Transcript Population Sample

BA 201
Lecture 11
Sampling Distributions
© 2001 Prentice-Hall, Inc.
Chap 7-1
Topics

Estimation Process
Point Estimates
Interval Estimates

Sampling Distribution of the Mean


© 2001 Prentice-Hall, Inc.
Chap 7-2
Population and Sample
Population
p.??
Sample
Use statistics to
summarize features
Use parameters to
summarize features
Inference on the population from the sample
© 2001 Prentice-Hall, Inc.
Chap 7-3
pp.??
Estimation Process
Population
Mean, , is
unknown
Random Sample
X  50
I conjecture
that the
population
mean, , is 50
Sample
© 2001 Prentice-Hall, Inc.
Chap 7-4
p.267
Point Estimates
Estimate Population
Parameters …
Mean
Proportion
Variance
Difference
© 2001 Prentice-Hall, Inc.

p
with Sample
Statistics
X
PS

1  2
2
S
2
X1  X 2
Chap 7-5
Another Point Estimate

Here is a link to some of the most recent poll
results
© 2001 Prentice-Hall, Inc.
Chap 7-6
p.?
Drawback of Point Estimates


Q. What is the probability that a point
estimate will equal to the true parameter that
is being estimated?
A. Zero. Theoretically, you will never obtain a
point estimate that equals the unknown
parameter.
© 2001 Prentice-Hall, Inc.
Chap 7-7
pp.??
Interval Estimation Process
Population
Mean, , is
unknown
Random Sample
X  50
I am 95%
confident that 
is between 40 &
60.
Sample
© 2001 Prentice-Hall, Inc.
Chap 7-8
p.267
Interval Estimates

Provides Range of Values
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


Take into consideration variation in sample
statistics from sample to sample
Based on observation from 1 sample
Give Information about Closeness to Unknown
Population Parameters
Stated in terms of level of confidence

© 2001 Prentice-Hall, Inc.
Never 100% sure
Chap 7-9
pp.??
Confidence Interval Estimates
Confidence
Intervals
Mean
 Known
© 2001 Prentice-Hall, Inc.
Proportion
 Unknown
Chap 7-10
Why Study Sampling
Distributions


Sample Statistics are Used to Estimate
Population Parameters
 E.g. X  50 estimates the population mean 
X
Problems: Different Sample Provides Different
Estimate



p.252
Large sample gives better estimate; large sample
costs more
How good is the estimate?
Approach to Solution: Theoretical Basis is
Sampling Distribution
© 2001 Prentice-Hall, Inc.
Chap 7-11
p.252
Sampling Distribution


Theoretical Probability Distribution of a
Sample Statistic
Sample Statistic is a Random Variable


Sample mean, sample proportion
Results from Taking All Possible Samples of
the Same Size
© 2001 Prentice-Hall, Inc.
Chap 7-12
pp. 256-261
When the Population is Normal
Central Tendency
Population Distribution
 X  10
X  X
Variation
X 
X
n
Sampling with
Replacement
© 2001 Prentice-Hall, Inc.
 X  50
Sampling Distributions
n4
X 5
n  16
 X  2.5
 X  50
X
Chap 7-13
When the Population is Not
pp.261-265
Normal
Population Distribution
Central Tendency
X  
Variation
X 

n
Sampling with
Replacement
© 2001 Prentice-Hall, Inc.
 X  10
 X  50
Sampling Distributions
n4
n  30
X 5
 X  1.8
 X  50
X
Chap 7-14
p.261
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
© 2001 Prentice-Hall, Inc.
Chap 7-15
Applet to Illustrate the CLT

Click here to access the applet that will
illustrate the Central Limit Theorem in action.
© 2001 Prentice-Hall, Inc.
Chap 7-16
p.265
How Large is Large Enough?

For Most Distributions, n>30

For Fairly Symmetric Distributions, n>15

For Normal Distribution, the Sampling
Distribution of the Mean is Always Normally
Distributed

This is a property of sampling from a normal
population distribution and is NOT a result of the
central limit theorem
© 2001 Prentice-Hall, Inc.
Chap 7-17
Summary
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
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Illustrated Estimation Process
Discussed Point Estimates
Addressed Interval Estimates
Discussed Sampling Distribution of the Sample
Mean
© 2001 Prentice-Hall, Inc.
Chap 7-18