Download Chap 6: Sampling Distributions

© 2000 Prentice-Hall, Inc.
Statistics for Business and
Economics
Sampling Distributions
Chapter 6
6-1
Learning Objectives
© 2000 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 Probability Problems Involving
Sampling Distributions
6-2
© 2000 Prentice-Hall, Inc.
Inferential Statistics
6-3
Statistical Methods
© 2000 Prentice-Hall, Inc.
Statistical
Methods
Descriptive
Statistics
6-4
Inferential
Statistics
Inferential Statistics
© 2000 Prentice-Hall, Inc.
1.
Involves:


2.
Estimation
Hypothesis
Testing
Purpose

Make Decisions
about Population
Characteristics
6-5
Population?
Inference Process
© 2000 Prentice-Hall, Inc.
6-6
Inference Process
© 2000 Prentice-Hall, Inc.
Population
6-7
Inference Process
© 2000 Prentice-Hall, Inc.
Population
Sample
6-8
Inference Process
© 2000 Prentice-Hall, Inc.
Population
Sample
statistic
(X)
6-9
Sample
Inference Process
© 2000 Prentice-Hall, Inc.
Estimates
& tests
Sample
statistic
(X)
6 - 10
Population
Sample
Estimators
© 2000 Prentice-Hall, Inc.
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
6 - 11
© 2000 Prentice-Hall, Inc.
Sampling Distributions
6 - 12
Sampling Distribution
© 2000 Prentice-Hall, Inc.
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
6 - 13
© 2000 Prentice-Hall, Inc.
Developing
Sampling 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
© 1984-1994 T/Maker Co.
6 - 14
Population
Characteristics
© 2000 Prentice-Hall, Inc.
Summary Measures
Population Distribution
N

X
i 1
N
i
.3
.2
.1
.0
 2.5
1
N

2


X


 i
i 1
N
6 - 15
 1.12
2
3
4
© 2000 Prentice-Hall, Inc.
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
6 - 16
© 2000 Prentice-Hall, Inc.
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
6 - 17
© 2000 Prentice-Hall, Inc.
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
6 - 18
P(X)
.3
.2
.1
.0
1.0 1.5 2.0 2.5 3.0 3.5 4.0
X
Summary Measures of
All Sample Means
© 2000 Prentice-Hall, Inc.
N
x 
 Xi
i 1
N
 X
N
x 
i 1
1.0  1.5    4.0

 2.5
16
 x 
2
i
N
1.0  2.5  1.5  2.5
2

6 - 19
2
16
   4.0  2.5
2
 0.79
Comparison
© 2000 Prentice-Hall, Inc.
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
6 - 20
Standard Error of Mean
© 2000 Prentice-Hall, Inc.
1. Standard Deviation of All Possible
Sample Means,X

Measures Scatter in All Sample Means,X
2. Less Than Pop. Standard Deviation
6 - 21
Standard Error of Mean
© 2000 Prentice-Hall, Inc.
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
6 - 22
© 2000 Prentice-Hall, Inc.
Properties of Sampling
Distribution of Mean
6 - 23
Properties of Sampling
Distribution of Mean
© 2000 Prentice-Hall, Inc.
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
6 - 24
Unbiasedness
© 2000 Prentice-Hall, Inc.
P(X)
Unbiased
A
C

6 - 25
Biased
X
Efficiency
© 2000 Prentice-Hall, Inc.
P(X) Sampling
distribution
of mean
B
Sampling
distribution
of median
A

6 - 26
X
Consistency
© 2000 Prentice-Hall, Inc.
P(X)
Larger
sample
size
B
Smaller
sample
size
A

6 - 27
X
© 2000 Prentice-Hall, Inc.
Sampling from
Normal Populations
6 - 28
© 2000 Prentice-Hall, Inc.
Sampling from
Normal Populations
Central Tendency
Population Distribution
= 10
x  
Dispersion

x 
n
Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =16
X = 2.5
X- = 50
6 - 29
X
X
© 2000 Prentice-Hall, Inc.
Standardizing Sampling
Distribution of Mean
Sampling
Distribution
X  x X  
Z


x
n
Standardized
Normal Distribution
X
= 1
X
6 - 30
X
 =0
Z
Thinking Challenge
© 2000 Prentice-Hall, Inc.
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?
6 - 31
© 1984-1994 T/Maker Co.
© 2000 Prentice-Hall, Inc.
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
6 - 32
-.50 0 .50
Z
© 2000 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
6 - 33
© 2000 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
Central Tendency
Population Distribution
= 10
x  
Dispersion

x 
n

Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =30
X = 1.8
X- = 50
6 - 34
X
X
© 2000 Prentice-Hall, Inc.
Central Limit Theorem
6 - 35
Central Limit Theorem
© 2000 Prentice-Hall, Inc.
6 - 36
Central Limit Theorem
© 2000 Prentice-Hall, Inc.
As
sample
size gets
large
enough
(n  30) ...
X
6 - 37
Central Limit Theorem
© 2000 Prentice-Hall, Inc.
As
sample
size gets
large
enough
(n  30) ...
sampling
distribution
becomes
almost
normal.
X
6 - 38
Central Limit Theorem
© 2000 Prentice-Hall, Inc.
As
sample
size gets
large
enough
(n  30) ...

x 
n
x  
6 - 39
sampling
distribution
becomes
almost
normal.
X
Conclusion
© 2000 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 Probability Problems Involving
Sampling Distributions
6 - 40
End of Chapter
Any blank slides that follow are
blank intentionally.