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© aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
1
Chapter 7
THE DISTRIBUTION
OF SAMPLE MEANS
© aSup-2007
THE DISTRIBUTION OF SAMPLE MEANS   
SAMPLING
?
© 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
© 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)!
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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)!
© 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
© 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
© 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
© 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
© 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
© 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
© 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

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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