Taufique Shaikh

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Sampling:Design and Procedures


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SamplingThe researcher generally seeks to draw conclusions about largenumber of individuals- i.e. population or universe.

Since researchers operate with limited time, energy, and economicresources, they rarely study each and every member of a givenpopulation.

Instead, researchers study only a sample that is a small number ofobservations from the population.

Through the sampling process, the researchers seek to generalize froma sample (a small group) to the entire population from which it was

taken (a large group).


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11-3

Basic Concepts in Samples and Sampling

Population:the entire group under study asdefined by research objectives. Sometimescalled the universe.

Researchers define populations in specific termssuch as heads of households, individual persontypes, families, types of retail outlets, etc.Population geographic location and time of studyare also considered.


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11-4

Basic Concepts in Samples and Sampling

Sample:a subset of the population that shouldrepresent the entire group

Sample unit:the basic level ofinvestigationconsumers, store managers, shelf-facings, teens, etc. The research objectiveshould define the sample unit

Census:an accounting of the completepopulation


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11-5

Reasons for Taking a Sample

Practical considerations such as cost andpopulation size

Inability of researcher to analyze large quantitiesof data potentially generated by a census

Samples can produce sound results if properrules are followed for the draw


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11-6

Basic Sampling Classifications

Probability samples:ones in which members ofthe population have a known chance (probability)of being selected

Non-probability samples:instances in which thechances (probability) of selecting members fromthe population are unknown


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11-7

Classification of Sampling Techniques

Fig. 11.2

Sampling Techniques

NonprobabilitySampling Techniques

ProbabilitySampling Techniques

Convenience

Sampling

Judgmental

Sampling

Quota

Sampling

Snowball

Sampling

Systematic

Sampling

Stratified

Sampling

Cluster

Sampling

Other Sampling

Techniques

Simple Random

Sampling


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Nonprobability Sampling

Techniques


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Convenience Sampling

Convenience sampling attempts to obtaina sample of convenient elements. Often,respondents are selected because theyhappen to be in the right place at the right

time.

use of students, and members of socialorganizations

people on the street interviews


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Judgmental Sampling

Judgmental sampling is a form of conveniencesampling in which the population elements areselected based on the judgment of the researcher.


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11-11

Quota Sampling

Quota sampling may be viewed as two-stage restricted judgmentalsampling.

The first stage consists of developing control categories, or quotas,of population elements.

In the second stage, sample elements are selected based onconvenience or judgment.

Population Samplecomposition composition

ControlCharacteristic Percentage Percentage Number

SexMale 48 48 480Female 52 52 520

____ ____ ____100 100 1000


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11-12

Snowball Sampling

In snowball sampling, an initial group ofrespondents is selected, usually at random.

After being interviewed, these respondents areasked to identify others who belong to the targetpopulation of interest.

Subsequent respondents are selected based onthe referrals.

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11-13

Probability

Sampling Techniques


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Random sampling gives each and every member ofpopulation an equal chance of being selected for the

sample.

One way to conduct a simple random sample is to assign anumber to each element in the population, write thesenumber on individual slips of paper, toss them into a hat,

and draw the required member of slip (the sample size, n)from the hat.

Sometimes the elements of the population are already numbered. Forexample population of drivers have driving license number, allemployees of a firm have employee number and all university Student

have student card number.

Simple Random Sampling

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11-15

Simple Random Sampling

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11-16

Systematic Sampling

The sample is chosen by selecting a random starting

point and then picking every ith element insuccession from the sampling frame.

The sampling interval, i, is determined by dividing the

population size N by the sample size n and roundingto the nearest integer.

For example, there are 100,000 elements in thepopulation and a sample of 1,000 is desired. In thiscase the sampling interval, i, is 100. A randomnumber between 1 and 100 is selected. If, forexample, this number is 23, the sample consists ofelements 23, 123, 223, 323, 423, 523, and so on.

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11-17

Systematic Sampling

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11-18

This method is used when the populationdistribution of items is skewed. It allows usto draw a more representative sample.Hence if there are more of certain type of

item in the population the sample has moreof this type and if there are fewer of anothertype, there are fewer in the sample.

Stratified Sampling


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A stratified random sample is obtained by dividing thepopulation into more homogeneous groups or strata from

which simple random samples are then taken. Examples ofcriteria for separating a population into strata are:

1. Sex : male, female

2. Age: under 20, 21-30, 31-40, 41-50, 51-60, over 603. Household income: under Rs. 8,000, Rs. 8,000-

19,999, Rs.20,000-50,000, over Rs. 50,000

Stratified Sampling

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11-20

Stratified Sampling


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A cluster sample is a simple random sample of group or

clusters of elements.

Suppose we want to estimate the average annualhousehold income in a large city.

A less expensive alternative would be to let each blockwithin the city represent a cluster. By reducing the distance,cluster sampling reduces the cost.

Cluster Sampling

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Cluster Sampling


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Sampling ErrorSampling error is an error that we expect to occur when wehave a statement about a population that is based only onthe observation contained in a sample taken from thepopulation.

We can use statistical inference to estimate the mean ofthe population, if we are willing to accepts less than 100%accuracy. The difference between the true (unknown)value of the population mean and its sample estimate isthe sampling error.

Then only way we can reduce the expected size of thiserror is to take a larger sample.


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Sampling Error

Population

70 80 93

86 85 90

56 52 67

40 78 57

89 49 48

99 72 3096 94

=

96 40 72

99 86 96

56 56 49

52 67 56

Sample A Sample B Sample C

Final Examination Grade


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Sampling Error

75.75 25.62 25.68

Final Examination Grade

96 40 72

99 86 96

56 56 49

52 67 56

303 249 273

Population

Sample A Sample B Sample C

70 80 93

86 85 90

56 52 67

40 78 57

89 49 48

99 72 30

96 94

= 71.55


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Non Sampling Error

Non sampling error is more serious, because

taking a larger sample wont diminish the size, orthe possibility of occurrence, of this error. Even

census can contain non sampling error.

Non sampling errors are reporting error, nonresponse error, data entry error.


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Sampling Distribution of Mean

Consider the population created by throwing

a fair die many times, with the randomvariable x indicating the number of spotsshowing on any one throw.

Find the probability distribution of therandom variable x. Its mean andVariance.

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Probability distribution of X

x 1 2 3 4 5 6P(x) 1/6 1/6 1/6 1/6 1/6 1/6


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Mean and Variance of X

)(xE )(xPx

6

16

6

15

6

14

6

13

6

12

6

11

5.3

)(2

xV )(.)( 2 xPx

6

15.36

6

15.35

6

15.34

6

15.33

6

15.32

6

15.31

222222

92.2


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XX

All Samples of Size Two and Their Means

Sample Sample X Sample


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XX

All Samples of Size Two and Their Means

Sample

1, 1 1.0 3, 1 2.0 5, 1 3.0

1, 2 1.5 3, 2 2.5 5, 2 3.5

1, 3 2.0 3, 3 3.0 5, 3 4.0

1, 4 2.5 3, 4 3.5 5, 4 4.5

1, 5 3.0 3, 5 4.0 5, 5 5.01, 6 3.5 3, 6 4.5 5, 6 5.5

2, 1 1.5 4, 1 2.5 6, 1 3.5

2, 2 2.0 4, 2 3.0 6, 2 4.0

2, 3 2.5 4, 3 3.5 6, 3 4.5

2, 4 3.0 4, 4 4.0 6, 4 5.0

2, 5 3.5 4, 5 4.5 6, 5 5.5

2, 6 4.0 4, 6 5.0 6, 6 6.0

Sample X Sample


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Sampling Distribution of

P( )

1.0 1/36

1.5 2/362.0 3/36

2.5 4/36

3.0 5/36

3.5 6/36

4.0 5/36

4.5 4/36

5.0 3/36

5.5 2/36

6.0 1/36

X X

X

Calculate the mean and

variance of this sampling Distribution.


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Mean and Variance of the sampling Distribution)(xE )(xPx

36

10.6.....

36

25.1

36

10.1

5.3

)(2

xV )(.)( 2 xPx X

36

15.36...........

36

15.35.1

36

15.30.1

222

46.1

X

92.22 46.1

2x

nx

22

nx

xn22

.

X

Z

X

XZ

n

XZ

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Central Limit Theorem

1. The random variablexhas a distribution

(which may or may not be normal) with mean and standard deviation .

2. Samples all of the same size nare randomly

selected from the population ofxvalues.

Given:

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Conclusions:

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1. The distribution of samplex will, as the

sample size increases, approach a normal

distribution.

Conclusions:

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distribution.2. The mean of the sample means will be the

population mean .

Conclusions:

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distribution.2. The mean of the sample means will be the

population mean .

3. The standard deviation of the sample means

will approach n

Conclusions:

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Practical Rules Commonly Used:

1. For samples of sizen

larger than 30, thedistribution of the sample means can be

approximated reasonably well by a normal

distribution. The approximation gets better as

the sample size nbecomes larger.

2. If the original population is itself normally

distributed, then the sample means will be

normally distributed for any sample size n(not just the values of nlarger than 30).

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Notation

the mean of the sample means

x=

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Notation


the standard deviation of sample mean

x=

x= n

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Notation


the standard deviation of sample mean

(often called standard error of the mean)

x=

x= n

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Example

A population consists of the five numbers

2,3,6,8, and 11. Consider all possible samples ofsize 2 that can be drawn with replacement fromthis population.

Find (a) the mean of the population,(b) the standard deviation of the population,(c) the mean of the sampling distribution of

means, and

(d) the standard deviation of the samplingdistribution of means (i.e., the standarderror of means).

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