Sampling Methods and
Sampling Distributions
Learning Objectives Explain Types of Samples
Describe the Properties of Estimators
Explain Sampling Distribution
Describe the Relationship betweenPopulations & Sampling Distributions
State the Central Limit Theorem
Solve Probability Problems Involving Sampling Distributions
Sampling
Methods
Types of SamplesType ofSample
ProbabilityNon
Probability
SimpleRandom
StratifiedSystematic Cluster
ChunkJudge-ment
Quota
Simple Random Sample
1.Each Population Element Has an Equal Chance of Being Selected
2.Selecting 1 Subject Does Not Affect Selecting Others
3.May Use Random Number Table, Lottery, ‘Fish Bowl’
Random Number Table
Column
00000 00001 11111 11111Row 12345 67890 12345 67890
01 49280 88924 35779 00283
02 61870 41657 07468 08612
03 43898 65923 25078 86129
Types of Samples
Type ofSample
ProbabilityNon
Probability
SimpleRandom
StratifiedSystematic Cluster
ChunkJudge-ment
Quota
Systematic Sample
1. Items of population arranged in some way- alphabetically, by date received
2.Every kth Element Is Selected After a Random Start within the First k Elements
3.Used in Telephone Surveys
© 1984-1994 T/Maker Co.
Types of SamplesType ofSample
ProbabilityNon
Probability
SimpleRandom
StratifiedSystematic Cluster
ChunkJudge-ment
Quota
Stratified Sample
1. Divide Population into Subgroups Mutually Exclusive
Collectively Exhaustive
At Least 1 Common Characteristic of Interest
All Students
Commuters Residents
Sample2. Select Simple Random Samples from Subgroups
Types of SamplesType ofSample
ProbabilityNon
Probability
SimpleRandom
StratifiedSystematic Cluster
ChunkJudge-ment
Quota
Cluster Sample
Divide Population into Clusters If Managers
are Elements, then Companies are Clusters
Select Clusters Randomly
Survey All or a Random Sample of Elements in Cluster
Companies (Clusters)
Sample
Types of Samples
Type ofSample
ProbabilityNon
Probability
SimpleRandom
StratifiedSystematic Cluster
ChunkJudge-ment
Quota
Nonprobability Samples
1.JudgmentUse Experience to Select Sample
e.g., Test Markets
2.QuotaSimilar to Stratified Sampling Except
No Random Sampling
3.Chunk (Convenience)Use Elements Most Available
Errors Due to Sampling
Sampling Error - occurs because sample is taken instead of census Errors are due to chance Equally likely to be too high or too low Improve by increasing sample size
Nonsampling Error - Bias A directional error Can not be reduced by increasing sample
size
Sampling Distributions
Statistical Methods
StatisticalMethods
DescriptiveStatistics
InferentialStatistics
Inferential Statistics
InvolvesEstimationHypothesis
Testing
PurposeMake Decisions
about Population Characteristics
Population?
Inference Process
Population
Sample
Sample Statistic (X, P )
Estimates & Tests
1. Random Variables Used to Estimate a Population Parameter
-Sample Mean, Sample Proportion, Sample Median
2. Sample Mean is an Estimator of Population Mean
If = 3 then 3 Is the Estimate of
3. Theoretical Basis Is Sampling Distribution
Estimators
X
Properties of Mean Unbiasedness
Mean of Sampling Distribution Equals Population Mean
Efficiency Sample Mean Comes Closer to Population Mean Than
Any Other Unbiased Estimator
Consistency As Sample Size Increases, Variation of Sample Mean
from Population Mean Decreases
Unbiasedness
x= x
Unbiased Biased
X
P(X)
CA
x C
Efficiency
x
Sampling Distribution of Median
Sampling Distribution
of Mean
X
P(X)
A
B
Consistency
x
Smaller Sample
Size
Larger Sample
Size
X
P(X)
A
B
Sampling Distribution
Theoretical Probability Distribution Random Variable is Sample Statistic
Sample Mean, Sample Proportion, etc.
Results from Drawing All Possible Samples of a Fixed Size
List of All Possible [X, P(X) ] PairsSampling Distribution of Mean
DevelopingSampling Distributions
Suppose There’s a Population ...
Population Size, N = 4
Random Variable, X, Is # Errors in Work
Values of X: 1, 2, 3, 4
Uniform Distribution
1 2.5 -1.5 2.252 2.5 -0.5 0.253 2.5 0.5 0.254 2.5 1.5 2.25
10 5
X (X -) (X - )2
10
42 5.
5
4112.
Population Mean and Standard Deviation
(# of errors)
Population Characteristics
Population DistributionSummary Measures
x
ii
N
x
i xi
N
X
N
X
N
1
2
1
2 5
112
.
.( )
.0
.1
.2
.3
1 2 3 4
Inference Process
Population
Sample
Sample Statistic (X, Ps )
Estimates & Tests
All Possible Samples of Size n = 2
16 Samples 16 Sample Means
Sample With Replacement
1st 2nd ObservationObs 1 2 3 4
1 1,1 1,2 1,3 1,4
2 2,1 2,2 2,3 2,4
3 3,1 3,2 3,3 3,4
4 4,1 4,2 4,3 4,4
1st 2nd ObservationObs 1 2 3 4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
Sampling Distribution of All Sample Means
1st 2nd ObservationObs 1 2 3 4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
16 Sample Means Sampling Distribution
X f p(X)
1.0 1 1/16
1.5 2 2/16
2.0 3 3/16
2.5 4 4/16
3.0 3 3/16
3.5 2 2/16
4.0 1 1/16
Sampling Distribution of All Sample Means
1st 2nd ObservationObs 1 2 3 4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
16 Sample Means Sampling Distribution
.0
.1
.2
.3
1.0 1.5 2.0 2.5 3.0 3.5 4.0X
P(X)
1.0 2.5 -1.5 2.251.5 2.5 -1.0 1.001.5 2.5 -1.0 1.002.0 2.5 -0.5 0.252.0 2.5 -0.5 0.252.0 2.5 -0.5 0.252.5 2.5 0.0 0.002.5 2.5 0.0 0.002.5 2.5 0.0 0.002.5 2.5 0.0 0.003.0 2.5 0.5 0.253.0 2.5 0.5 0.253.0 2.5 0.5 0.253.5 2.5 1.0 1.003.5 2.5 1.0 1.004.0 2.5 1.5 2.25
40 10.00
X x x)2(X-x)(X-
x N 10 15 4 0
162.5. . .i
NX
i14016
Summary Measures of All Possible Sample Means
x
ii
N
X
N
1
10 15 4 0
162 5
. . ..
x
i xi
N
X
N
( )2
1
79. ( ) ( ) ( )2 2 2
10 2 5 15 2 5 4 0 2 5
16
. . . . . .
1016
Comparison of Population & Sampling Distribution
Population Sampling Distribution
.0
.1
.2
.3
1 2 3 4
P(X)
.0
.1
.2
.3
1 1.5 2 2.5 3 3.5 4
X
P(X)
x 2 5.
x .79
x 2 5.
x 112.
Standard Error of Mean(Standard Deviation of the Sampling Distribution of Means)
Standard Deviation of All Possible Sample Means,X Measures Scatter in All Sample Means,X
Less Than Population Standard Deviation Formula
(Sampling With Replacement)
x
i xi
N
X
N
( )2
1
n
x = x
x n
Sampling is done with replacementor
Population is infiniteor
n/N < .05
Sampling Distribution of the Sample Means Summary
Sampling from Normal Populations
Central Tendency
Dispersion
Sampling With Replacement
Population Distribution
Sampling Distributionn =16n =4
x x
xx
n
X = 50
X= 10
X
X = 50- X
x= 2.5x = 5
Z = 0
z= 1
Z
Standardizing Sampling Distribution of Mean
Sampling Distribution
Standardized Normal Distribution
ZX x
x
n
X x
x
X
X
X
Thinking Challenge
You’re an operations analyst for AT&T. Long-distance telephone calls are normally distribution with x = 8 min. & x = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? © 1984-1994 T/Maker Co.
Sampling Distribution Solution*
ZX
nx
x
8 2 8
2 2550
..
nx 2 25Z
X x
7 8 850
..
Sampling Distribution
8
X = .4
7.8 8.2 X
Standardized Normal Distribution
0
Z = 1
Z
.1915.1915
-.50 .50
.3830
Sampling from Normal Populations
Central Tendency
Dispersion
Sampling With Replacement
Population Distribution
Sampling Distributionn =16
x= 2.5n =4
x = 5
x x
xx
n
X = 50
X= 10
X
X = 50- X
Sampling from Non-Normal Populations
Central Tendency
Dispersion
Sampling With Replacement
Population Distribution
Sampling Distributionn =30x =1.8
n = 4x= 5
x x
xx
n
X = 50
X= 10
X
X = 50- X
Central Limit TheoremFor a population with a mean u and a standard deviation, the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed assuming that the sample size is sufficiently large.
X
Central Limit Theorem
As sample size gets large enough ( 30) ...
sampling distribution becomes almost normal.
Central Limit Theorem
The sampling distribution of means is a normal distribution if population is normally distributed
Even if population is not normally distributed, the sampling distribution of means is approximated by a normal distribution for large n (n>30)
X
Central Limit Theorem
As sample size gets large enough ( 30) ...
sampling distribution becomes almost normal.
Proportions
Categorical Variable (e.g., Gender) % Population Having a Characteristic If Two Outcomes, Binomial Distribution
Possess - Don’t Possess Characteristic
Sample Proportion Formula:
PX
n
number of successes
sample size
Approximated by Normal Distribution n·p 5 n·(1 - p) 5
Mean
Standard Error
Sampling Distribution of Proportion
Sampling Distribution
.0
.1
.2
.3
.0 .2 .4 .6 .8 1.0
P
P(Ps)
P
p
nwhere p = Population
Proportion
P
p p
1( )
Z = 0
z= 1
Z
Standardizing Sampling Distribution of Proportion
Sampling Distribution
Standardized Normal Distribution
P
ZP P p
p p
n
P
P
( )1
P P
Thinking Challenge
You’re manager of a bank. 40% of depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?
© 1984-1994 T/Maker Co.
.87
Solution* P(.40 P .43)
Sampling Distribution
.3078
ZZ= 0
Z = 1
Standardized Normal Distribution
n·p 5n·(1 - p) 5
PP = .40
P = .0346
.43
ZP p
p p
n
( )
. .
. ( . ).
1
43 40
40 1 40
200
87
Sampling from Finite Populations
Modify Standard Error if Sample Size (n) Is Large Relative to Population Size (N) n > .05·N (or n/N > .05)
Use Finite Population Correction (fpc) Factorfor Standard Errors if n/N > .05
xx
n
P
p p
n
1( )N n
N
1
N n
N
1)
( )(
x = x
x n
xx
n
N n
N
1
Sampling is done with replacementor
Population is infiniteor
n/N < .05
Sampling is without replaacementand
Population is finiteand
n/N > .05
Sampling Distribution of the Sample Means Summary
Thinking Challenge
You’re manager of a bank. 40% of all 1000 depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ?
© 1984-1994 T/Maker Co.
ZZ= 0
Z = 1
.97
Solution* P(.40 P .43)
Sampling Distribution
.3340
Standardized Distribution
ZP p
p p
n
N n
N
( )
. .
. ( . ).
1
1
43 40
40 1 40
200
1000 200
1000 1
97
PP = .40
P = .0310
.43
Selecting a Sample Size
Selecting a Sample Size
The Degree of Cofidence Selected
The Maximum Allowable Error The Population Standard
Deviation
Sample Size for Means
E is the allowable error z is the z score associated with degree of confidence is the population standard deviation
nz
E
z
E
2 2
2
2 n
z
E
z
E
2 2
2
2
The marketing manager would like to estimate the population mean annual usage of home heating oil to within 50 gallons of the true value and desires to be 95% confident of correctly estimating the true mean. Based on a previous study taken last year,the marketing manager feels that the standard deviation can be estimated as 325 gallons. What is the sample size need to obtain these results?
n
z
E
2 2
2
2
2
2196 325 384 105 625
2500162 31
. ( )
(50)
( . )( , ).
nhomes need to be sampled
Confidence = 95%E = 50 = 325
z = 1.96
Sample Size for Proportions
E is the maximum allowable errorz is the z value associated with the degree of confidencep is the estimated proportion
n
p p z
E
1 2
2
A political pollister would like to estimate the proportion of voters who will vote for the Democratic candidate in a presidential campaign. The pollster would like 95% confidence that her prediction is correct to within .04 of the true proportion. What sample size is needed?
Confidence = 95%E = .04p = unknown use p = .5
np P z
E
( ) . ( . )( . )
(. ).
1 5 1 5 196
04600 25
2
2
2
2
n = 601 voters
Conclusion Examined Sampling Methods
Described the Properties of Estimators
Explained Sampling Distribution
Described the Relationship between Populations & Sampling Distributions
Stated the Central Limit Theorem
Solved Probability Problems Involving Sampling Distributions
End of Chapter
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