Transcript Basic Business Statistics, 9th Edition

STATISTICS FOR
MANAGERS
University of Management and Technology
1925 North Lynn Street
Arlington, VA 22209
Voice: (703) 516-0035 Fax: (703) 516-0985
Website: www.umtweb.edu
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
1 of 26
Chapter 7, STAT125
CHAPTER 7
Sampling Distributions
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
2 of 26
Chapter 7, STAT125
Chapter Topics
Sampling Distribution of the Mean
The Central Limit Theorem
Sampling Distribution of the Proportion
Sampling from Finite Population
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
3 of 26
Chapter 7, STAT125
Why Study Sampling Distributions
Sample Statistics are Used to Estimate Population
Parameters
E.g., X  50 estimates the population mean

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
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
4 of 26
Chapter 7, STAT125
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
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
5 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
Suppose There is a Population …
Population Size N=4
Random Variable, X,
is Age of Individuals
Values of X: 18, 20,
22, 24 Measured in
Years
B
C
D
A
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
6 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
(continued)
Summary Measures for the Population Distribution
N
 

P(X)
Xi
i 1
.3
N

18  20  22  24
.2
 21
4
N
 
X
i
i 1
N
© Prentice Hall 2003
Basic Business Statistics


.1
0
2
 2 .2 3 6
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Visit UMT online at www.umtweb.edu
7 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
(continued)
All Possible Samples of Size n=2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
16 Sample Means
20 20,18 20,20 20,22 20,24
1st 2nd Observation
Obs 18 20 22 24
22 22,18 22,20 22,22 22,24
18 18 19
20
21
24 24,18 24,20 24,22 24,24
20 19 20
21
22
22 20 21
22
23
24 21 22
23
24
16 Samples Taken
with Replacement
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
8 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
.3
20 19 20 21 22
.2
22 20 21 22 23
.1
24 21 22 23 24
0
© Prentice Hall 2003
Basic Business Statistics
PX

_
18 19
20 21 22 23
Visit UMT online at www.umtweb.edu
24
X
9 of 26
Chapter 7, STAT125
Developing Sampling
Distributions
(continued)
Summary Measures of Sampling Distribution
N
X 
X
i 1
18  19  19 
i

N

 21
16
N
X 
 24
X
i
 X

2
i 1
N
18  21 
2
 19  21  
2
  24  21 
2
 1.58
16
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
10 of 26
Chapter 7, STAT125
Comparing the Population with
Its Sampling Distribution
Population
N=4
  21
  2.236
.3
PX
Sample Means Distribution
n=2
 X  21  X  1.58

.3
.2
.2
.1
.1
0
A
B
C
D
(18)
(20)
(22)
(24)
© Prentice Hall 2003
Basic Business Statistics
X
0
PX

_
18 19
20 21 22 23
Visit UMT online at www.umtweb.edu
24
X
11 of 26
Chapter 7, STAT125
Properties of Summary Measures
X  
I.e.,
X is unbiased
Standard Error (Standard Deviation) of the Sampling
Distribution  X is Less Than the Standard Error of Other
Unbiased Estimators
For Sampling with Replacement or without Replacement
from Large or Infinite Populations:
X 

n
As n increases, 
© Prentice Hall 2003
Basic Business Statistics
X
decreases
Visit UMT online at www.umtweb.edu
12 of 26
Chapter 7, STAT125
Unbiasedness (
f
X  
)
X 
Unbiased

© Prentice Hall 2003
Basic Business Statistics
Biased
X
Visit UMT online at www.umtweb.edu
X
13 of 26
Chapter 7, STAT125
Less Variability
Standard Error (Standard Deviation) of the Sampling
Distribution  is Less Than the Standard Error of Other
X
Unbiased Estimators
f
X 
Sampling
Distribution
of Median
© Prentice Hall 2003
Basic Business Statistics
Sampling
Distribution of
Mean

Visit UMT online at www.umtweb.edu
X
14 of 26
Chapter 7, STAT125
Effect of Large Sample
For sam pling w ith replacem ent:
f
A s n increases, 
X 
Basic Business Statistics
decreases
Larger
sample size
Smaller
sample size
© Prentice Hall 2003
X

Visit UMT online at www.umtweb.edu
X
15 of 26
Chapter 7, STAT125
When the Population is Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 
© Prentice Hall 2003
Basic Business Statistics

n
Sampling Distributions
n4
n  16
X 5
 X  2.5
Visit UMT online at www.umtweb.edu  X  50
X
16 of 26
Chapter 7, STAT125
When the Population is
Not Normal
Population Distribution
Central Tendency
  10
X  
  50
Variation
X 

n
Sampling Distributions
n4
n  30
X 5

X
 1.8
X
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
 X  50 Chapter 7,17STAT125
of 26
Central Limit Theorem
Sampling
Distribution
Becomes
Almost
Normal
Regardless
of Shape of
Population
As Sample
Size Gets
Large
Enough
X
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
18 of 26
Chapter 7, STAT125
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
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
19 of 26
Chapter 7, STAT125
Example:
 8
 =2
n  25
P  7.8  X  8.2   ?
 7.8  8
X  X
8.2  8 
P  7.8  X  8.2   P 



X
2 / 25 
 2 / 25
 P   .5  Z  .5   .3830
Standardized
Normal Distribution
Sampling Distribution
2
X 
 .4
25
Z 1
.1 9 1 5
7 .8
8 .2
X  8
© Prentice Hall 2003
Basic Business Statistics
X
 0 .5
Visit UMT online at www.umtweb.edu  Z  0
0 .5
Z
20 of 26
Chapter 7, STAT125
Population Proportions
 p
Categorical Variable
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a Characteristic
Sample Proportion Provides an Estimate
pS 
X
n

 p
num ber of successes
sam ple size
If Two Outcomes, X Has a Binomial Distribution
Possess or do not possess characteristic
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
21 of 26
Chapter 7, STAT125
Sampling Distribution of
Sample Proportion
Approximated by
Normal Distribution
np  5
n 1  p   5
Sampling Distribution
f(ps)
Mean:
p  p
S
Standard error:

pS

p 1  p 
.3
.2
.1
0
0
.2
.4
.6
8
1
ps
n
p = population proportion
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
22 of 26
Chapter 7, STAT125
Standardizing Sampling
Distribution of Proportion
Z 
p S   pS

pS

pS  p
p 1  p 
n
Standardized
Normal Distribution
Sampling Distribution

Z 1
pS
p
© Prentice Hall 2003
Basic Business Statistics
S
pS
Visit UMT online at www.umtweb.edu
Z  0
Z
23 of 26
Chapter 7, STAT125
n  200
Example:
p  .4
P  p S  .43   ?


p S   pS

P  p S  .43   P

 p
S


Standardized
Normal Distribution
Sampling Distribution



.43  .4 
 P  Z  .87   .8078
.4  1  .4  

200

Z 1
pS
 p .4 3
S
© Prentice Hall 2003
Basic Business Statistics
pS
Visit UMT online at www.umtweb.edu
0 .8 7
Z
24 of 26
Chapter 7, STAT125
Sampling from Finite Sample
Modify Standard Error if Sample Size (n) is Large Relative to
Population Size (N )
n  .0 5 N
o r n / N  .0 5
Use Finite Population Correction Factor (fpc)
Standard Error with FPC
X 
P 
S
© Prentice Hall 2003
Basic Business Statistics

N n
n
N 1
p 1  p 
N n
n
N 1
Visit UMT online at www.umtweb.edu
25 of 26
Chapter 7, STAT125
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
© Prentice Hall 2003
Basic Business Statistics
Visit UMT online at www.umtweb.edu
26 of 26
Chapter 7, STAT125