Sampling and Sampling Distributions: Part 2 Sample size and the sampling distribution of Sampling...

Sampling and Sampling Distributions: Part 2

•Sample size and the sampling distribution of•Sampling distribution of •Sampling methods

x

p

Relationship Between the Sample SizeRelationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of xx

Suppose we select a simple random sample of 100Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered.applicants instead of the 30 originally considered.

EE( ) = ( ) = regardless of the sample size. In regardless of the sample size. In ourour example,example, E E( ) remains at 990.( ) remains at 990.

xxxx

Whenever the sample size is increased, the standardWhenever the sample size is increased, the standard error of the mean is decreased. With the increaseerror of the mean is decreased. With the increase in the sample size to in the sample size to nn = 100, the standard error of the = 100, the standard error of the mean is decreased to:mean is decreased to:

xx

808.0

100x

n

80

8.0100

xn


( ) 990E x ( ) 990E x xx

14.6x 14.6x With With nn = 30, = 30,

8x 8x With With nn = 100, = 100,

Recall that when Recall that when nn = 30, = 30, PP(980 (980 << << 1000) = .5034. 1000) = .5034.xx


We follow the same steps to solve for We follow the same steps to solve for PP(980 (980 << << 1000) 1000) when when nn = 100 as we showed earlier when = 100 as we showed earlier when nn = 30. = 30.

xx

Now, with Now, with nn = 100, = 100, PP(980 (980 << << 1000) = .7888. 1000) = .7888.xx

Because the sampling distribution with Because the sampling distribution with nn = 100 has a = 100 has a smaller standard error, the values of have lesssmaller standard error, the values of have less variability and tend to be closer to the populationvariability and tend to be closer to the population mean than the values of with mean than the values of with nn = 30. = 30.

xx

xx


xx10001000980980 990990

Area Area = .7888= .7888

SamplingSamplingDistributionDistribution

ofof xx8x 8x

x

A simple random sampleA simple random sampleof of nn elements is selected elements is selected

from the population.from the population.

Population Population with proportionwith proportion

pp = ? = ?

Making Inferences about a Population Making Inferences about a Population ProportionProportion

The sample data The sample data provide a value for provide a value for

thethesample sample

proportionproportion . .

pp

The value of is usedThe value of is usedto make inferencesto make inferences

about the value of about the value of pp..

pp

Sampling Distribution ofSampling Distribution ofpp

E p p( ) E p p( )


where:where:pp = the population proportion = the population proportion

The The sampling distribution of sampling distribution of is the probability is the probabilitydistribution of all possible values of the sampledistribution of all possible values of the sampleproportion .proportion .pp

pp

ppExpected Value ofExpected Value of

pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

is referred to as the is referred to as the standard error standard error of theof theproportionproportion..

p p


Finite PopulationFinite Population Infinite PopulationInfinite Population

ppStandard Deviation ofStandard Deviation of

The sampling distribution of can be approximatedThe sampling distribution of can be approximated by a normal probability distribution whenever theby a normal probability distribution whenever the sample size is large.sample size is large.

pp

The sample size is considered large whenever The sample size is considered large whenever thesethese conditions are satisfied:conditions are satisfied:

npnp >> 5 5 nn(1 – (1 – pp) ) >> 5 5andand


For values of For values of pp near .50, sample sizes as near .50, sample sizes as small as 10small as 10permit a normal approximation.permit a normal approximation.

With very small (approaching 0) or very large With very small (approaching 0) or very large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.


For our example, with n = 30 and p = .72, the normal probability distribution is an acceptable approximation because:

Sampling Distribution of Sampling Distribution of for the Proportionfor the Proportion of Applicants Wanting On-Campus Housingof Applicants Wanting On-Campus Housing

pp

nn(1 - (1 - pp) = 30(.28) = 8.4 ) = 30(.28) = 8.4 >> 5 5

andand

npnp = 30(.72) = 21.6 = 30(.72) = 21.6 >> 5 5

p

.72(1 .72).082

30

p

.72(1 .72).082

30

( ) .72E p ( ) .72E p pp


pp


of of pp

What is the probability that a simple random sampleWhat is the probability that a simple random sample

of 30 applicants will provide an estimate of the populationof 30 applicants will provide an estimate of the population

proportion of applicants desiring on-campus housing thatproportion of applicants desiring on-campus housing that

is within plus or minus .05 of the actual populationis within plus or minus .05 of the actual population

proportion?proportion?

In other words, what is the probability that will beIn other words, what is the probability that will be

between .67 and .77?between .67 and .77?

pp


pp


pp

Step 1: Step 1: Calculate the Calculate the zz-value at the -value at the upperupper endpoint of endpoint of the interval.the interval.

zz = (.77 = (.77 .72)/.082 = .61 .72)/.082 = .61

PP((zz << .61) = .7291 .61) = .7291

Step 2:Step 2: Find the area under the curve to the left of the Find the area under the curve to the left of the upperupper endpoint. endpoint.

Cumulative Probabilities forCumulative Probabilities for the Standard Normal the Standard Normal

DistributionDistributionz .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

. . . . . . . . . . .

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

. . . . . . . . . . .


pp

.77.77.72.72

Area = .7291Area = .7291

pp


pp


of of pp

.082p .082p

p


pp

Step 3: Step 3: Calculate the Calculate the zz-value at the -value at the lowerlower endpoint of endpoint of the interval.the interval.

Step 4:Step 4: Find the area under the curve to the left of the Find the area under the curve to the left of the lowerlower endpoint. endpoint.

zz = (.67 = (.67 .72)/.082 = - .61 .72)/.082 = - .61

PP((zz << -.61) = -.61) = PP((zz >> .61) .61)

= .2709= .2709= 1 = 1 . 7291 . 7291

= 1 = 1 PP((zz << .61) .61)

..6767 .72.72

Area = .2709Area = .2709


pp

pp


of of pp

.082p .082p

PP(.67 (.67 << << .77) = .4582 .77) = .4582pp


pp

Step 5: Step 5: Calculate the area under the curve betweenCalculate the area under the curve between the lower and upper endpoints of the interval.the lower and upper endpoints of the interval.

PP(-.61 (-.61 << zz << .61) = .61) = PP((zz << .61) .61) PP((zz << -.61) -.61)

= .7291 = .7291 .2709 .2709= .4582= .4582

The probability that the sample proportion of applicantsThe probability that the sample proportion of applicantswanting on-campus housing will be within +/-.05 of thewanting on-campus housing will be within +/-.05 of theactual population proportion :actual population proportion :


pp

.77.77.67.67 .72.72

Area = .4582Area = .4582

pp


of of pp

.082p .082p

Sampling Methods

• Stratified Random Sampling• Cluster Sampling• Systematic Sampling• Convenience Sampling• Judgment Sampling

The population is first divided into groups ofThe population is first divided into groups of elements called elements called stratastrata.. The population is first divided into groups ofThe population is first divided into groups of elements called elements called stratastrata..

Stratified Random SamplingStratified Random Sampling

Each element in the population belongs to one andEach element in the population belongs to one and only one stratum.only one stratum. Each element in the population belongs to one andEach element in the population belongs to one and only one stratum.only one stratum.

Best results are obtained when the elements withinBest results are obtained when the elements within each stratum are as much alike as possibleeach stratum are as much alike as possible (i.e. a (i.e. a homogeneous grouphomogeneous group).).

Best results are obtained when the elements withinBest results are obtained when the elements within each stratum are as much alike as possibleeach stratum are as much alike as possible (i.e. a (i.e. a homogeneous grouphomogeneous group).).

Cluster SamplingCluster Sampling

The population is first divided into separate groupsThe population is first divided into separate groups of elements called of elements called clustersclusters.. The population is first divided into separate groupsThe population is first divided into separate groups of elements called of elements called clustersclusters..

Ideally, each cluster is a representative small-scaleIdeally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).version of the population (i.e. heterogeneous group). Ideally, each cluster is a representative small-scaleIdeally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).version of the population (i.e. heterogeneous group).

A simple random sample of the clusters is then takenA simple random sample of the clusters is then taken.. A simple random sample of the clusters is then takenA simple random sample of the clusters is then taken..

All elements within each sampled (chosen) clusterAll elements within each sampled (chosen) cluster form the sample.form the sample. All elements within each sampled (chosen) clusterAll elements within each sampled (chosen) cluster form the sample.form the sample.

Cluster SamplingCluster Sampling

AdvantageAdvantage: The close proximity of elements can be: The close proximity of elements can be cost effective (i.e. many sample observations can becost effective (i.e. many sample observations can be obtained in a short time).obtained in a short time).

AdvantageAdvantage: The close proximity of elements can be: The close proximity of elements can be cost effective (i.e. many sample observations can becost effective (i.e. many sample observations can be obtained in a short time).obtained in a short time).

DisadvantageDisadvantage: This method generally requires a: This method generally requires a larger total sample size than simple or stratifiedlarger total sample size than simple or stratified random sampling.random sampling.

DisadvantageDisadvantage: This method generally requires a: This method generally requires a larger total sample size than simple or stratifiedlarger total sample size than simple or stratified random sampling.random sampling.

ExampleExample: A primary application is area sampling,: A primary application is area sampling, where clusters are city blocks or other well-definedwhere clusters are city blocks or other well-defined areas.areas.

ExampleExample: A primary application is area sampling,: A primary application is area sampling, where clusters are city blocks or other well-definedwhere clusters are city blocks or other well-defined areas.areas.

Systematic SamplingSystematic Sampling

If a sample size of If a sample size of nn is desired from a population is desired from a population containing containing NN elements, we might sample one elements, we might sample one element for every element for every nn//NN elements in the population. elements in the population.

If a sample size of If a sample size of nn is desired from a population is desired from a population containing containing NN elements, we might sample one elements, we might sample one element for every element for every nn//NN elements in the population. elements in the population.

We randomly select one of the first We randomly select one of the first nn//NN elements elements from the population list.from the population list. We randomly select one of the first We randomly select one of the first nn//NN elements elements from the population list.from the population list.

We then select every We then select every nn//NNth element that follows inth element that follows in the population list.the population list. We then select every We then select every nn//NNth element that follows inth element that follows in the population list.the population list.

Systematic SamplingSystematic Sampling

This method has the properties of a simple randomThis method has the properties of a simple random sample, especially if the list of the populationsample, especially if the list of the population elements is a random ordering.elements is a random ordering.

This method has the properties of a simple randomThis method has the properties of a simple random sample, especially if the list of the populationsample, especially if the list of the population elements is a random ordering.elements is a random ordering.

AdvantageAdvantage: The sample usually will be easier to: The sample usually will be easier to identify than it would be if simple random samplingidentify than it would be if simple random sampling were used.were used.

AdvantageAdvantage: The sample usually will be easier to: The sample usually will be easier to identify than it would be if simple random samplingidentify than it would be if simple random sampling were used.were used.

ExampleExample: Selecting every 100: Selecting every 100thth listing in a telephone listing in a telephone book after the first randomly selected listingbook after the first randomly selected listing ExampleExample: Selecting every 100: Selecting every 100thth listing in a telephone listing in a telephone book after the first randomly selected listingbook after the first randomly selected listing

Convenience SamplingConvenience Sampling

It is a It is a nonprobability sampling techniquenonprobability sampling technique. Items are. Items are included in the sample without known probabilitiesincluded in the sample without known probabilities of being selected.of being selected.

It is a It is a nonprobability sampling techniquenonprobability sampling technique. Items are. Items are included in the sample without known probabilitiesincluded in the sample without known probabilities of being selected.of being selected.

ExampleExample: A professor conducting research might use: A professor conducting research might use student volunteers to constitute a sample.student volunteers to constitute a sample. ExampleExample: A professor conducting research might use: A professor conducting research might use student volunteers to constitute a sample.student volunteers to constitute a sample.

The sample is identified primarily by The sample is identified primarily by convenienceconvenience.. The sample is identified primarily by The sample is identified primarily by convenienceconvenience..

Convenience SamplingConvenience Sampling

The aThe advantage dvantage of convenience of convenience sampling is that Sample sampling is that Sample selection and data collection are selection and data collection are relatively easy. The relatively easy. The disadvantage:disadvantage: It is impossible It is impossible to determine how to determine how representative of the population representative of the population the sample is.the sample is.

Judgment SamplingJudgment Sampling

The person most knowledgeable on the subject of theThe person most knowledgeable on the subject of the study selects elements of the population that he orstudy selects elements of the population that he or she feels are most representative of the population.she feels are most representative of the population.

The person most knowledgeable on the subject of theThe person most knowledgeable on the subject of the study selects elements of the population that he orstudy selects elements of the population that he or she feels are most representative of the population.she feels are most representative of the population.

It is a It is a nonprobability sampling techniquenonprobability sampling technique.. It is a It is a nonprobability sampling techniquenonprobability sampling technique..

ExampleExample: A reporter might sample three or four: A reporter might sample three or four senators, judging them as reflecting the generalsenators, judging them as reflecting the general opinion of the senate.opinion of the senate.

ExampleExample: A reporter might sample three or four: A reporter might sample three or four senators, judging them as reflecting the generalsenators, judging them as reflecting the general opinion of the senate.opinion of the senate.

Judgment SamplingJudgment Sampling

AdvantageAdvantage: It is a relatively easy way of selecting a: It is a relatively easy way of selecting a sample.sample. AdvantageAdvantage: It is a relatively easy way of selecting a: It is a relatively easy way of selecting a sample.sample.

DisadvantageDisadvantage: The quality of the sample results: The quality of the sample results depends on the judgment of the person selecting thedepends on the judgment of the person selecting the sample.sample.

DisadvantageDisadvantage: The quality of the sample results: The quality of the sample results depends on the judgment of the person selecting thedepends on the judgment of the person selecting the sample.sample.

Sampling and Sampling Distributions: Part 2 Sample size and the sampling distribution of Sampling...

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