Download - WordPress.com

Sampling Methods
Beberapa istilah
1.
2.
3.
4.
Populasi
Sampel
Parameter
Statistik
Some of reason for the sampling are :
1. To contact the whole population would
be time consuming
2. The cost of studying all the items in a
population may be prohibitive
3. The physical impossibility of checking
all items in the population.
4. The destructive nature of some tests
5. The sample results are adequate
Methods to Select a Sample
a. Simple random sampling → a sample selected
so that each item or person in the population
has the same chance of being included.
b. Systematic random sampling → a random
starting point is selected, and then every kth
member of the population is selected.
c. Stratified random sampling → a population is
divided into subgroups, called strata, and a
sample is randomly selected from each
stratum.
d. Cluster sampling → a population is divided into
clusters using naturally occurring geographic
or other boundaries. Then, clusters are
randomly selected and a sample is collected
by randomly selecting from each cluster.
Sampel random sederhana memillih sampel
dengan metode yang memberikan kesempatan
yang sama kepada setiap calon anggota sampel
dari anggota populasinya untuk menjadi anggota
sampel. Misalnya dalam suatu proses recruiting
karyawan baru terdapat 4 orang pelamar yaitu
A,B,C, dan D. Dari pelamar ini akan dipilih 2
pelamar untuk mengikuti tes wawancara.
Kombinasi yang mungkin sebanyak 2 yang dipilih
dari 4 pelamar dengan kesempatan yang sama
kepada setiap pelamar adalah pasangan
AB,AC,AD,BC,CD. Karena terdapat 6 pasangan yang
mungkin terjadi, maka masing-masing mempunyai
kesempatan probabiltas yang sama yaitu 1/6
Sampel sistematis memilih anggota sampel dari
suatu populasi dengan interval sama, biasanya
diukur dengan ukuran waktu, urutan, ranking atau
tempat. Misal kita menginginkan informasi
mengenai penghasilan rata-rata pedagang kaki
lima dengan menggunakan interval urutan,
terlebih dahulu ditentukan urutan yang ke berapa
dari anggota populasi yang dipilih menjadi
anggota sampel.Misalnya pemilihan menggunakan
daftar nama pedagang kaki lima, kemudian akan
dipilih secara random dimulai dari urutan kelima
sebagai data pertama. Pengambilan anggota sampel
berikutnya diambil pedagang kaki lima pada urutan kelima
berikutnya pada daftar tersebut.
Sampel bertingkat memilih anggota sampel
dengan cara membagi populasi menjadi beberapa
lapisan, disebut strata, secara acak. Misalnya
pada suatu penelitian untuk mengetahui minat
masyarakat terhadap penggunaan ATM.
masyarakat yang akan diteliti dibagi menjadi
beberapa lapisan, misalnya pedagang,pegawai
negeri, pegawai swasta. Anggota sampel yang
digunakan dalam penelitian merupakan
penjumlahan dari anggota masyarakat yang
bekerja sebagai pedagang, pegawai negeri,dan
pegawai swasta dipilih secara acak.
Sampel berkelompok memilih sampel dengan
membuat populasi menjadi beberapa kelompok
misal dalam satu penelitian bertujuan mengetahui
pola perubahan pengeluaran masyarakat di kota
Bengkulu. Maka kita bagi kota Bengkulu
menjadi beberapa lokasi pemilihan sampel, yaitu
Kec.Gading Cempaka, Kec. Selebar, Kec. Muara
Bangkahulu, Kec Teluk Segara dst. Pada masingmasing kecamatan dipilih beberapa keluarga
secara acak untuk
membentuk sampel yang diperlukan.
Sampling Distribution of The Sample Mean
Sample distribution of the sample mean → a probability
distribution of all possible sample means of a given
sample size
Example :
Tartus industries has seven production employees.The
hourly of each employee are given in table :
Employee
Hourly Earnings
Joe
$7
Sam
$7
Sue
$8
Bob
$8
Jan
$7
Art
$8
ted
$9
The Questions
a.What is the populations mean?
b.What is the sampling distribution of the
sample mean for the samples of size 2?
c.What the mean of the sampling
distributions ?
d.What the observasions can be made
about the population and the sampling
distribution?
The Answer
a.What is the populations mean?
μ = $7 + $7+ $8 +$8 + $7 + $8 + $9 = $7.71
7
b. What is the sampling distribution of the sample
mean for the samples of size 2 ?
NCn = N!
=
7!
= 21
n!(N-n)! 2!(7-2)!
c. What the mean of the sampling distributions ?
μx = Sum of all sample means
Total number of samples
= $7.00+$7.50+…+$8.50
= $7.71
21
d. What the observasions can be made about the
population and the sampling distribution?
Population Values
Distribution of sample mean
40
30
20
10
0
40
30
20
10
0
7
8
9
7
7.5
8
8.5
9
The Central Limit Theorem
Central limit theorem → if all samples of a particular
size are selected from any population, the sampling
distribution of the sample mean is approximately a
normal distribution. This approximation improves with
large samples.
ESTIMATION AND CONFIDENCE INTERVALS
Point Estimate → the statistic, computed from
sample information, which is used to estimate the
population parameter.
Confidence Interval → a range of value
constructed from sample data so that the
population parameter is likely to occur within
that range at a specified probability. The specified
probability is called the level of confidence.
Confidence interval for the population mean
(n > 30) = X + z s
√n
Example :
The American Management Assosiation
wishes to have information on the income
of middle managers in the retail idustry.
A random sample of 256 managers reveals
a sample mean of $45,420. The standard
deviation of this sample is $2,050.
The assosiation would like answer to the
following
Questions :
a.What is the population mean?
b.What is a reasonable range of values for
the population mean?
c.What do these results mean?
The Answers :
a. What is the population mean ?
in this case, we do not know. We do
know the sample mean is $45,420.
Hence, our best estimate of the unknown
population value is the corresponding
sample statistic. Thus the sample mean
of $45, 420 is a point estimate of the
unknown population mean.
b.What is a reasonable range of values for the
population mean?
The association decides to use the 95 percent
level of confidence. To determine the
corresponding confidence interval we use :
X+z s
= $45,420 + 1.96 $ 2,050
√n
√ 256
= $45,420 + $251
the usual practice is to round these endpoints to
$45,169 and $45,671. these endpoints called
the confidence limits. The degree of confidence
or the level of confidence is 95 percent and the
confidence interval is from $45,169 to $45,671.
c. What do these results mean?
Suppose we select many samples of 256
managers, perhaps several hundred. For
each sample,we compute the mean and
the standard deviation and then construct
a 95 percent confidence interval, such as
we did in the previous section. We could
expect about 95% of these confidence
intervals to contain the population mean.
About 5% of the intervals would not
contain the population mean annual
income, which is μ.
A Confidence Interval for Proportion
Proportion → the fraction, ratio, or percent
indicating the part of the sample or the population
having a particular trait of interest.
Sample Proportion : p = x
n
Confidence Interval for a Population Proportion
p + zσp
Standar Error of The Sample Proportion
σp = √ p(1-p)
n
Confidence Interval for A Population Proportion
p + z = √ p(1-p)
n
Example :
The union representing the Bottle Blowers of
America (BBA) is considering a proposal to merge
with the teamsters union. According to BBA union
Bylaws, at least threefourths of the union
Membership must approve any merger. A random
Sample of 2,000 current BBA members reveals
1,600 plan to vote for the merger proposal.
Questions:
What is the estimate of the population proportion ?
Develop a 95 % confidence interval for the
Population proportion. Basing your decision on this
Sample information, can you conclude that the
Necessary of BBA member favor the merger, why?
Answer :
First, calculate the sample proportion from formula
p = x = 1,600 = 0,80
n
2,000
Thus, we estimate that 80% of the population favor the
merger proposal. The z value corresponding to
the 95% level of confidence is 1.96
p + z √ p(1-p) = 0.80 + 1.96√ 0.80(1-0.80)
n
2,000
= 0.80 + 0.018
the endpoints of the confidence interval are .782
and .818. the lower endpoint is greater than .75. Hence, we
conclude that the merger proposal will likely pass because
the interval estimate includes values greater than 75% of
the unions memberships.