Chapter 7 STATISTICS in PRACTICE
Transcript of Chapter 7 STATISTICS in PRACTICE
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Chapter 7Chapter 7
STATISTICSSTATISTICS in in PRACTICEPRACTICE� 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.
� 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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Chapter 7 Chapter 7
Sampling and Sampling DistributionsSampling and Sampling Distributions
� 7.1 The Electronics Associates Sampling Problem
� 7.2 Simple Random Sampling
� 7.3 Point Estimation
� 7.4 Introduction to Sampling Distributions
� 7.5 Sampling Distribution of x
� 7.6 Sampling Distribution of p
� 7.7 Properties of Point Estimators
� 7.8 Other Sampling Methods
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The purpose of statistical inference is to obtaininformation about a population from informationcontained in a sample.
The purpose of The purpose of statistical inferencestatistical inference is to obtainis to obtaininformation about a population from informationinformation about a population from informationcontained in a sample.contained in a sample.
Statistical InferenceStatistical Inference
A population is the set of all the elements of interest.A A populationpopulation is the set of all the elements of interest.is the set of all the elements of interest.
A sample is a subset of the population.A A samplesample is a subset of the population.is a subset of the population.
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The sample results provide only estimates of thevalues of the population characteristics.
The sample results provide only The sample results provide only estimatesestimates of theof thevalues of the population characteristics.values of the population characteristics.
A parameter is a numerical characteristic of apopulation.
A A parameterparameter is a numerical characteristic of ais a numerical characteristic of apopulation.population.
With proper sampling methods, the sample resultscan provide “good” estimates of the populationcharacteristics.
With With proper sampling methodsproper sampling methods, the sample results, the sample resultscan provide can provide ““goodgood”” estimates of the populationestimates of the populationcharacteristics.characteristics.
Statistical InferenceStatistical Inference
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7.1 7.1 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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7.2 Simple Random Sampling:7.2 Simple Random Sampling:
Finite PopulationFinite Population� Finite populations are often defined by lists
such as:
•Organization membership roster
•Credit card account numbers
•Inventory product numbers
� 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:Simple Random Sampling:
Finite PopulationFinite Population
� In large sampling projects, computer-
generated random numbers are often used to
automate the sample selection process.
Excel provides a function for generating
random numbers in its worksheets.
� 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 SamplingSimple 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:Simple Random Sampling:
Infinite PopulationInfinite 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:Simple Random Sampling:
Infinite PopulationInfinite 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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s is the point estimator of the population standarddeviation σ.
ss is the is the point estimatorpoint estimator of the population standardof the population standarddeviation deviation σσ..
In point estimation we use the data from the sample to compute a value of a sample statistic that servesas an estimate of a population parameter.
In In point estimationpoint estimation we use the data from the sample we use the data from the sample to compute a value of a sample statistic that servesto compute a value of a sample statistic that servesas an estimate of a population parameter.as an estimate of a population parameter.
7.3 Point Estimation7.3 Point Estimation
We refer to as the point estimator of the populationmean µ.
We refer to We refer to as the as the point estimatorpoint estimator of the populationof the populationmean mean µµ..
xx
is the point estimator of the population proportion p.is the is the point estimatorpoint estimator of the population proportion of the population proportion pp..pp
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Point EstimationPoint Estimation
� Example: to estimate the population mean, the
population standard deviation and population
proportion.
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Sampling ErrorSampling 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.
� 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 ErrorSampling Error
� The sampling errors are:
| |p p−| |p p− for sample proportionfor sample proportion
| |s σ−| |s σ− for sample standard deviationfor sample standard deviation
| |x µ−| |x µ− for sample meanfor sample mean
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Example: St. AndrewExample: St. Andrew’’ss
St. Andrew’s College
receives 900 applications
annually from
prospective students.
The application form
contains a variety of information
including the individual’s scholastic aptitude
test (SAT) score and whether or not the
individual desires on-campus housing.
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Example: St. AndrewExample: St. Andrew’’ss
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.
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Example: St. AndrewExample: St. Andrew’’ss
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 CensusConducting 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
ixµ = =∑
2( )80
900
ix µσ
−= =∑
Conducting a CensusConducting 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 SamplingSimple Random Sampling
� The applicants were numbered, from 1 to 900, as
their applications arrived.
� She decides a sample of 30 applicants will be used.
� 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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� Taking a Sample of 30 Applicants
Simple Random Sampling:Simple Random Sampling:
Using a Random Number TableUsing a Random Number Table
• 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.
• Because the finite population has 900 elements, we
will need 3-digit random numbers to randomly
select applicants numbered from 1 to 900.
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� Taking a Sample of 30 Applicants
Simple Random Sampling:Simple Random Sampling:
Using a Random Number TableUsing 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.)
• We will continue to draw random numbers until
we have selected 30 applicants for our sample.
• 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.
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� Use of Random Numbers for Sampling
Simple Random Sampling:Simple Random Sampling:
Using a Random Number TableUsing a Random Number Table
744744436436865865790790835835902902
190190836836
. . . and so on. . . and so on
33--DigitDigitRandom NumberRandom Number
ApplicantApplicantIncluded in SampleIncluded in Sample
No. 436No. 436No. 865No. 865No. 790No. 790No. 835No. 835
Number exceeds 900Number exceeds 900
No. 190No. 190No. 836No. 836
No. 744No. 744
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� Sample Data
Simple Random Sampling:Simple Random Sampling:
Using a Random Number TableUsing a Random Number Table
11 744 744 Conrad HarrisConrad Harris 10251025 YesYes
22 436436 Enrique RomeroEnrique Romero 950950 YesYes
33 865865 Fabian Fabian AvanteAvante 10901090 NoNo
44 790790 LucilaLucila CruzCruz 11201120 YesYes
55 835835 Chan ChiangChan Chiang 930930 NoNo.. .. .. .. ..
3030 498498 Emily MorseEmily Morse 10101010 NoNo
No.No.RandomRandomNumberNumber ApplicantApplicant
SATSATScoreScore
Live OnLive On--CampusCampus
.. .. .. .. ..
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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:Simple Random Sampling:
Using a ComputerUsing a Computer
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29,910997
30 30
ixx = = =∑
2( ) 163,99675.2
29 29
ix xs
−= = =∑
Point EstimationPoint Estimation
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
� s as Point Estimator of σσσσ
� as Point Estimator of µµµµx
–� p as Point Estimator of p
.6820/30 ==p
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PopulationPopulationParameterParameter
PointPointEstimatorEstimator
PointPointEstimateEstimate
ParameterParameterValueValue
µµ = Population mean= Population meanSAT score SAT score
990990 997997
σσ = Population std.= Population std.deviation for deviation for SAT score SAT score
8080 s s = Sample std.= Sample std.deviation fordeviation forSAT score SAT score
75.275.2
pp = Population pro= Population pro--portion wantingportion wantingcampus housing campus housing
.72.72 .68.68
Summary of Point EstimatesSummary of Point Estimates
Obtained from a Simple Random SampleObtained from a Simple Random Sample
= Sample mean= Sample meanSAT score SAT score
xx
= Sample pro= Sample pro--portion wantingportion wantingcampus housing campus housing
pp
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7.4 Sampling Distribution7.4 Sampling Distribution
� Example: Relative Frequency Histogram of Sample
Mean Values from 500 Simple Random Samples of
30 each.
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Sampling DistributionSampling 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 toThe value of is used tomake inferences aboutmake inferences about
the value of the value of µµ..
xx The sample data The sample data provide a value forprovide a value forthe sample meanthe sample mean ..xx
A simple random sampleA simple random sampleof of nn elements is selectedelements is selectedfrom the population.from the population.
Population Population with meanwith mean
µµ = ?= ?
7.5 Sampling Distribution of 7.5 Sampling Distribution of x
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The sampling distribution of is the probability
distribution of all possible values of the sample
mean .
Sampling Distribution ofSampling Distribution of
where:
µµµµ = the population mean
EE( ) = ( ) = µµxx
Expected Value of x
x
x
x
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Finite Population Infinite Population
σσ
xn
N n
N=
−
−( )
1σ
σx
n
N n
N=
−
−( )
1σ
σx
n=σ
σx
n=
• is referred to as the standard error of the
mean.
xσxσ
• A finite population is treated as being
infinite if n/N < .05.
• is the finite correction factor.)1/()( −− NnN )1/()( −− NnN
Standard Deviation of x
Sampling Distribution ofSampling Distribution of x
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If we use a large (If we use a large (nn >> 30) simple random sample, the30) simple random sample, thecentral limit theoremcentral limit theorem enables us to conclude that theenables us to conclude that thesampling distribution of can be approximated bysampling distribution of can be approximated bya normal distribution.a normal distribution.
xx
When the simple random sample is small (When the simple random sample is small (nn < 30),< 30),the sampling distribution of can be consideredthe sampling distribution of can be considerednormal only if we assume the population has anormal only if we assume the population has anormal distribution.normal distribution.
xx
Form of the Sampling Distribution ofForm of the Sampling Distribution of 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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8014.6
30x
n
σσ = = =
8014.6
30x
n
σσ = = =
( ) 990E x =( ) 990E x =
xx
SamplingSamplingDistributionDistribution
of of xx
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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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?
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
x
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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 ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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Cumulative Probabilities forCumulative Probabilities forthe Standard Normal Distributionthe 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 ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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xx
990990
SamplingSamplingDistributionDistribution
of of xx
14.6xσ =14.6xσ =
10001000
Area = .7517Area = .7517
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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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 ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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xx
980980 990990
Area = .2483Area = .2483
SamplingSamplingDistributionDistribution
of of xx
14.6xσ =14.6xσ =
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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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) = .5034xx
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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xx10001000980980 990990
Area = .5034Area = .5034
SamplingSamplingDistributionDistribution
of of xx
14.6xσ =14.6xσ =
Sampling Distribution ofSampling Distribution of for SAT Scoresfor SAT Scoresx
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� Suppose we select a simple random sample
of 100 applicants instead of the 30 originally
considered.
Relationship Between the Sample SizeRelationship Between the Sample Size
and the Sampling Distribution ofand the Sampling Distribution of x
� 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
� E( ) = m regardless of the sample size. In
our example, E( ) remains at 990.
x
x
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( ) 990E x =( ) 990E x =
xx
14.6xσ =14.6xσ =
With With nn = 30,= 30,
8xσ = 8xσ =
With With nn = 100,= 100,
Relationship Between the Sample SizeRelationship Between the Sample Size
and the Sampling Distribution ofand the Sampling Distribution of x
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Relationship Between the Sample SizeRelationship Between the Sample Size
and the Sampling Distribution ofand the Sampling Distribution of x
� Recall that when n = 30, P(980 < < 1000) = .5034.x
� We follow the same steps to solve for P(980 < < 1000)when n = 100 as we showed earlier when n = 30.
x
� Now, with n = 100, P(980 < < 1000) = .7888.x
� Because the sampling distribution with n = 100 has asmaller standard error, the values of have lessvariability and tend to be closer to the populationmean than the values of with n = 30.
x
x
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xx
10001000980980 990990
Area = .7888Area = .7888
SamplingSamplingDistributionDistribution
of of xx
8xσ =8xσ =
Relationship Between the Sample SizeRelationship Between the Sample Size
and the Sampling Distribution ofand the Sampling Distribution of x
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Sampling DistributionSampling Distribution
� Example: Relative Frequency Histogram of
Sample Proportion Values from 500 Simple
Random Samples of 30 each.
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A simple random sampleA simple random sampleof of nn elements is selectedelements is selectedfrom the population.from the population.
Population Population with proportionwith proportion
pp = ?= ?
� Making Inferences about a Population Proportion
The sample data The sample data provide a value for theprovide a value for thesample proportionsample proportion ..pp
The value of is usedThe value of is usedto make inferencesto make inferencesabout the value of about the value of pp..
pp
7.6 Sampling Distribution of 7.6 Sampling Distribution of pp
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E p p( ) =E p p( ) =
where:
p = the population proportion
The sampling distribution of p is the probability
distribution of all possible values of the sample
proportion p .
Expected Value of p
Sampling Distribution ofSampling Distribution of p
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σ p
p p
n
N n
N=
− −
−
( )1
1σ p
p p
n
N n
N=
− −
−
( )1
1σ p
p p
n=
−( )1σ p
p p
n=
−( )1
is referred to as the standard error of
the proportion.
σ pσ p
Sampling Distribution ofSampling Distribution of
Finite Population Infinite Population
Standard Deviation of p
p
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The sampling distribution of can be approximatedThe sampling distribution of can be approximatedby a normal distribution whenever the sample size by a normal distribution whenever the sample size is large.is large.
pp
The sample size is considered large whenever theseThe sample size is considered large whenever theseconditions are satisfied:conditions are satisfied:
npnp >> 55 nn(1 (1 –– pp) ) >> 55andand
Form of the Sampling Distribution ofForm of the Sampling Distribution of p
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For values of For values of pp near .50, sample sizes as small as 10near .50, sample sizes as small as 10
permit a normal approximationpermit 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 samples are , much larger samples are
needed.needed.
Form of the Sampling Distribution ofForm of the Sampling Distribution of p
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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 ofSampling 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:
nn(1 (1 -- pp) = 30(.28) = 8.4 ) = 30(.28) = 8.4 >> 55
and
npnp = 30(.72) = 21.6 = 30(.72) = 21.6 >> 55
Sampling Distribution ofSampling Distribution of p
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pp
SamplingSamplingDistributionDistribution
of of pp
Sampling Distribution ofSampling Distribution of p
082.30
)72.1(72.=
−=pσ
72.)( =pE
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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 ofSampling 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 ofSampling Distribution of p
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.77.77.72.72
Area = .7291Area = .7291
pp
SamplingSamplingDistributionDistribution
of of pp
.082pσ = .082pσ =
Sampling Distribution ofSampling 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 ofSampling Distribution of p
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.67.67 .72.72
Area = .2709Area = .2709
pp
SamplingSamplingDistributionDistribution
of of pp
.082pσ = .082pσ =
Sampling Distribution ofSampling Distribution of p
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PP(.67 (.67 << << .77) = .4582.77) = .4582pp
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 ofSampling Distribution of p
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.77.77.67.67 .72.72
Area = .4582Area = .4582
pp
SamplingSamplingDistributionDistribution
of of pp
.082pσ = .082pσ =
Sampling Distribution ofSampling Distribution of p
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Point EstimatorsPoint Estimators
� Notations:
� θθθθ = the population parameter of interest.
For example, population mean, population standard
deviation, population proportion, and so on.
� θθθθ = the sample statistic or point estimator of θθθθ .
Represents the corresponding sample statistic such as
the sample mean, sample standard deviation, and
sample proportion.
� The notation θθθθ is the Greek letter theta.
� the notation θθθθ is pronounced “theta-hat.”^
^
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7.7 Properties of Point Estimators7.7 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.
ConsistencyConsistency
EfficiencyEfficiency
UnbiasedUnbiased
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Properties of Point EstimatorsProperties 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.
UnbiasedUnbiased
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Properties of Point EstimatorsProperties of Point Estimators
� Unbised
The sample statistic θθθθ is unbiased estimator of
the population parameter θθθθ if
E(θθθθ)=θθθθ^
where
E(θθθθ)=the expected value of the sample statistic θθθθ^ ^
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Properties of Point EstimatorsProperties of Point Estimators
� Examples of Unbiased and Biased Point
Estimators
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Properties of Point EstimatorsProperties 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.
EfficiencyEfficiency
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Properties of Point EstimatorsProperties of Point Estimators
� Example: Sampling Distributions of Two
Unbiased Point Estimators.
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Properties of Point EstimatorsProperties 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.
ConsistencyConsistency
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7.8 Other Sampling Methods7.8 Other Sampling Methods
� Stratified Random Sampling
� Cluster Sampling
� Systematic Sampling
� Convenience Sampling
� Judgment Sampling
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The population is first divided into groups ofelements called strata.
The population is first divided into groups ofThe population is first divided into groups ofelements called elements called stratastrata..
Stratified Random SamplingStratified Random Sampling
Each element in the population belongs to one andonly one stratum.
Each element in the population belongs to one andEach element in the population belongs to one andonly one stratum.only one stratum.
Best results are obtained when the elements withineach stratum are as much alike as possible(i.e. a homogeneous group).
Best results are obtained when the elements withinBest results are obtained when the elements withineach stratum are as much alike as possibleeach stratum are as much alike as possible(i.e. a (i.e. a homogeneous grouphomogeneous group).).
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Stratified Random SamplingStratified Random Sampling
� Diagram for Stratified Random Sampling
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Stratified Random SamplingStratified Random Sampling
A simple random sample is taken from each stratum.A simple random sample is taken from each stratum.A simple random sample is taken from each stratum.
Formulas are available for combining the stratumsample results into one population parameterestimate.
Formulas are available for combining the stratumFormulas are available for combining the stratumsample results into one population parametersample results into one population parameterestimate.estimate.
Advantage: If strata are homogeneous, this methodis as “precise” as simple random sampling but witha smaller total sample size.
AdvantageAdvantage: If strata are homogeneous, this method: If strata are homogeneous, this methodis as is as ““preciseprecise”” as simple random sampling but withas simple random sampling but witha smaller total sample size.a smaller total sample size.
Example: The basis for forming the strata might bedepartment, location, age, industry type, and so on.
ExampleExample: The basis for forming the strata might be: The basis for forming the strata might bedepartment, location, age, industry type, and so on.department, location, age, industry type, and so on.
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Cluster SamplingCluster Sampling
The population is first divided into separate groupsof elements called clusters.
The population is first divided into separate groupsThe population is first divided into separate groupsof elements called of elements called clustersclusters..
Ideally, each cluster is a representative small-scaleversion of the population (i.e. heterogeneous group).
Ideally, each cluster is a representative smallIdeally, each cluster is a representative small--scalescaleversion of the population (i.e. heterogeneous group).version of the population (i.e. heterogeneous group).
A simple random sample of the clusters is then taken.A simple random sample of the clusters is then taken.A simple random sample of the clusters is then taken.
All elements within each sampled (chosen) clusterform the sample.
All elements within each sampled (chosen) clusterAll elements within each sampled (chosen) clusterform the sample.form the sample.
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Cluster SamplingCluster Sampling
� Diagram for Cluster Sampling
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Cluster SamplingCluster Sampling
Advantage: The close proximity of elements can becost effective (i.e. many sample observations can beobtained in a short time).
AdvantageAdvantage: The close proximity of elements can be: The close proximity of elements can becost effective (i.e. many sample observations can becost effective (i.e. many sample observations can beobtained in a short time).obtained in a short time).
Disadvantage: This method generally requires alarger total sample size than simple or stratifiedrandom sampling.
DisadvantageDisadvantage: This method generally requires a: This method generally requires alarger total sample size than simple or stratifiedlarger total sample size than simple or stratifiedrandom sampling.random sampling.
Example: A primary application is area sampling,where clusters are city blocks or other well-definedareas.
ExampleExample: A primary application is area sampling,: A primary application is area sampling,where clusters are city blocks or other wellwhere clusters are city blocks or other well--defineddefinedareas.areas.
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Systematic SamplingSystematic Sampling
If a sample size of n is desired from a populationcontaining N elements, we might sample oneelement for every n/N elements in the population.
If a sample size of If a sample size of nn is desired from a populationis desired from a populationcontaining containing NN elements, we might sample oneelements, we might sample oneelement for every element for every nn//NN elements in the population.elements in the population.
We randomly select one of the first n/N elementsfrom the population list.
We randomly select one of the first We randomly select one of the first nn//NN elementselementsfrom the population list.from the population list.
We then select every n/Nth element that follows inthe population list.
We then select every We then select every nn//NNth element that follows inth element that follows inthe population list.the population list.
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Systematic SamplingSystematic Sampling
This method has the properties of a simple randomsample, especially if the list of the populationelements is a random ordering.
This method has the properties of a simple randomsample, especially if the list of the populationelements is a random ordering.
Advantage: The sample usually will be easier toidentify than it would be if simple random samplingwere used.
AdvantageAdvantage: The sample usually will be easier to: The sample usually will be easier toidentify than it would be if simple random samplingidentify than it would be if simple random samplingwere used.were used.
Example: Selecting every 100th listing in a telephonebook after the first randomly selected listing
ExampleExample: Selecting every 100: Selecting every 100thth listing in a telephonelisting in a telephonebook after the first randomly selected listingbook after the first randomly selected listing
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Convenience SamplingConvenience Sampling
It is a nonprobability sampling technique. Items areincluded in the sample without known probabilitiesof being selected.
It is a It is a nonprobabilitynonprobability sampling techniquesampling technique. Items are. Items areincluded in the sample without known probabilitiesincluded in the sample without known probabilitiesof being selected.of being selected.
Example: A professor conducting research might usestudent volunteers to constitute a sample.
ExampleExample: A professor conducting research might use: A professor conducting research might usestudent volunteers to constitute a sample.student volunteers to constitute a sample.
The sample is identified primarily by convenience.The sample is identified primarily by The sample is identified primarily by convenienceconvenience..
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Advantage: Sample selection and data collection arerelatively easy.
AdvantageAdvantage: Sample selection and data collection are: Sample selection and data collection arerelatively easy.relatively easy.
Disadvantage: It is impossible to determine howrepresentative of the population the sample is.
DisadvantageDisadvantage: It is impossible to determine how: It is impossible to determine howrepresentative of the population the sample is.representative of the population the sample is.
Convenience SamplingConvenience Sampling
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Judgment SamplingJudgment Sampling
The person most knowledgeable on the subject of thestudy selects elements of the population that he orshe feels are most representative of the population.
The person most knowledgeable on the subject of theThe person most knowledgeable on the subject of thestudy selects elements of the population that he orstudy selects elements of the population that he orshe feels are most representative of the population.she feels are most representative of the population.
It is a nonprobability sampling technique.It is a It is a nonprobabilitynonprobability sampling techniquesampling technique..
Example: A reporter might sample three or foursenators, judging them as reflecting the generalopinion of the senate.
ExampleExample: A reporter might sample three or four: A reporter might sample three or foursenators, judging them as reflecting the generalsenators, judging them as reflecting the generalopinion of the senate.opinion of the senate.
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Judgment SamplingJudgment Sampling
Advantage: It is a relatively easy way of selecting asample.
AdvantageAdvantage: It is a relatively easy way of selecting a: It is a relatively easy way of selecting asample.sample.
Disadvantage: The quality of the sample resultsdepends on the judgment of the person selecting thesample.
DisadvantageDisadvantage: The quality of the sample results: The quality of the sample resultsdepends on the judgment of the person selecting thedepends on the judgment of the person selecting thesample.sample.