means for n = 1, n = 2, n = 3, n = 4, n = 5 nCr = n! r! (n-r)!
- 17. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 17 SAMPLING DISTRIBUTION … is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population CENTRAL LIMIT THEOREM For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity
- 18. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 18 The STANDARD ERROR OF MEAN The value we will be working with is the standard deviation for the distribution of sample means, and it called the σM Remember the sampling error There typically will be some error between the sample and the population The σM measures exactly how much difference should be expected on average between sample mean M and the population mean μ
- 19. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 19 The MAGNITUDE of THE σM Determined by two factors: ○The size of the sample, and ○The standard deviation of the population from which the sample is selected
- 20. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 20 A population of scores is normal with μ = 100 and σ = 15 ○ Describe the distribution of sample means for samples size n = 25 and n =100 LEARNING CHECK
- 21. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 21 PROBABILITY AND THE DISTRIBUTION OF SAMPLE MEANS The primary use of the standard distribution of sample means is to find the probability associated with any specific sample Because the distribution of sample means present the entire set of all possible Ms, we can use proportions of this distribution to determine probabilities
- 22. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 22 EXAMPLE The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. If you take a random sample of n = 16 students, what is the probability that sample mean will be greater that M = 540? σM = σ √n = 25 z = M - μ σM = 1.6 z = 1.6 Area C p = .0548
- 23. © aSup-2007 THE DISTRIBUTION OF SAMPLE MEANS 23 The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. We are going to determine the exact range of values that is expected for sample mean 95% of the time for sample of n = 25 students See Example 7.3 on Gravetter’s book page 207 LEARNING CHECK