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統計學
Fall 2003
授課教師：統計系余清祥
日期：2003年11月18日
第十週：抽樣與抽樣分配
Slide 1
Chapter 7
Sampling and Sampling Distributions







Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of x
Sampling Distribution of p
n = 100
Properties of Point Estimators
Other Sampling Methods
n = 30
Slide 2
Statistical Inference



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The purpose of statistical inference is to obtain
information about a population from information
contained in a sample.
A population is the set of all the elements of interest.
A sample is a subset of the population.
The sample results provide only estimates of the
values of the population characteristics.
A parameter is a numerical characteristic of a
population.
With proper sampling methods, the sample results
will provide “good” estimates of the population
characteristics.
Slide 3
什麼是統計?

統計學是研究定義問題、運用資料蒐
集、整理、陳示、分析與推論等科學
方法, 在不確定(Uncertainty)情況下, 做
出合理決策的科學。
Slide 4
為什麼要抽樣？
為什麼只看一部份的母體？
普查(Census)：逐一檢查母體的所有個體
。例如：戶口普查、工商業普查。
普查需要較長的時間、較多的經費與人
力，往往只有政府負擔得起。(政府也是
每十年普查一次，其他時間輔以問卷調
查、公務統計等等彌補資料的不足。)
有時抽樣是唯一可行的方法。

Slide 5
抽樣的實例
品質管制(Quality Control)
為確保品質，產品出廠時須經過檢查。但
逐一檢查耗費過多的時間及金錢，通常
每一批抽一個(或幾個)檢查。
毀滅性抽樣(如鞭炮、罐頭等等產品)
 健康檢查
抽血、切片或抹片檢查

Slide 6
對樣本的要求

因為我們將從樣本推測出母體的原貌，
抽出的部分必須能反映全體的特性，也
就是說樣本需能代表母體。
 樣本代表性！！！
 最忌諱「瞎子摸象」
Slide 7
Simple Random Sampling

Finite Population
• A simple random sample from a finite population
of size N is a sample selected such that each
possible sample of size n has the same probability
of being selected.
• Replacing each sampled element before selecting
subsequent elements is called sampling with
replacement.
• Sampling without replacement is the procedure
used most often.
• In large sampling projects, computer-generated
random numbers are often used to automate the
sample selection process.
Slide 8
Simple Random Sampling

Infinite Population
• A simple random sample from an infinite
population is a sample selected such that the
following conditions are satisfied.
• Each element selected comes from the same
population.
• Each element is selected independently.
• The population is usually considered infinite if it
involves an ongoing process that makes listing or
counting every element impossible.
• The random number selection procedure cannot
be used for infinite populations.
Slide 9
Point Estimation




In point estimation we use the data from the sample
to compute a value of a sample statistic that serves as
an estimate of a population parameter.
We refer to x as the point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion
p.
Slide 10
Sampling Error



The absolute difference between an unbiased point
estimate and the corresponding population
parameter is called the sampling error.
Sampling error is the result of using a subset of the
population (the sample), and not the entire
population to develop estimates.
The sampling errors are:
| x   | for sample mean
|s -  | for sample standard deviation
| p  p | for sample proportion
Slide 11
Example: St. Andrew’s
St. Andrew’s University receives 900 applications
annually from prospective students. The application
forms contain a variety of information including the
individual’s scholastic aptitude test (SAT) score and
whether or not the individual desires on-campus
housing.
Slide 12
Example: St. Andrew’s
The director of admissions would like to know the
following information:
• the average SAT score for the applicants, and
• the proportion of applicants that want to live on
campus.
We will now look at three alternatives for obtaining
the desired information.
• Conducting a census of the entire 900 applicants
• Selecting a sample of 30 applicants, using a
random number table
• Selecting a sample of 30 applicants, using
computer-generated random numbers
Slide 13
Example: St. Andrew’s

Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
x


i
900
 990
• Population Standard Deviation
(x  )
2

i
900
 80
• Applicants Wanting On-Campus Housing
• Population Proportion
648
p
 .72
900
Slide 14
Example: St. Andrew’s

Take a Sample of 30Applicants Using a Random
Number Table
Since the finite population has 900 elements, we
will need 3-digit random numbers to randomly select
applicants numbered from 1 to 900.
We will use the last three digits of the 5-digit
random numbers in the third column of a random
number table. The numbers we draw will be the
numbers of the applicants we will sample unless
• the random number is greater than 900 or
• the random number has already been used.
We will continue to draw random numbers until we
have selected 30 applicants for our sample.
Slide 15
Example: St. Andrew’s

Use of Random Numbers for Sampling
3-Digit
Applicant
Random Number Included in Sample
744
No. 744
436
No. 436
865
No. 865
790
No. 790
835
No. 835
902
Number exceeds 900
190
No. 190
436
Number already used
etc.
etc.
Slide 16
Example: St. Andrew’s

Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
30
685
Applicant
SAT Score
Connie Reyman
1025
William Fox
950
Fabian Avante
1090
Eric Paxton
1120
Winona Wheeler
1015
.
.
Kevin Cossack
965
On-Campus
Yes
Yes
No
Yes
No
.
No
Slide 17
Example: St. Andrew’s

Take a Sample of 30 Applicants Using ComputerGenerated Random Numbers
• Excel provides a function for generating random
numbers in its worksheet.
• 900 random numbers are generated, one for each
applicant in the population.
• Then we choose the 30 applicants corresponding
to the 30 smallest random numbers as our sample.
• Each of the 900 applicants have the same
probability of being included.
Slide 18
Using Excel to Select
a Simple Random Sample

Formula Worksheet
1
2
3
4
5
6
7
8
9
A
B
C
Applicant
Number
1
2
3
4
5
6
7
8
SAT
Score
1008
1025
952
1090
1127
1015
965
1161
On-Campus
Housing
Yes
No
Yes
Yes
Yes
No
Yes
No
D
Random
Number
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
Note: Rows 10-901 are not shown.
Slide 19
Using Excel to Select
a Simple Random Sample

Value Worksheet
1
2
3
4
5
6
7
8
9
A
B
C
D
Applicant
Number
1
2
3
4
5
6
7
8
SAT
Score
1008
1025
952
1090
1127
1015
965
1161
On-Campus
Housing
Yes
No
Yes
Yes
Yes
No
Yes
No
Random
Number
0.41327
0.79514
0.66237
0.00234
0.71205
0.18037
0.71607
0.90512
Note: Rows 10-901 are not shown.
Slide 20
Using Excel to Select
a Simple Random Sample

Value Worksheet (Sorted)
1
2
3
4
5
6
7
8
9
A
B
C
D
Applicant
Number
12
773
408
58
116
185
510
394
SAT
Score
1107
1043
991
1008
1127
982
1163
1008
On-Campus
Housing
No
Yes
Yes
No
Yes
Yes
Yes
No
Random
Number
0.00027
0.00192
0.00303
0.00481
0.00538
0.00583
0.00649
0.00667
Note: Rows 10-901 are not shown.
Slide 21
Example: St. Andrew’s

Point Estimates
• x as Point Estimator of 
x

x
30
i

29,910
 997
30
• s as Point Estimator of 
(x  x )

s

2
i
29
• p as Point Estimator of p
163,996
 75.2
29
p  20 30  .68

Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
Slide 22
Sampling Distribution of x

Process of Statistical Inference
Population
with mean
=?
The value of x is used to
make inferences about
the value of .
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x.
Slide 23
Sampling Distribution of x


The sampling distribution of x is the probability
distribution of all possible values of the sample
mean x.
Expected Value of x
E( x ) = 
where:
 = the population mean
Slide 24
Sampling Distribution of x

Standard Deviation of x
Finite Population

N n
x  ( )
n N 1
Infinite Population
x 

n
• A finite population is treated as being infinite if
n/N < .05.
• ( N  n) / ( N  1) is the finite correction factor.
•  x is referred to as the standard error of the mean.
Slide 25
Sampling Distribution of x

If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
sampling distribution of x can be approximated by a
normal probability distribution.

When the simple random sample is small (n < 30), the
sampling distribution of x can be considered normal
only if we assume the population has a normal
probability distribution.
Slide 26
Example: St. Andrew’s

Sampling Distribution of
x
for the SAT Scores

80
x 

 14.6
n
30
E ( x )    990
x
Slide 27
Example: St. Andrew’s

Sampling Distribution of x for the SAT Scores
What is the probability that a simple random
sample of 30 applicants will provide an estimate of
the population mean SAT score that is within plus or
minus 10 of the actual population mean  ?
In other words, what is the probability that x
will be between 980 and 1000?
Slide 28
Example: St. Andrew’s

Sampling Distribution of x for the SAT Scores
Sampling
distribution
of x
Area = .2518
Area = .2518
x
980 990 1000
Using the standard normal probability table with
z = 10/14.6= .68, we have area = (.2518)(2) = .5036
Slide 29
Sampling Distribution of p

The sampling distribution of p is the probability
distribution of all possible values of the sample
proportion p

Expected Value of p
E ( p)  p
where:
p = the population proportion
Slide 30
Sampling Distribution of p

Standard Deviation of p
Finite Population
p 
p(1  p) N  n
n
N 1
Infinite Population
p 
p(1  p)
n
•  p is referred to as the standard error of the
proportion.
Slide 31
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
.72(1  .72)
p 
 .082
30
E( p )  .72
The normal probability distribution is an acceptable
approximation since np = 30(.72) = 21.6 > 5 and
n(1 - p) = 30(.28) = 8.4 > 5.
Slide 32
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
What is the probability that a simple random
sample of 30 applicants will provide an estimate of
the population proportion of applicants desiring oncampus housing that is within plus or minus .05 of
the actual population proportion?
In other words, what is the probability that p
will be between .67 and .77?
Slide 33
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
Sampling
distribution
of p
Area = .2291
Area = .2291
0.67 0.72 0.77
p
For z = .05/.082 = .61, the area = (.2291)(2) = .4582.
The probability is .4582 that the sample proportion
will be within +/-.05 of the actual population
proportion.
Slide 34
Properties of Point Estimators

Before using a sample statistic as a point estimator,
statisticians check to see whether the sample statistic
has the following properties associated with good
point estimators.
• Unbiasedness
• Efficiency
• Consistency
Slide 35
Properties of Point Estimators

Unbiasedness
If the expected value of the sample statistic is
equal to the population parameter being estimated,
the sample statistic is said to be an unbiased
estimator of the population parameter.
Slide 36
Properties of Point Estimators

Efficiency
Given the choice of two unbiased estimators of
the same population parameter, we would prefer to
use the point estimator with the smaller standard
deviation, since it tends to provide estimates closer to
the population parameter.
The point estimator with the smaller standard
deviation is said to have greater relative efficiency
than the other.
Slide 37
Properties of Point Estimators

Consistency
A point estimator is consistent if the values of the
point estimator tend to become closer to the
population parameter as the sample size becomes
larger.
Slide 38
Other Sampling Methods
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Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling
Slide 39
Stratified Random Sampling
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The population is first divided into groups of
elements called strata.
Each element in the population belongs to one and
only one stratum.
Best results are obtained when the elements within
each stratum are as much alike as possible (i.e.
homogeneous group).
A simple random sample is taken from each stratum.
Formulas are available for combining the stratum
sample results into one population parameter
estimate.
Slide 40
分層隨機抽樣(Stratified Random Sampling)
第一層
○○○○○○
○○○○○
○ ○
第二層
XXXXX
XXXX
X
第三層



○○○○
○○
XXX
XX


抽樣
Slide 41
Stratified Random Sampling


Advantage: If strata are homogeneous, this method
is as “precise” as simple random sampling but with a
smaller total sample size.
Example: The basis for forming the strata might be
department, location, age, industry type, etc.
Slide 42
Cluster Sampling

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The population is first divided into separate groups
of elements called clusters.
Ideally, each cluster is a representative small-scale
version of the population (i.e. heterogeneous group).
A simple random sample of the clusters is then taken.
All elements within each sampled (chosen) cluster
form the sample.
… continued
Slide 43
集體隨機抽樣(Cluster Random Sampling)
○○○○○○○
×××
△△△△△
A
○○○○○○○
×××
△△△△△
B
○○○○○○○
×××
△△△△△
C
抽出A、D
○○○○○○○
×××
△△△△△

○○○○○○○
×××
△△△△△
D
○○○○○○○
×××
△△△△△
E
○○○○○○○
×××
△△△△△
F
○○○○○○○
×××
△△△△△
Slide 44
Cluster Sampling



Advantage: The close proximity of elements can be
cost effective (I.e. many sample observations can be
obtained in a short time).
Disadvantage: This method generally requires a
larger total sample size than simple or stratified
random sampling.
Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.
Slide 45
Systematic Sampling




If a sample size of n is desired from a population
containing N elements, we might sample one element
for every n/N elements in the population.
We randomly select one of the first n/N elements
from the population list.
We then select every n/Nth element that follows in
the population list.
This method has the properties of a simple random
sample, especially if the list of the population
elements is a random ordering.
… continued
Slide 46
系統抽樣(Systematic Sampling)
∣○●○○∣○●○○∣○●○○∣○●○○∣○●○○∣母體
↓
●●●●●
樣本
Slide 47
Systematic Sampling


Advantage: The sample usually will be easier to
identify than it would be if simple random sampling
were used.
Example: Selecting every 100th listing in a telephone
book after the first randomly selected listing.
Slide 48
Convenience Sampling (便利抽樣)





It is a nonprobability sampling technique. Items are
included in the sample without known probabilities
of being selected.
The sample is identified primarily by convenience.
Advantage: Sample selection and data collection are
relatively easy.
Disadvantage: It is impossible to determine how
representative of the population the sample is.
Example: A professor conducting research might use
student volunteers to constitute a sample.
Slide 49
Judgment Sampling (立意抽樣)





The person most knowledgeable on the subject of the
study selects elements of the population that he or
she feels are most representative of the population.
It is a nonprobability sampling technique.
Advantage: It is a relatively easy way of selecting a
sample.
Disadvantage: The quality of the sample results
depends on the judgment of the person selecting the
sample.
Example: A reporter might sample three or four
senators, judging them as reflecting the general
opinion of the senate.
Slide 50
End of Chapter 7
Slide 51