Chapter 7
Student Lecture Notes
7-1
Basic Business Statistics
(9th Edition)
Chapter 7
Sampling Distributions
© 2004 Prentice-Hall, Inc.
Chap 7-1
Chapter Topics
„
Sampling Distribution of the Mean
„
The Central Limit Theorem
„
Sampling Distribution of the Proportion
„
Sampling from Finite Population
Chap 7-2
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Why Study Sampling
Distributions
„
„
Sample Statistics are Used to Estimate
Population Parameters
„ E.g., X = 50 estimates the population mean µ
Problem: Different Samples Provide Different
Estimates
„
„
„
Large sample gives better estimate; large sample
costs more
How good is the estimate?
Approach to Solution: Theoretical Basis is
Sampling Distribution
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Chap 7-3
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-2
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
„
Chap 7-4
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Developing Sampling
Distributions
„
Suppose There is a Population …
„
Population Size N=4
B
C
Random Variable, X,
is Age of Individuals
Measured in Years
„
Values of X : 18, 20,
22, 24
A
„
D
Chap 7-5
© 2004 Prentice-Hall, Inc.
Developing Sampling
Distributions
(continued)
Summary Measures for the Population Distribution
N
µ=
∑X
i =1
P(X)
i
.3
N
18 + 20 + 22 + 24
=
= 21
4
N
σ =
∑(X
i =1
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i
N
−µ)
.2
.1
0
2
= 2.236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Chap 7-6
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
Developing Sampling
Distributions
7-3
(continued)
All Possible Samples of Size n=2
1st
Obs
18
2nd Observation
20
22
24
16 Sample Means
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
1st 2nd Observation
Obs 18 20 22 24
24 24,18 24,20 24,22 24,24
18 18 19 20 21
20 19 20 21 22
Nn = 42 = 16
Samples Taken with
Replacement
22 20 21 22 23
24 21 22 23 24
Chap 7-7
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Developing Sampling
Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
.3
18 18 19 20 21
P( X )
.2
20 19 20 21 22
.1
22 20 21 22 23
0
24 21 22 23 24
_
18 19
20 21 22 23
X
24
Chap 7-8
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Developing Sampling
Distributions
(continued)
Summary Measures of Sampling Distribution
Nn
µX =
∑X
i =1
N
n
i
=
Nn
σX =
∑( X
i =1
i
18 + 19 + 19 + L + 24
= 21
16
− µX )
2
Nn
(18 − 21) + (19 − 21)
2
=
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16
2
+ L + ( 24 − 21)
2
= 1.58
Chap 7-9
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-4
Comparing the Population with
Its Sampling Distribution
Population
N=4
µ = 21
P( X )
Sample Means Distribution
n=2
σ = 2.236
µ X = 21
.3
.3
.2
.2
.1
.1
0
A
B
C
(18)
(20)
(22)
D X
0
P( X )
σ X = 1.58
_
18 19
20 21 22 23
24
X
(24)
Chap 7-10
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Properties of Summary Measures
„
µX = µ
„
„
„
I.e., X is unbiased
Standard Error (Standard Deviation) of the
Sampling Distribution σ X is Less Than the
Standard Error of Other Unbiased Estimators
For Sampling with Replacement or without
Replacement from Large or Infinite Populations:
σX =
„
σ
n
As n increases,
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σX
decreases
Chap 7-11
Unbiasedness ( µ X = µ )
f (X )
Unbiased
µ
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µX
X
Chap 7-12
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-5
Less Variability
Standard Error (Standard Deviation) of the
Sampling Distribution σ X is Less Than the
Standard Error of Other Unbiased Estimators
f ( X ) Sampling
Distribution
of Median
Sampling
Distribution of
Mean
µ
X
Chap 7-13
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Effect of Large Sample
For sampling with replacement:
As n increases, σ X decreases
f (X )
Larger
sample size
Smaller
sample size
µ
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X
Chap 7-14
When the Population is Normal
Population Distribution
Central Tendency
µX = µ
Variation
σX =
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σ
n
σ = 10
µ = 50
Sampling Distributions
n=4
n = 16
σX =5
σ X = 2.5
µ X = 50
X
Chap 7-15
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-6
When the Population is
Not Normal
Population Distribution
Central Tendency
σ = 10
µX = µ
µ = 50
Variation
σX =
σ
n
Sampling Distributions
n=4
n = 30
σX =5
σ X = 1.8
µ X = 50
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X
Chap 7-16
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
X
Chap 7-17
© 2004 Prentice-Hall, Inc.
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 Regardless of the Sample Size
„
This is a property of sampling from a normal
population distribution and is NOT a result of the
central limit theorem
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Chap 7-18
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
Example: µ = 8
σ =2
7-7
n = 25
P ( 7.8 < X < 8.2 ) = ?
⎛ 7.8 − 8 X − µ X 8.2 − 8 ⎞
P ( 7.8 < X < 8.2 ) = P ⎜
<
<
⎟
σX
2 / 25 ⎠
⎝ 2 / 25
= P ( −.5 < Z < .5 ) = .3830
Standardized
Normal Distribution
Sampling Distribution
2
= .4
25
σX =
σZ =1
.1915
7.8
8.2
−0.5
X
µX = 8
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0.5
Z
µZ = 0
Chap 7-19
( p)
Population Proportion
Categorical Variable
„
„
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a
Characteristic ( p )
„
Sample Proportion Provides an Estimate
„
„
pS =
X number of successes
=
n
sample size
If Two Outcomes, X Has a Binomial
„
Distribution
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„
Possess or do not possess characteristic
Chap 7-20
Sampling Distribution of
Sample Proportion
„
Approximated by
Normal Distribution
„
„
np ≥ 5
n (1 − p ) ≥ 5
.3
.2
.1
0
Mean:
µp = p
„
Sampling Distribution
f(ps)
0
.2
.4
.6
8
1
ps
S
„
Standard error:
„
σp =
S
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p (1 − p )
n
p = population proportion
Chap 7-21
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-8
Standardizing Sampling
Distribution of Proportion
Z≅
pS − µ p S
σp
=
S
pS − p
p (1 − p )
n
Standardized
Normal Distribution
Sampling Distribution
σp
σZ =1
S
µp
µZ = 0
pS
S
Z
Chap 7-22
© 2004 Prentice-Hall, Inc.
n = 200
Example:
p = .4
⎛
⎜ p −µ
S
pS
<
P ( pS < .43) = P ⎜
⎜ σ pS
⎜⎜
⎝
⎞
⎟
.43 − .4 ⎟
= P ( Z < .87 ) = .8078
.4 (1 − .4 ) ⎟
⎟⎟
200 ⎠
Standardized
Normal Distribution
Sampling Distribution
σp
P ( pS < .43) = ?
σZ =1
S
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µ pS .43
pS
0 .87
Z
Chap 7-23
Sampling from Finite Population
(CD ROM Topic)
„
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 Factor (FPC)
Standard Error with FPC
σ N −n
„
σX =
n N −1
„
© 2004 Prentice-Hall, Inc.
σP =
S
p (1 − p ) N − n
n
N −1
Chap 7-24
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.
Chapter 7
Student Lecture Notes
7-9
Chapter Summary
„
„
„
„
Discussed Sampling Distribution of the Sample
Mean
Described the Central Limit Theorem
Discussed Sampling Distribution of the Sample
Proportion
Described Sampling from Finite Populations
© 2004 Prentice-Hall, Inc.
Chap 7-25
Statistics for Managers Using Microsoft Excel, 2/e
© 1999 Prentice-Hall, Inc.