Download Chap 6: Sampling Distributions

© 1998 Prentice-Hall, Inc.
Chapter 6
Sampling Distributions
6-1
Learning Objectives
© 1998 Prentice-Hall, Inc.
1. Describe the properties of estimators
2. Explain sampling distribution
3. Describe the relationship between
populations & sampling distributions
4. State the Central Limit Theorem
5. Solve a probability problem involving
sampling distributions
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Inferential Statistics
6-3
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Types of
Statistical Applications
Statistical
Methods
Descriptive
Statistics
6-4
Inferential
Statistics
Inferential Statistics
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1.
Involves


Estimation
Hypothesis
testing
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Inferential Statistics
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1.
Involves


Estimation
Hypothesis
testing
6-6
Population?
Inferential Statistics
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1.
Involves


2.
Estimation
Hypothesis
testing
Purpose

Make decisions
about population
characteristics
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Population?
Inference Process
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6-8
Inference Process
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Population
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Inference Process
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Population
Sample
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Inference Process
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Population
Sample
statistic
(X)
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Sample
Inference Process
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Estimate
& test
population
parameter
Sample
statistic
(X)
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Population
Sample
Estimators
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1. Random variables used to estimate a
population parameter

Sample mean, sample proportion, sample
median
2. Example: Sample meanx is an
estimator of population mean 

Ifx = 3 then 3 is the estimate of 
3. Theoretical basis is sampling distribution
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Sampling Distributions
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Sampling Distribution
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1. Theoretical probability distribution
2. Random variable is sample statistic

Sample mean, sample proportion etc.
3. Results from drawing all possible
samples of a fixed size
4. List of all possible [x, P(x) ] pairs

Sampling distribution of mean
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Developing
Sampling Distributions
Suppose there’s a
population ...
Population size, N = 4
Random variable, x,
is # televisions owned
Values of x: 1, 2, 3, 4
Equally distributed
(p=1/4)
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© 1984-1994 T/Maker Co.
© 1998 Prentice-Hall, Inc.
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Population
Characteristics
Population
Characteristics
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Summary Measures
N

 Xi
i 1
N
 2.5
N

 aX i   f
2
i 1
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N
 112
.
Population
Characteristics
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Summary Measures
N
   X i  p( X i  i )  2.5
i 1

 ( X i   )  p( X
N
i 1
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2
i
 i )  1.12
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Population
Characteristics
Summary Measures
  2.5
  1.12
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Population Distribution
.3
.2
.1
.0
1
2
3
4
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Let’s Draw All Possible
Samples of Size n = 2
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Let’s Draw All Possible
Samples of Size n = 2
16 Samples
1st 2nd Observation
Obs 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
Sample with replacement
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Let’s Draw All Possible
Samples of Size n=2
16 Samples
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1st 2nd Observation
Obs 1
2
3
4
1 1,1 1,2 1,3 1,4
1 1.0 1.5 2.0 2.5
2 2,1 2,2 2,3 2,4
2 1.5 2.0 2.5 3.0
3 3,1 3,2 3,3 3,4
3 2.0 2.5 3.0 3.5
4 4,1 4,2 4,3 4,4
4 2.5 3.0 3.5 4.0
Sample with replacement
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Sampling Distribution
of All Sample Means
16 Sample Means
Sampling
Distribution
1st 2nd Observation
Obs 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
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P(X)
.3
.2
.1
.0
1.0 1.5 2.0 2.5 3.0 3.5 4.0
X
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Summary Measures of
All Sample Means (n=16)
n
x 
X
i 1
n
 X
n
x 

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i 1
i
1.0  1.5    4.0

 2.5
16
 x 
2
i
n
1.0  2.52  1.5  2.52    4.0  2.52
16
 0.79
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Comparison of Population
& Sampling Distribution
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Comparison of Population
& Sampling Distribution
Population
.3
.2
.1
.0
P(X)
1
2
3
  2.5
  112
.
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4
© 1998 Prentice-Hall, Inc.
Comparison of Population
& Sampling Distribution
Population
.3
.2
.1
.0
Sampling Distribution
P(X)
.3
.2
.1
.0
P(X)
1
2
3
4
X
1 1.5 2 2.5 3 3.5 4
  2.5
 x  2.5
  112
.
 x  0.79
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Standard Error of Mean
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1. Standard deviation of all possible
sample means,x

Measures scatter in all sample means,x
2. Less than pop. standard deviation
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Standard Error of Mean
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1. Standard deviation of all possible
sample means,x

Measures scatter in all sample means,x
2. Less than pop. standard deviation
3. Formula (sampling with replacement)

x 
n
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Properties of Sampling
Distribution of Mean
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Properties of Sampling
Distribution of Mean
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1. Unbiasedness

Mean of sampling distribution equals
population mean
2. Efficiency

Sample mean comes closer to population
mean than any other unbiased estimator
3. Consistency

As sample size increases, variation of sample
mean from population mean decreases
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Unbiasedness
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P(X)
Unbiased
A
C

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Biased
X
Efficiency
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P(X) Sampling
distribution
of mean
B
Sampling
distribution
of median
A

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X
Consistency
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P(X)
Larger
sample
size
B
Smaller
sample
size
A

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X
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Sampling from
Normal Populations
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Sampling from
Normal Populations
Central Tendency
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Sampling from
Normal Populations
Central Tendency
x  
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Sampling from
Normal Populations
Central Tendency
x  
Dispersion

x 
n
Sampling with
replacement
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Sampling from
Normal Populations
Central Tendency
x  
Population
= 10
Distribution
Dispersion

x 
n
Sampling with
replacement
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 = 50
X
© 1998 Prentice-Hall, Inc.
Sampling from
Normal Populations
Population
= 10
Distribution
Central Tendency
x  
Dispersion

x 
n
Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =16
X = 2.5
X- = 50
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X
X
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Standardizing Sampling
Distribution of Mean
Suppose you want to make
probability statements about
the sampling distribution...
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Standardizing Sampling
Distribution of Mean
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Sampling
Distribution
X
X
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X
Standardizing Sampling
Distribution of Mean
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Sampling
Distribution
Standardized
Normal Distribution
X
= 1
X
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X
 =0
Z
Standardizing Sampling
Distribution of Mean
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Sampling
Distribution
X  x X  
Z


x
n
Standardized
Normal Distribution
X
= 1
X
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X
 =0
Z
Thinking Challenge
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You’re an operations
analyst for AT&T. Longdistance telephone calls
are normally distribution
with  = 8 min. &  = 2
min. If you select random
samples of 25 calls, what
percentage of the sample
means would be between
7.8 & 8.2 minutes?
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Alone
© 1984-1994 T/Maker Co.
Group Class
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Sampling Distribution
Solution*
X   7.8  8
Z

  .50
 n 2 25
Sampling
Distribution
X   8.2  8
Z

 .50
Standardized
 n 2 25
Normal Distribution
X = .4
=1
.3830
.1915 .1915
7.8 8 8.2 X
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-.50 0 .50
Z
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Sampling from
Non-Normal Populations
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Sampling from
Non-Normal Populations
Central Tendency
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Sampling from
Non-Normal Populations
Central Tendency
x  
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© 1998 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
Central Tendency
x  
Dispersion

x 
n

Sampling with
replacement
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© 1998 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
Central Tendency
x  
Population
= 10
Distribution
Dispersion

x 
n

Sampling with
replacement
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 = 50
X
© 1998 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
Population
= 10
Distribution
Central Tendency
x  
Dispersion

x 
n

Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =30
X = 1.8
X- = 50
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X
X
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Central Limit Theorem
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Central Limit Theorem
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6 - 55
Central Limit Theorem
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As
sample
size gets
large
enough
(n  30) ...
X
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Central Limit Theorem
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As
sample
size gets
large
enough
(n  30) ...
sampling
distribution
becomes
almost
normal.
X
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Central Limit Theorem
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As
sample
size gets
large
enough
(n  30) ...

x 
n
x  
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sampling
distribution
becomes
almost
normal.
X
Conclusion
© 1998 Prentice-Hall, Inc.
1. Described the properties of estimators
2. Explained sampling distribution
3. Described the relationship between
populations & sampling distributions
4. Stated the Central Limit Theorem
5. Solved a probability problem involving
sampling distributions
6 - 59
This Class...
© 1998 Prentice-Hall, Inc.
Please take a moment to answer the
following questions in writing:
1. What was the most important thing you
learned in class today?
2. What do you still have questions about?
3. How can today’s class be improved?
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End of Chapter
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