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Chapter 7Sampling and Sampling Distributions
Simple Random SamplingPoint EstimationIntroduction to Sampling DistributionsSampling Distribution of Sampling Distribution ofSampling Methods
xxpp nn = 100 = 100
nn = 30 = 30
2
Statistical Inference
The purpose of statistical inference is to obtain information about a population from information contained in a sample.A population is the set of all the elements of interest.A ________ is a subset of the population.The sample results provide only ________ of the values of the population characteristics.A parameter is a numerical characteristic of a population.With proper sampling methods, the sample results will provide “good” estimates of the population characteristics.
3
Simple Random Sampling
Finite Population– A simple random sample 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.
– Replacing each sampled element before selecting subsequent elements is called sampling with ___________.
4
Simple Random Sampling
Finite Population– Sampling without ___________ is the
procedure used most often.– In large sampling projects, computer-
generated random numbers are often used to automate the sample selection process.
5
Infinite 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.
Simple Random Sampling
6
Infinite Population– The population is usually considered infinite
if it involves an ongoing process that makes listing or counting every element impossible.
– The random number selection procedure cannot be used for infinite populations.
Simple Random Sampling
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Point Estimation
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.We refer to as the _____________ of the population mean .s is the point estimator of the population standard deviation . is the point estimator of the population proportion p.
xx
pp
8
Point Estimation
When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased.
9
Sampling Error
The absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error.Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates.The sampling errors are:
for sample mean for sample standard
deviationfor sample proportion
| |x | |x
| |s | |s
| |p p| |p p
10
Example: St. Andrew’s
St. Andrew’s College receives 900 applicationsannually from prospective students. The
applicationforms contain a variety of information including
theindividual’s scholastic aptitude test (SAT) score
andwhether or not the individual desires on-campus housing.
11
Example: St. Andrew’s
The director of admissions would like to know the
following information:– the average SAT score for the applicants,
and– the proportion of applicants that want to
live on campus.
12
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
computer-generated random numbers
13
Taking a Census of the 900 Applicants– SAT Scores
• Population Mean
• Population Standard Deviation
ix 990900
ix 990900
ix
2( )80
900
ix
2( )80
900
Example: St. Andrew’s
14
Example: St. Andrew’s
Taking a Census of the 900 Applicants– Applicants Wanting On-Campus Housing
• Population Proportion
p648
.72900
p648
.72900
15
Example: St. Andrew’s
Take a Sample of 30 ApplicantsUsing a Random Number Table
Since the finite population has ____ 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 (________________).
16
Example: St. Andrew’s
Take a Sample of 30 ApplicantsUsing a Random Number Table
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 _______ applicants for our sample.
17
Use of Random Numbers for Sampling
3-Digit Applicant Random Number Included in Sample
744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds
900 190 No. 190 436 Number already
used etc. etc.
Example: St. Andrew’s
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Sample Data
RandomNo. Number Applicant SAT Score On-Campus 1 744 Connie Reyman 1025
Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120
Yes 5 835 Winona Wheeler 1015
No . . . . . 30 685 Kevin Cossack 965 No
Example: St. Andrew’s
19
Example: St. Andrew’s
Take a Sample of 30 ApplicantsUsing Computer-Generated Random Numbers– Excel provides a function for generating
random numbers in its worksheet.– 900 random numbers are generated, one for
each applicant in the population.– Then we choose the ____ applicants
corresponding to the 30 smallest random numbers as our sample.
– Each of the 900 applicants have the same probability of being included.
20
Using Excel to Selecta Simple Random Sample
Formula WorksheetA B C D
1Applicant Number
SAT Score
On-Campus Housing
Random Number
2 1 1008 Yes =RAND()3 2 1025 No =RAND()4 3 952 Yes =RAND()5 4 1090 Yes =RAND()6 5 1127 Yes =RAND()7 6 1015 No =RAND()8 7 965 Yes =RAND()9 8 1161 No =RAND()
Note: Rows 10-901 are not shown.Note: Rows 10-901 are not shown.
21
Using Excel to Selecta Simple Random Sample
n Value WorksheetValue WorksheetA B C D
1Applicant Number
SAT Score
On-Campus Housing
Random Number
2 1 1008 Yes 0.413273 2 1025 No 0.795144 3 952 Yes 0.662375 4 1090 Yes 0.002346 5 1127 Yes 0.712057 6 1015 No 0.180378 7 965 Yes 0.716079 8 1161 No 0.90512
Note: Rows 10-901 are not shown.Note: Rows 10-901 are not shown.
22
Using Excel to Selecta Simple Random Sample
Value Worksheet (Sorted)A B C D
1Applicant Number
SAT Score
On-Campus Housing
Random Number
2 12 1107 No 0.000273 773 1043 Yes 0.001924 408 991 Yes 0.003035 58 1008 No 0.004816 116 1127 Yes 0.005387 185 982 Yes 0.005838 510 1163 Yes 0.006499 394 1008 No 0.00667
Note: Rows 10-901 are not shown.Note: Rows 10-901 are not shown.
23
Point Estimates– as Point Estimator of
– s as Point Estimator of
– as Point Estimator of p
xx
pp
ixx29,910
99730 30
ixx29,910
99730 30
ix x
s2( ) 163,996
75.229 29
ix x
s2( ) 163,996
75.229 29
p 20 30 .68 p 20 30 .68
Example: St. Andrew’s
24
Example: St. Andrew’s
Point Estimates
Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.
25
Sampling Distribution of
Process of Statistical Inference
Population Population with meanwith mean
= ?= ?
Population Population with meanwith mean
= ?= ?
A simple random sampleA simple random sampleof of nn elements is selected elements is selected
from the population.from the population.
xx
The sample data The sample data provide a value forprovide a value for
the sample meanthe sample mean . .
The sample data The sample data provide a value forprovide a value for
the sample meanthe sample mean . .xx
The value of is used toThe value of is used tomake inferences aboutmake inferences about
the value of the value of ..
The value of is used toThe value of is used tomake inferences aboutmake inferences about
the value of the value of ..
xx
26
The sampling distribution of is the probability distribution of all possible values of the sample mean .Expected Value of
E( ) =
where: = the population mean
What does this imply?– The sample mean (from a simple random
sample) is an unbiased estimator of population mean.
Sampling Distribution of
xx
xxxx
xx
xx
27
Sampling Distribution of
: Standard Deviation of: Standard Deviation of
• referred to as the referred to as the standard error of the standard error of the meanmean..
Finite PopulationFinite Population Infinite Infinite Population Population
• A finite population is treated as being A finite population is treated as being infinite if infinite if nn//NN << .05. .05.
• is the finite correction is the finite correction factor.factor.
x n
N nN
( )1
x n
N nN
( )1
x n
x n
( ) / ( )N n N 1( ) / ( )N n N 1
xx
xx
x x
28
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 probability distribution.
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 probability distribution.
xx
xx
Sampling Distribution of xx
29
Sampling Distribution of for the SAT Scores
Example: St. Andrew’s
xn
8014.6
30
xn
8014.6
30
E x( ) 990E x( ) 990
xx
xx
30
Sampling Distribution of for the SAT ScoresWhat is the probability that a simple
random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean ?
In other words, what is the probability that will be between _____ and ______?
xx
Example: St. Andrew’s
xx
31
Example: St. Andrew’s
Sampling Distribution of for the SAT Scores
xx
Sampling distribution of
Sampling distribution of xx
10001000980980 990990
Area = ?Area = ?
xx
32
Sampling Distribution of for the SAT Scores
Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2517)(2) =
The probability is _______ that the sample mean will be within +/-10 of the actual population mean.
Example: St. Andrew’s
xx
33
Example: St. Andrew’s
Sampling Distribution of for the SAT Scores
xx
Sampling distribution of
Sampling distribution of xx
10001000980980 990990
Area = 2(.2517) = .5034Area = 2(.2517) = .5034
xx
34
The sampling distribution of is the probability distribution of all possible values of the sample proportion .
Expected Value of
where:p = the population proportion
Sampling Distribution of
pp
pp
pp
E p p( ) E p p( )
pp
35
Standard Deviation of
Finite Population Infinite Population
– is referred to as the standard error of the proportion.
Sampling Distribution of
pp pn
N nN
( )11
pp pn
N nN
( )11
pp pn
( )1 pp pn
( )1
p p
pp
pp
36
The sampling distribution of can be approximated by a normal probability distribution whenever the sample size is large.The sample size is considered large whenever these conditions are satisfied:
np > 5 and n(1 – p) > 5
Sampling Distribution of
pp
pp
37
n For values of For values of pp near .50, sample sizes as small near .50, sample sizes as small as 10 permit a normal approximation.as 10 permit a normal approximation.
n With very small (approaching 0) or large With very small (approaching 0) or large (approaching 1) values of (approaching 1) values of pp, much larger , much larger samples are needed.samples are needed.
ppSampling Distribution of
38
Example: St. Andrew’s
Sampling Distribution of for Applicants Desiring On-campus Housing
The normal probability distribution is an acceptable approximation because:
np = 30(.72) = 21.6 > 5 and
n(1 - p) = 30(.28) = 8.4 > 5.
pp
39
Sampling Distribution of for Applicants Desiring On-campus Housing
Example: St. Andrew’s
p
.72(1 .72).082
30
p
.72(1 .72).082
30
( ) .72E p ( ) .72E p
pp
40
Sampling Distribution of for Applicants Desiring On-campus Housing
What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion?
In other words, what is the probability that will be between ______ and ______?
pp
Example: St. Andrew’s
pp
41
Example: St. Andrew’s
Sampling Distribution of for Applicants Desiring On-campus Housing
Sampling distribution of
Sampling distribution of
0.770.770.670.67 0.720.72
Area = ?Area = ?
pp
pp
pp
42
Sampling Distribution of for Applicants Desiring On-campus Housing
For z = .05/.082 = .61, the area = (.2291)(2) = ______.The probability is _______ that the sample proportion will be within +/-.05 of the actual population proportion.
Example: St. Andrew’s
pp
43
Example: St. Andrew’s
Sampling Distribution of for In-State Residents
Sampling distribution of
Sampling distribution of
0.770.770.670.67 0.720.72
Area = 2(.2291) = .4582Area = 2(.2291) = .4582pp
pp
pp
44
Sampling Methods
Stratified Random SamplingCluster SamplingSystematic SamplingConvenience SamplingJudgment Sampling
45
Stratified Random Sampling
The population is first divided into groups of elements called strata.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. homogeneous group).A simple random sample is taken from each stratum.Example: The basis for forming the strata might be department, location, age, industry type, etc.
46
Stratified Random Sampling
Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size.Formulas are available for combining the stratum sample results into one population parameter estimate.
47
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.Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.
… continued
48
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.
49
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.This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.
… … continuedcontinued
50
Systematic Sampling
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.
51
Convenience Sampling
It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.The sample is identified primarily by convenience.Advantage: Sample selection and data collection are relatively easy.Disadvantage: It is impossible to determine how representative of the population the sample is. Example: A professor conducting research might use student volunteers to constitute a sample.
52
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.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.Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.
53
End of Chapter 7