Distribution of sampling means

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THE DISTRIBUTION OF SAMPLE MEANS

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Chapter 7


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SAMPLING

?

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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.

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Prinsip-Prinsip Sampling Jika kita ingin mengambil dua individu

untuk memperkirakan usia rata-rata dari empat individu.

4C2= 6 AB, AC, AD, BC, BD, CD

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nCr =n!

r! (n-r)!

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n = 2A = 18 B = 20 C = 23 D = 25

SAMPLE M μ M - μAB 19 21,5 -2,5AC 20,5 21,5 -1,5AD 21,5 21,5 0BC 21,5 21,5 0BD 22,5 21,5 1,5CD 24 21,5 2,5

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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.

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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

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Prinsip-Prinsip Sampling Jika kita ingin mengambil tiga individu

untuk memperkirakan usia rata-rata dari empat individu.

4C3= 4 ABC, ABD, ACD, BCD

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n = 3A = 18 B = 20 C = 23 D = 25

SAMPLE M μ M - μABC 20,33 21,5 -1,17ACD 21 21,5 -0,5ACD 22 21,5 -0,5BCD 22,67 21,5 1,17

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Principle-TWO

The greater sample size, the more accurate will be estimate of the true population

mean

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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

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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)

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THE DISTRIBUTION OF SAMPLE MEANS Faktor-faktor yang mempengaruhi

kesimpulan yang ditarik dari sampelPrinsip-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.

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THE DISTRIBUTION OF SAMPLE MEANS Faktor-faktor yang mempengaruhi

kesimpulan yang ditarik dari sampelPrinsip-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).

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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

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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)!

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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

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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

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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

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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

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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

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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

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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

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Distribution of sampling means

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