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Statistics
Sampling and Sampling Distribution
1/101
STATISTICS in PRACTICE
MeadWestvaco Corporation’s products include textbook paper, magazine paper, and office products. MeadWestvaco’s internal consulting group uses sampling to provide information
that enables the company to obtain significant productivity benefits and remain competitive.
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STATISTICS in PRACTICE
Managers need reliable and accurate information about the timberlands and forests to evaluate the company’s ability to meet its future raw material needs.
Data collected from sample plots throughout the forests are the basis for learning about the population of trees owned by the company.
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Contents
The Electronics Associates Sampling Problem Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of p Properties of Point Estimators Other Sampling Methods
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The purpose of statistical inference is to obtain information about a population from information contained in a sample.
Statistical Inference
A population is the set of all the elements of interest. A sample is a subset of the population.
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The sample results provide only estimates of the values of the population characteristics.
A parameter is a numerical characteristic of a population.
With proper sampling methods, the sample results can provide “good” estimates of the population characteristics.
Statistical Inference
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The Electronics Associates Sampling Problem
Often the cost of collecting information from a sample is substantially less than from a population,
Especially when personal interviews must be conducted to collect the information.
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Simple Random Sampling:Finite Population
Finite populations are often defined by lists such as: Organization membership roster Credit card account numbers Inventory product numbers
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Simple Random Sampling:Finite Population
A simple random sample of size n from a
finite population of size N is a sample
selected such that each possible sample of
size n has the same probability of being
selected.
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Simple Random Sampling:Finite Population
Sampling without replacement is the procedure used most often.
Replacing each sampled element before selecting subsequent elements is called sampling with replacement.
10/101
Simple Random Sampling Random Numbers: the numbers in the table are
random, these four-digit numbers are equally likely.
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Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.
Simple Random Sampling:Infinite Population
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Simple Random Sampling:Infinite Population
A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied.
Each element selected comes from the same population. Each element is selected independently.
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Simple Random Sampling:Infinite Population
The random number selection procedure
cannot be used for infinite populations.
In the case of infinite populations, it is impossible to obtain a list of all elements in the population.
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In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter.
Point Estimation
We refer to as the point estimator of the population mean .
x
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s is the point estimator of the population standard deviation .
Point Estimation
is the point estimator of the population proportion p.
p
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Point Estimation
Example: to estimate the population mean, the population standard deviation and population proportion.
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Sampling Error
The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.
When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased.
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Sampling Error
Statistical methods can be used to make probability statements about the size of the sampling error.
Sampling error is the result of using a subset of the population (the sample), and not the entire population.
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Sampling Error
The sampling errors are:
| |p p for sample proportion
| |s for sample standard deviation
| |x for sample mean
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Example: St. Andrew’s St. Andrew’s College
receives 900 applications annually from prospective students. The application form contains a variety of informationincluding the individual’s scholastic aptitudetest (SAT) score and whether or not the individual desires on-campus housing.
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Example: St. Andrew’s The director of admissions
would like to know the
following information: the average SAT score
for the 900 applicants,
and the proportion of applicants that want to
live on campus.22/101
Example: St. Andrew’s
We will now look at three
alternatives for obtaining
The desired information. Conducting a census of the entire 900 applicants Selecting a sample of 30
applicants, using a random number table Selecting a sample of 30 applicants, using Excel
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Conducting a Census
If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3.
We will assume for the moment that conducting a census is practical in this example.
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990900
ix
2( )80
900ix
Conducting a Census
648.72
900p
Population Mean SAT Score
Population Standard Deviation for SAT Score
Population Proportion Wanting On-Campus Housing
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Simple Random Sampling
Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours.
Now suppose that the necessary data on the current year’s applicants were not yet entered in the college’s database.
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Simple Random Sampling
Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours.
Now suppose that the necessary data on the current year’s applicants were not yet
entered in the college’s database.
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Simple Random Sampling
The applicants were numbered, from 1 to 900, as their applications arrived.
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Taking a Sample of 30 Applicants
Simple Random Sampling:Using a Random Number Table
Because the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900.
We will use the last three digits of the 5-digit random numbers in the third column of the textbook’s random number table , and continue into the fourth column as needed.29/101
Taking a Sample of 30 Applicants
Simple Random Sampling:Using a Random Number Table
The numbers we draw will be the numbers of
the applicants we will sample unless the
random number is greater than 900 or
the random number has already been used. We will continue to draw random numbers
until we have selected 30 applicants for our sample.
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Simple Random Sampling:Using a Random Number Table
(We will go through all of column 3 and part of column 4 of the random number table, encountering in the process five numbers greater than 900 and one duplicate, 835.)
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Use of Random Numbers for Sampling
Simple Random Sampling:Using a Random Number Table
744436865790835902190836
. . . and so on
3-DigitRandom Number
ApplicantIncluded in Sample
No. 436No. 865No. 790No. 835
Number exceeds 900No. 190No. 836
No. 744
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Sample Data
Simple Random Sampling:Using a Random Number Table
1 744 Conrad Harris 1025 Yes2 436 Enrique Romero 950 Yes3 865 Fabian Avante 1090 No
4 790 Lucila Cruz 1120 Yes5 835 Chan Chiang 930 No. . . . .
30 498 Emily Morse 1010 No
No.RandomNumber Applicant
SAT Score
Live On-Campus
. . . . .
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Taking a Sample of 30 Applicants
• Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample.
• For example, Excel’s function = RANDBETWEEN(1,900) can be used to generate random numbers between 1 and 900.
• Computers can be used to generate random numbers for selecting random samples.
Simple Random Sampling:Using a Computer
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29,910997
30 30ix
x
2( ) 163,99675.2
29 29ix x
s
Point Estimation
s as Point Estimator of
as Point Estimator of i i
i
wxx
w
– p as Point Estimator of p
.6820/30 p
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Point Estimation
Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.
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PopulationParameter
PointEstimator
PointEstimate
ParameterValue
m = Population mean SAT score
990 997
s = Population std. deviation for SAT score
80 s = Sample std. deviation for SAT score
75.2
p = Population pro- portion wanting campus housing
.72 .68
Summary of Point EstimatesObtained from a Simple Random Sample
= Sample mean SAT score
x
= Sample pro- portion wanting campus housing
p
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Sampling Distribution Example: Relative Frequency Histogram of Sample
Mean Values from 500 Simple Random Samples of
30 each.
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Sampling Distribution Example: Relative Frequency Histogram of Sample
Proportion Values from 500 Simple Random
Samples of 30 each.
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Process of Statistical Inference
The value of is used tomake inferences aboutthe value of m.
x The sample data provide a value forthe sample mean .x
A simple random sampleof n elements is selectedfrom the population.
Population with meanm = ?
Sampling Distribution of i i
i
wxx
w
The sampling distribution of is the probability
distribution of all possible values of the sample
mean .
Sampling Distribution of
where: = the population mean
E( ) = x
Expected Value ofi i
i
wxx
w
i i
i
wxx
w
i i
i
wxx
w
i i
i
wxx
w
41/101
Finite Population Infinite Population
x n
N nN
( )1
x n
• A finite population is treated as being infinite if n/N < .05.
Standard Deviation ofx
Sampling Distribution of x
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• is referred to as the standard error of the mean.
x
• is the finite correction factor.
)1/()( NnN
Sampling Distribution of x
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Form of the Sampling Distribution of i i
i
wxx
w
If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal distribution.
x
When the simple random sample is small (n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal distribution.
x
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Central Limit Theorem Illustration of The Central Limit Theorem
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Relationship Between the Sample Size and the Sampling Distribution of Sample Mean
A Comparison of The Sampling Distributions of Sample Mean for Simple Random Samples of n = 30 and n = 100.
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Sampling Distribution of for SAT Scores
i i
i
wxx
w
8014.6
30x
n
( ) 990E x x
SamplingDistributionof x
47/71
What is the probability that a simple
random sample of 30 applicants will provide
an estimate of the population mean SAT score
that is within +/-10 of the actual population mean ?
In other words, what is the probability that
will be between 980 and 1000?
Sampling Distribution of for SAT Scores
i i
i
wxx
w
i i
i
wxx
w
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Step 1: Calculate the z-value at the upper endpoint of the interval.
z = (1000 - 990)/14.6= .68
P(z < .68) = .7517
Step 2: Find the area under the curve to the left of the upper endpoint.
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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Cumulative Probabilities for the Standard Normal Distribution
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
. . . . . . . . . . .
.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
. . . . . . . . . . .
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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x990
SamplingDistributionof x
14.6x
1000
Area = .7517
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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Step 3: Calculate the z-value at the lower endpoint of the interval.
Step 4: Find the area under the curve to the left of the lower endpoint.
z = (980 - 990)/14.6= - .68
P(z < -.68) = P(z > .68)
= .2483= 1 - . 7517= 1 - P(z < .68)
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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x980 990
Area = .2483
SamplingDistributionof x
14.6x
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval.
P(-.68 < z < .68) = P(z < .68) - P(z < -.68)= .7517 - .2483= .5034
The probability that the sample mean SAT score will be between 980 and 1000 is:
P(980 < < 1000) = .5034x
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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x1000980 990
Area = .5034
SamplingDistributionof x
14.6x
Sampling Distribution of for SAT Scores
i i
i
wxx
w
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Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.
Relationship Between the Sample Size and the Sampling Distribution of i i
i
wxx
w
E( ) = m regardless of the sample size.
in our example, E( ) remains at 990.
i i
i
wxx
w
i i
i
wxx
w
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Relationship Between the Sample Size and the Sampling Distribution of i i
i
wxx
w
Whenever the sample size is increased, the standard error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to:
8.0100
80x
n
σσ
xσ
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Relationship Between the Sample Size and the Sampling Distribution of i i
i
wxx
w
( ) 990E x x
14.6x With n = 30,
8x With n = 100,
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Recall that when n = 30, P(980 < < 1000) = .5034.i i
i
wxx
w
We follow the same steps to solve for P(980 < < 1000) when n = 100 as we showed earlier when n = 30.
i i
i
wxx
w
i i
i
wxx
w
Relationship Between the Sample Size and the Sampling Distribution of
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Relationship Between the Sample Size and the Sampling Distribution of
Now, with n = 100, P(980 < < 1000) = .7888.i i
i
wxx
w
Because the sampling distribution with n = 100 has a smaller standard error, the values of have less variability and tend to be closer to the population mean than the values of with n = 30.
i i
i
wxx
w
i i
i
wxx
w
i i
i
wxx
w
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x1000980 990
Area = .7888
SamplingDistributionof x
8x
Relationship Between the Sample Size and the Sampling Distribution of i i
i
wxx
w
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Sampling Distribution
Example: Relative Frequency Histogram of Sample Proportion Values from 500 Simple Random Samples of 30 each.
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A simple random sampleof n elements is selected
from the population.
Population with proportion
p = ?
Making Inferences about a Population Proportion
The sample data provide a value for thesample proportion .p
The value of is usedto make inferences
about the value of p.
p
Sampling Distribution of p
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E p p( )
where:p = the population proportion
The sampling distribution of p is the probabilitydistribution of all possible values of the sampleproportion p .
Expected Value of p
Sampling Distribution of p
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pp p
nN nN
( )11
pp p
n ( )1
is referred to as the standard error of the proportion.
p
Sampling Distribution of
Finite Population Infinite Population
Standard Deviation of p
p
The sampling distribution of can be approximated by a normal distribution whenever the sample size is large.
p
The sample size is considered large whenever these conditions are satisfied:
np > 5 n(1 – p) > 5and
Form of the Sampling Distribution of p
For values of p near .50, sample sizes as small as 10 permit a normal approximation.
With very small (approaching 0) or very
large (approaching 1) values of p, much larger samples are needed.
pForm of the Sampling Distribution of
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Recall that 72% of the prospective students
applying to St. Andrew’s College desire
on-campus housing.
Example: St. Andrew’s College
Sampling Distribution of p
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Example: St. Andrew’s College
Sampling Distribution of
What is the probability that a simple random
sample of 30 applicants will provide an estimate
of the population proportion of applicant
desiring on-campus housing that is within plus or
minus .05 of the actual population proportion?
p
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For our example, with n = 30 and p = .72,
the normal distribution is an acceptable
approximation because:
n(1 - p) = 30(.28) = 8.4 > 5and
np = 30(.72) = 21.6 > 5
Sampling Distribution of p
p
SamplingDistribution
of p
Sampling Distribution of p
082.30
)72.1(72.
p
72.)( pE
Step 1: Calculate the z-value at the upper endpoint of the interval.
z = (.77 - .72) /.082 = .61
P(z < .61) = .7291
Step 2: Find the area under the curve to the left of the upper endpoint.
Sampling Distribution of p
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Cumulative Probabilities for the Standard Normal Distribution
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
. . . . . . . . . . .
.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
. . . . . . . . . . .
Sampling Distribution of p
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.77.72
Area = .7291
p
SamplingDistribution
of p
.082p
Sampling Distribution of p
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Step 3: Calculate the z-value at the lower endpoint of the interval.
Step 4: Find the area under the curve to the left of the lower endpoint.
z = (.67 - .72) /.082 = - .61
P(z < -.61) = P(z > .61)
= .2709= 1 - . 7291= 1 - P(z < .61)
Sampling Distribution of p
75/101
.67 .72
Area = .2709
p
SamplingDistribution
of p.082p
Sampling Distribution of p
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P(.67 < < .77) = .4582p
Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval.
P(-.61 < z < .61) = P(z < .61) - P(z < -.61)= .7291 - .2709= .4582
The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion :
Sampling Distribution of p
77/101
.77.67 .72
Area = .4582
p
SamplingDistribution
of p.082p
Sampling Distribution of p
78/101
Point Estimators Notations: θ = the population parameter of interest.
For example, population mean, population standard deviation, population proportion, and so on.
^
79/101
Point Estimators Notations: θ = the sample statistic or point estimator
of θ .
Represents the corresponding sample statistic such as the sample mean, sample standard deviation, and sample proportion.
The notation θ is the Greek letter theta. the notation θ is pronounced “theta-hat.”
^
^
80/101
Properties of Point Estimators
Before using a sample statistic as a point estimator, statisticians check to see whether the sample statistic has the following properties associated with good point estimators.
Consistency
Efficiency
Unbiased
81/101
Properties of Point Estimators
If the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an unbiased estimator of the population parameter.
Unbised
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Properties of Point Estimators
Unbised
The sample statistic θ is unbiased estimator of the population parameter θ if E(θ)= θ
^
where
E(θ)=the expected value of the sample statistic θ
^
^
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Properties of Point Estimators Examples of Unbiased and Biased Point
Estimators
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Properties of Point Estimators
Given the choice of two unbiased estimators of the same population parameter, we would prefer to use the point estimator with the smaller standard deviation, since it tends to provide estimates closer to the population parameter.
The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other.
Efficiency
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Properties of Point Estimators Example: Sampling Distributions of Two
Unbiased Point Estimators.
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Properties of Point Estimators
A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger.
Consistency
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Other Sampling Methods
Stratified Random Sampling(分層隨機抽樣 ) Cluster Sampling(部落抽樣) Systematic Sampling(系統抽樣) Convenience Sampling(便利抽樣)
Judgment Sampling(判斷抽樣)
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The population is first divided into groups of elements called strata.
Stratified Random Sampling
Each element in the population belongs to one and only one stratum. Best results are obtained when the elements within each stratum are as much alike as possible (i.e. a homogeneous group).
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Stratified Random Sampling
Diagram for Stratified Random Sampling
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Stratified Random Sampling A simple random sample is taken from each stratum. Formulas are available for combining the stratum sample results into one population parameter estimate.
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Stratified Random Sampling Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size.
Example: The basis for forming the strata might be department, location, age, industry type, and so on.
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Cluster Sampling
The population is first divided into separate groups of elements called clusters. Ideally, each cluster is a representative small- scale version of the population (i.e. heterogeneous group). A simple random sample of the clusters is then taken. All elements within each sampled (chosen) cluster form the sample.
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Cluster Sampling
Diagram for Cluster Sampling
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Cluster Sampling
Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be obtained in a short time). Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling.
Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.
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Systematic Sampling
If a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population. We randomly select one of the first n/N elements from the population list. We then select every n/Nth element that follows in the population list.
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Systematic Sampling This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.
Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used.
Example: Selecting every 100th listing in a telephone book after the first randomly selected listing 97/101
Convenience Sampling It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.
Example: A professor conducting research might use student volunteers to constitute a sample.
The sample is identified primarily by convenience.
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Advantage: Sample selection and data collection are relatively easy.
Disadvantage: It is impossible to determine how representative of the population the sample is.
Convenience Sampling
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Judgment Sampling The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population. It is a nonprobability sampling technique.
Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.
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Judgment Sampling Advantage: It is a relatively easy way of selecting a sample.
Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample.
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