Statistics Sampling and Sampling Distribution 1/101.

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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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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Sampling without replacement is the procedure used most often.

Replacing each sampled element before selecting subsequent elements is called sampling with replacement.

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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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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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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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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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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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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



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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(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


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


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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• 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 proportion wanting campus housing

.72 .68

Summary of Point EstimatesObtained from a Simple Random Sample

= Sample mean SAT score

x

= Sample proportion 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

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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

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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?


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.


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

. . . . . . . . . . .


i i

i

wxx

w

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x990


14.6x

1000

Area = .7517


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)


i i

i

wxx

w

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x980 990

Area = .2483


14.6x


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


i i

i

wxx

w

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x1000980 990

Area = .5034


14.6x


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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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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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


8x


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


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pp p

nN nN

( )11

pp p

n ( )1

is referred to as the standard error of the proportion.

p


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


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Example: St. Andrew’s College


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


p

SamplingDistribution

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.


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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

. . . . . . . . . . .


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.77.72

Area = .7291

p


of p

.082p


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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)


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.67 .72

Area = .2709

p


of p.082p


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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 :


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.77.67 .72

Area = .4582

p


of p.082p


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Point Estimators Notations: θ = the population parameter of interest.

For example, population mean, population standard deviation, population proportion, and so on.

^

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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.”

^

^

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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

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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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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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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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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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Statistics Sampling and Sampling Distribution 1/101.

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