商务统计学课件第7章 出版社： 中国人民大学出版社
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.2 .1
18 19 20 21 22 23 24
0
X
0
18 19 20 21 22 23
24
_
X
Chap 7-10
A
B
C
D
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-3
Sampling Distributions
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
Smaller sample size
Larger sample size
μ
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
x
Chap 7-16
If the Population is not Normal
We can apply the Central Limit Theorem:
16 Sample Means
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
Chap 7-7
24
24,18
24,20
24,22
24,24
16 possible samples (sampling with replacement)
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
n↑
the sampling distribution becomes almost normal regardless of shape of population
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-2
Sampling Distributions
Sampling Distributions
Sampling Distribution of the Mean
A B C D
Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years)
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Comparing the Population with its Sampling Distribution
μ 21
P(X) .3
.2 .1
Population N=4
σ 2.236
Sample Means Distribution n=2
μX 21
_
σ X 1.58
P(X) .3
Learning Objectives
In this chapter, you learn:
The concept of the sampling distribution To compute probabilities related to the sample mean and the sample proportion The importance of the Central Limit Theorem To distinguish between different survey sampling methods To evaluate survey worthiness and survey errors
Chap 7-14
Sampling Distribution Properties
μx μ
(i.e.
Normal Population Distribution
μ
x is unbiased )
x
Sampling Distribution is also normal (and has the same mean)
Z
( X μX ) σX
( X μ) σ n
where:
μ = population mean
X = sample mean
σ = population standard deviation
n = sample size
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-5
Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
X μ
N
i
P(x)
.3 .2 .1 0
18 19 20 21 22 23 24
18 20 22 24 21 4
Business Statistics, A First Course
4Biblioteka Baiduh Edition
Chapter 7 Sampling Distributions
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-1
16 Sample Means
1st 2nd Observation Obs 18 20 22 24
Sample Means Distribution _
P(X) .3 .2 .1 0
18 19 20 21 22 23 24
18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Developing a Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)
σX
σ n
Chap 7-12
Note that the standard error of the mean decreases as the sample size increases
Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution:
μx
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
x
Chap 7-15
Sampling Distribution Properties
(continued)
As n increases,
σ x decreases
σ (X i μ) 2 N 2.236
x
A
B
C
D
Uniform Distribution
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 7-6
Developing a Sampling Distribution
μX μ
and
σX
σ n
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-13
Z-value for Sampling Distribution of the Mean
Z-value for the sampling distribution of X :
Chap 7-11
Standard Error of the Mean
Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with
(continued)
Now consider all possible samples of size n=2
1st Obs 18 20 22 2nd Observation 18 18,18 20,18 22,18 20 18,20 20,20 22,20 22 18,22 20,22 22,22 24 18,24 20,24 22,24
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-4
Developing a Sampling Distribution
Assume there is a population … Population size N=4
μx μ
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
and
σx
σ n
Chap 7-17
Central Limit Theorem
As the sample size gets large enough…
μX
X
N
i
18 19 21 24 21 16
σX
( X i μ X )2 N (18 - 21) 2 (19 - 21) 2 (24 - 21) 2 1.58 16
Chap 7-9
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
_
X
Chap 7-8
(no longer uniform)
Developing a Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
18 19 20 21 22 23 24
0
X
0
18 19 20 21 22 23
24
_
X
Chap 7-10
A
B
C
D
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-3
Sampling Distributions
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
Smaller sample size
Larger sample size
μ
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
x
Chap 7-16
If the Population is not Normal
We can apply the Central Limit Theorem:
16 Sample Means
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
Chap 7-7
24
24,18
24,20
24,22
24,24
16 possible samples (sampling with replacement)
Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
n↑
the sampling distribution becomes almost normal regardless of shape of population
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-2
Sampling Distributions
Sampling Distributions
Sampling Distribution of the Mean
A B C D
Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years)
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Comparing the Population with its Sampling Distribution
μ 21
P(X) .3
.2 .1
Population N=4
σ 2.236
Sample Means Distribution n=2
μX 21
_
σ X 1.58
P(X) .3
Learning Objectives
In this chapter, you learn:
The concept of the sampling distribution To compute probabilities related to the sample mean and the sample proportion The importance of the Central Limit Theorem To distinguish between different survey sampling methods To evaluate survey worthiness and survey errors
Chap 7-14
Sampling Distribution Properties
μx μ
(i.e.
Normal Population Distribution
μ
x is unbiased )
x
Sampling Distribution is also normal (and has the same mean)
Z
( X μX ) σX
( X μ) σ n
where:
μ = population mean
X = sample mean
σ = population standard deviation
n = sample size
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-5
Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
X μ
N
i
P(x)
.3 .2 .1 0
18 19 20 21 22 23 24
18 20 22 24 21 4
Business Statistics, A First Course
4Biblioteka Baiduh Edition
Chapter 7 Sampling Distributions
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-1
16 Sample Means
1st 2nd Observation Obs 18 20 22 24
Sample Means Distribution _
P(X) .3 .2 .1 0
18 19 20 21 22 23 24
18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Developing a Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)
σX
σ n
Chap 7-12
Note that the standard error of the mean decreases as the sample size increases
Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Properties of the sampling distribution:
μx
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
x
Chap 7-15
Sampling Distribution Properties
(continued)
As n increases,
σ x decreases
σ (X i μ) 2 N 2.236
x
A
B
C
D
Uniform Distribution
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 7-6
Developing a Sampling Distribution
μX μ
and
σX
σ n
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-13
Z-value for Sampling Distribution of the Mean
Z-value for the sampling distribution of X :
Chap 7-11
Standard Error of the Mean
Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with
(continued)
Now consider all possible samples of size n=2
1st Obs 18 20 22 2nd Observation 18 18,18 20,18 22,18 20 18,20 20,20 22,20 22 18,22 20,22 22,22 24 18,24 20,24 22,24
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Chap 7-4
Developing a Sampling Distribution
Assume there is a population … Population size N=4
μx μ
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
and
σx
σ n
Chap 7-17
Central Limit Theorem
As the sample size gets large enough…
μX
X
N
i
18 19 21 24 21 16
σX
( X i μ X )2 N (18 - 21) 2 (19 - 21) 2 (24 - 21) 2 1.58 16
Chap 7-9
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc.
_
X
Chap 7-8
(no longer uniform)
Developing a Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution: