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Types of Non probabilityTypes of Non probability

SamplingSampling Judgment SamplingJudgment Sampling. Sample items are. Sample items are

selected by using a researchers personalselected by using a researchers personaljudgment judgment

Convenience Sampling.Convenience Sampling. SelectingSelectingsample items that are close at hand orsample items that are close at hand orotherwise easy to obtainotherwise easy to obtain

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Quota Sampling.Quota Sampling. Resea rchers det ermin eResea rchers det ermin et he p ercentag e o f t he ta rg et population t hat t he p ercentag e o f t he ta rg et population t hat po ssesses t he cha ra ct eri sti cs o f int erest an dpo ssesses t he cha ra ct eri sti cs o f int erest an dt hen sp eci fy t he num ber o f t hese in divi dual st hen sp eci fy t he num ber o f t hese in divi dual s

to be in clu ded in t he sampl e to ref l ect t hei rto be in clu ded in t he sampl e to ref l ect t hei rp ropo rtion in t he population p ropo rtion in t he population

Snow ball Sampling.Snow ball Sampling. Initial respon dent sInitial respon dent s

p rovi de nam es o f a dditional respon dent s to p rovi de nam es o f a dditional respon dent s to in clu de in a sampl e. Resea rchers u se t hi sin clu de in a sampl e. Resea rchers u se t hi sreferral m et ho d w hen pot ential respon dent sreferral m et ho d w hen pot ential respon dent sa re di ff i cult to lo cat e becau se t hey a re a a re di ff i cult to lo cat e becau se t hey a re a

tin y pa rt o f t he enti re population tin y pa rt o f t he enti re population

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Simple Data ViewingSimple Data Viewing

Frequency distributionFrequency distribution

The distribution of data into classes orThe distribution of data into classes or

groups along with their frequencies.groups along with their frequencies.

Probability distributionProbability distribution

The distribution of data into classes orThe distribution of data into classes orgroups along with their probabilities.groups along with their probabilities.

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xx f f

2222 11

2323 222424 33

2525 22

2626 11

xx f f f (x)f (x)

2222 11 1/91/9

2323 22 2/92/9

2424 33 3/93/9

2525 22 2/92/9

2626 11 1/91/9

Frequency distribution probability distribution

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0

0.5

1

1.5

2

2.5

3

22 23 24 25 26

Frequency histogram

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Types of frequency curvesTypes of frequency curves

Symmetric distributionSymmetric distribution

Asymmetric or skewed distributionAsymmetric or skewed distribution

Extremely skewed or j shaped distributionExtremely skewed or j shaped distribution U shaped distributionU shaped distribution

Uniform distributionUniform distribution

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Basic StatisticsBasic StatisticsPopulationPopulation SampleSample

MeanMean QQ X X

VarianceVariance WW22 ss22

Standard DeviationStandard Deviation WW ssSizeSize NN nn

X=x/n

S²= (x- x)²/n S²= x²/n (x/n) ²S²= f (x- x)²/ f S²= f x²/ f (f x/ f ) ²

X=f x /f

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Sampling Distribution of a Sample MeanSampling Distribution of a Sample Mean

Random Variable isRandom Variable is Sample StatisticSample Statistic::

Sample Mean, Sample ProportionSample Mean, Sample Proportion

A A

sampling distribution is a sampling distribution is a probability distribution of aprobability distribution of arandom variable along with their probabilities.random variable along with their probabilities.

A A sampling distribution of the sample mean sampling distribution of the sample mean isis thethedistribution of sample means ( by taking all possibledistribution of sample means ( by taking all possible

samples of same size) along with their probabilities.samples of same size) along with their probabilities. A A sampling distribution sampling distribution has their own mean and standardhas their own mean and standard

deviation.deviation.

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R andom variable,R andom variable, X X ,,

isis AgeAge of individualsof individuals

Values of Values of X X : : 18, 20, 22, 2418, 20, 22, 24

measured in yearsmeasured in years

EVERYONE is one of these 4EVERYONE is one of these 4ages in this populationages in this population

© 1984-1994 T/Maker Co.

DevelopingSampling Distributions

A

B C

D

Suppose there¶s a

population...

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2362

214

24222018

1

2

1

.N

X

N

X

N

i i

N

i i

!§

!

!

!

§

!

!

!

QW

Q

Population Characteristics

Summary Measure PopulationDistribution

.3

.2

.1

0A B C D

(18) (20) (22) (24)

Uniform Distribution

P(X)

X

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Sample #Sample # samplessamples meansmeans

11 18,1818,18 1818

22 18,2018,20 1919

33 18,2218,22 2020

44 18,2418,24 2121

55 20,1820,18 1919

66 20,2020,20 2020

77 20,2220,22 2121

88 20,2420,24 2222

Sample #Sample # samplessamples meansmeans

99 22,1822,18 2020

1010 22,2022,20 2121

1111 22,2222,22 2222

1212 22,2422,24 2323

1313 24,1824,18 2121

1414 24,2024,20 2222

1515 24,2224,22 2323

1616 24,2424,24 2424

Samples Taken with ReplacementThere will be (N )n samples i, e 42=16

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Sampling Distribution of mean

xx f f f ( x )f ( x )

1818 11 1/161/16

1919 22 2/162/16

2020 33 3/163/16

2121 44 4/164/16

2222 33 3/163/16

2323 22 2/162/16

2424 11 1/161/16

totaltotal 1616 11

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18 19 20 21 22 23 240

.1

.2

.3

P(X)

X

Sample Means

Distribution

Sampling Distribution

of All Sample Means

_

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18 19 20 21 22 23 240

.1

.2

.3P(X)

X

Sample Means Distributionn = 2

Comparing the Population with its

Sampling Distribution

A B C D

(18) (20) (22) (24)

0

.1

.2

.3

Population

Q= 21, W= 2.236P(X)

X

21!x

Q 581 .x

!W

_

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Mean and standard deviation of Mean and standard deviation of

sampling distributionsampling distribution Mean of the sampling distributionMean of the sampling distribution

µµxx=x f (x)=x f (x)

Variance of the sampling distributionVariance of the sampling distribution

²²xx==xx22 f (x)f (x)-- {x f (x)} {x f (x)} 22

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sampling distributionsampling distributionxx f (x)f (x) x f( x)x f( x) xx22 f (x)f (x)

1818 1/161/16 18/1618/16

1919 2/162/16 38/1638/162020 3/163/16 60/1660/16

2121 4/164/16 84/1684/16

2222 3/163/16 66/1666/162323 2/162/16 46/1646/16

2424 1/161/16 24/1624/16

totaltotal 11 336/16336/16

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Properties of sampling distributionProperties of sampling distribution

µµxx=µ=µ²²xx= = ²² (with replacement )(with replacement )

nn Shape of the distributionShape of the distribution If the population is normal then the sampling distributionIf the population is normal then the sampling distribution

of mean will also be normal regardless of sample sizeof mean will also be normal regardless of sample size If the population sampled is non normal, then forIf the population sampled is non normal, then for

sufficiently large sample size, the sampling distributionsufficiently large sample size, the sampling distributionof mean will approximate the normal distribution.of mean will approximate the normal distribution.

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Standardized form of a randomStandardized form of a random

variablevariable Standardized form of random variable canStandardized form of random variable can

be obtained by subtracting its mean frombe obtained by subtracting its mean fromit and dividing the difference by itsit and dividing the difference by itsstandard deviation.standard deviation.

z= z= r.v r.v meanmean

st devst dev

z= z= xx--µµx x ==xx--µµ

x x /n/n

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Example 14.8Example 14.8

N=5 consists of 0,3,6,9,12 values aN=5 consists of 0,3,6,9,12 values asample of size n=3 is randomly selectedsample of size n=3 is randomly selectedwithout replacement .without replacement .

Find sampling distribution & verify itsFind sampling distribution & verify itsproperties.properties.

Solution : the number of samples of sizeSolution : the number of samples of sizen=3 which can be drawn without n=3 which can be drawn without replacement isreplacement is NNCCnn= = 55CC33=10=10

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Sample #Sample # SamplesSamples MeansMeans

11 0,3,60,3,6 33

22 0,3,90,3,9 44

33 0,3,120,3,12 55

44 0,6,90,6,9 55

55 0,6,120,6,12 66

66 0,9,120,9,12 77

77 3,6,93,6,9 66

88 3,6,123,6,12 77

99 3,9,123,9,12 88

1010 6,9,126,9,12 99

All possible sampleand their mean

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Sampling distribution of meanSampling distribution of meanwithout replacement without replacement

xx f f f( x )f( x )33 11 1/101/10

44 11 1/101/10

55 22 2/102/1066 22 2/102/10

77 22 2/102/10

88 11 1/101/10

99 11 1/101/10

totaltotal 1010 11

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xx f f f( x )f( x )

33 11 1/101/10

44 11 1/101/10

55 22 2/102/10

66 22 2/102/10

77 22 2/102/10

88 11 1/101/10

99 11 1/101/10

totaltotal 1010 11

x f (x)x f (x) xx22f ( x )f ( x )

3/103/10 9/109/10

4/104/10 16/1016/10

10/1010/10 50/1050/10

12/1012/10 72/1072/10

14/1014/10 98/1098/10

8/108/10 64/1064/10

9/109/10 81/1081/10

60/1060/10 390/10390/10

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Mean and standard deviationMean and standard deviation

Mean of the sampling distributionMean of the sampling distribution

µµxx=x f (x)=60/10=6=x f (x)=60/10=6

Variance of the sampling distributionVariance of the sampling distribution

²²xx==xx22 f (x)f (x)-- {x f (x)} {x f (x)} 22

=390/10=390/10 (60/10)²(60/10)²=3=3

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populationpopulation µ=x/N=30/5=6µ=x/N=30/5=6

²= x²/N²= x²/N-- (x/N)²(x/N)²

=18=18

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Properties of sampling distributionProperties of sampling distribution

µµxx=µ=µ²²xx==²² NN--nn ( without replacement )( without replacement )

n n-1fpc= N-n/n-1If n 5% of the population size use fpc.

Shape of the distribution:- Standardized random variable:-

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