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© 2003 Prentice-Hall, Inc. Chap 7-1
Basic Business Statistics (9th Edition)
Chapter 7Sampling Distributions
© 2003 Prentice-Hall, Inc. Chap 7-2
Chapter Topics
Sampling Distribution of the Mean
The Central Limit Theorem
Sampling Distribution of the Proportion
Sampling from Finite Population
© 2003 Prentice-Hall, Inc. Chap 7-3
Why Study Sampling Distributions
Sample Statistics are Used to Estimate Population Parameters E.g., 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
50X
© 2003 Prentice-Hall, Inc. Chap 7-4
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
© 2003 Prentice-Hall, Inc. Chap 7-5
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 inYears A
B C
D
© 2003 Prentice-Hall, Inc. Chap 7-6
1
2
1
18 20 22 2421
4
2.236
N
ii
N
ii
X
N
X
N
.3
.2
.1
0 A B C D (18) (20) (22) (24)
Uniform Distribution
P(X)
X
Developing Sampling Distributions
(continued)
Summary Measures for the Population Distribution
© 2003 Prentice-Hall, Inc. Chap 7-7
1st 2nd Observation Obs 18 20 22 24
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
24 24,18 24,20 24,22 24,24
All Possible Samples of Size n=2
16 Samples Taken with Replacement
16 Sample Means1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Developing Sampling Distributions
(continued)
© 2003 Prentice-Hall, Inc. Chap 7-8
1st 2nd Observation Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Sampling Distribution of All Sample Means
18 19 20 21 22 23 240
.1
.2
.3
X
Sample Means
Distribution
16 Sample Means
_
Developing Sampling Distributions
(continued)
P X
© 2003 Prentice-Hall, Inc. Chap 7-9
1
2
1
2 2 2
18 19 19 2421
16
18 21 19 21 24 211.58
16
N
ii
X
N
i Xi
X
X
N
X
N
Summary Measures of Sampling Distribution
Developing Sampling Distributions
(continued)
© 2003 Prentice-Hall, Inc. Chap 7-10
Comparing the Population with Its Sampling
Distribution
18 19 20 21 22 23 240
.1
.2
.3
X
Sample Means Distribution
n = 2
A B C D (18) (20) (22) (24)
0
.1
.2
.3
PopulationN = 4
X_
21 2.236 21 1.58X X P X P X
© 2003 Prentice-Hall, Inc. Chap 7-11
Properties of Summary Measures
I.e., is unbiased
Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators
For Sampling with Replacement or without Replacement from Large or Infinite Populations:
As n increases, decreases
X
X
Xn
X
X
© 2003 Prentice-Hall, Inc. Chap 7-12
Unbiasedness ( )
BiasedUnbiased
X X
X
f X
© 2003 Prentice-Hall, Inc. Chap 7-13
Less Variability
Sampling Distribution of Median Sampling
Distribution of Mean
X
f X
Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators
X
© 2003 Prentice-Hall, Inc. Chap 7-14
Effect of Large Sample
Larger sample size
Smaller sample size
X
f X
For sampling with replacement:
As increases, decreasesXn
© 2003 Prentice-Hall, Inc. Chap 7-15
When the Population is Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
Xn
X50X
4
5X
n
16
2.5X
n
50
10
© 2003 Prentice-Hall, Inc. Chap 7-16
When the Population isNot Normal
Central Tendency
Variation
Population Distribution
Sampling Distributions
X
Xn
X50X
4
5X
n
30
1.8X
n
50
10
© 2003 Prentice-Hall, Inc. Chap 7-17
Central Limit Theorem
As Sample Size Gets Large Enough
Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X
© 2003 Prentice-Hall, Inc. Chap 7-18
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
© 2003 Prentice-Hall, Inc. Chap 7-19
Example:
8 =2 25
7.8 8.2 ?
n
P X
Sampling Distribution
Standardized Normal
Distribution2
.425
X 1Z
8X 8.2 Z0Z
0.5
7.8 8 8.2 87.8 8.2
2 / 25 2 / 25
.5 .5 .3830
X
X
XP X P
P Z
7.8 0.5
.1915
X
© 2003 Prentice-Hall, Inc. Chap 7-20
Population Proportions Categorical Variable
E.g., Gender, Voted for Bush, College Degree
Proportion of Population Having a Characteristic
Sample Proportion Provides an Estimate
If Two Outcomes, X Has a Binomial
Distribution Possess or do not possess characteristic
number of successes
sample sizeS
Xp
n
p
p
© 2003 Prentice-Hall, Inc. Chap 7-21
Sampling Distribution ofSample Proportion
Approximated by Normal Distribution
Mean:
Standard error: p = population
proportion
Sampling Distributionf(ps)
.3
.2
.1 0
0 . 2 .4 .6 8 1ps
5np 1 5n p
Spp
1Sp
p p
n
© 2003 Prentice-Hall, Inc. Chap 7-22
Standardizing Sampling Distribution of Proportion
1S
S
S p S
p
p p pZ
p p
n
Sampling Distribution
Standardized Normal
Distribution
Sp 1Z
Sp Sp Z0Z
© 2003 Prentice-Hall, Inc. Chap 7-23
Example: 200 .4 .43 ?Sn p P p
.43 .4.43 .87 .8078
.4 1 .4
200
S
S
S pS
p
pP p P P Z
Sampling Distribution
Standardized Normal
DistributionSp
1Z
Sp
Sp Z0.43 .87
© 2003 Prentice-Hall, Inc. Chap 7-24
Sampling from Finite Sample
Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N ) Use Finite Population Correction Factor (fpc)
Standard Error with FPC
1X
N n
Nn
1
1SP
p p N n
n N
.05 or / .05n N n N
© 2003 Prentice-Hall, Inc. Chap 7-25
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