1 1 slide 統計學 fall 2003 授課教師：統計系余清祥日期： 2003 年 11 月 18 日...

1 1 Slide Slide

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

2 2 Slide Slide

Chapter 7Chapter 7Sampling and Sampling DistributionsSampling and Sampling Distributions

Simple Random SamplingSimple Random Sampling Point EstimationPoint Estimation Introduction to Sampling DistributionsIntroduction to Sampling Distributions Sampling Distribution of Sampling Distribution of Sampling Distribution ofSampling Distribution of Properties of Point EstimatorsProperties of Point Estimators Other Sampling MethodsOther Sampling Methods

xxpp nn = 100 = 100

nn = 30 = 30

3 3 Slide Slide

Statistical InferenceStatistical Inference

The purpose of The purpose of statistical inferencestatistical inference is to obtain is to obtain information about a population from information about a population from information contained in a sample.information contained in a sample.

A A populationpopulation is the set of all the elements of is the set of all the elements of interest.interest.

A A samplesample is a subset of the population. is a subset of the population. The sample results provide only The sample results provide only estimatesestimates of of

the values of the population characteristics.the values of the population characteristics. A A parameterparameter is a numerical characteristic of a is a numerical characteristic of a

population.population. With With proper sampling methodsproper sampling methods, the sample , the sample

results will provide “good” estimates of the results will provide “good” estimates of the population characteristics.population characteristics.

4 4 Slide Slide

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。

5 5 Slide Slide

為什麼要抽樣？為什麼要抽樣？為什麼只看一部份的母體？為什麼只看一部份的母體？普查普查 ((Census)Census) ：：逐一檢查母體的所有個逐一檢查母體的所有個體。例如：戶口普查、工商業普查。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。力，往往只有政府負擔得起。 (( 政府也是政府也是每十年普查一次，其他時間輔以問卷調每十年普查一次，其他時間輔以問卷調查、公務統計等等彌補資料的不足。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。有時抽樣是唯一可行的方法。

6 6 Slide Slide

抽樣的實例抽樣的實例品質管制品質管制 ((Quality Control)Quality Control)

為確保品質，產品出廠時須經過檢查。但為確保品質，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，通常逐一檢查耗費過多的時間及金錢，通常每一批抽一個每一批抽一個 (( 或幾個或幾個 )) 檢查。檢查。

毀滅性抽樣毀滅性抽樣 (( 如鞭炮、罐頭等等產品如鞭炮、罐頭等等產品 )) 健康檢查健康檢查抽血、切片或抹片檢查抽血、切片或抹片檢查

7 7 Slide Slide

對樣本的要求對樣本的要求因為我們將從樣本推測出母體的原貌，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。就是說樣本需能代表母體。樣本代表性！！！樣本代表性！！！

最忌諱「瞎子摸象」最忌諱「瞎子摸象」

8 8 Slide Slide

Simple Random SamplingSimple Random Sampling

Finite PopulationFinite Population

• A A simple random sample from a finite simple random sample from a finite population population of size of size NN is a sample selected is a sample selected such that each possible sample of size such that each possible sample of size nn has has the same probability of being selected.the same probability of being selected.

• Replacing each sampled element before Replacing each sampled element before selecting subsequent elements is called selecting subsequent elements is called sampling with replacementsampling with replacement..

• Sampling without replacementSampling without replacement is the is the procedure used most often.procedure used most often.

• In large sampling projects, computer-In large sampling projects, computer-generated generated random numbersrandom numbers are often used are often used to automate the sample selection process.to automate the sample selection process.

9 9 Slide Slide

Infinite PopulationInfinite Population

• A simple random sample from an infinite A simple random sample from an infinite population is a sample selected such that population is a sample selected such that the following conditions are satisfied.the following conditions are satisfied.• Each element selected comes from the Each element selected comes from the

same population.same population.• Each element is selected independently.Each element is selected independently.

• The population is usually considered infinite The population is usually considered infinite if it involves an ongoing process that makes if it involves an ongoing process that makes listing or counting every element listing or counting every element impossible.impossible.

• The random number selection procedure The random number selection procedure cannot be used for infinite populations.cannot be used for infinite populations.

Simple Random SamplingSimple Random Sampling

10 10 Slide Slide

Point EstimationPoint Estimation

In In point estimationpoint estimation we use the data from the we use the data from the sample to compute a value of a sample sample to compute a value of a sample statistic that serves as an estimate of a statistic that serves as an estimate of a population parameter.population parameter.

We refer to as the We refer to as the point estimatorpoint estimator of the of the population mean population mean ..

ss is the is the point estimatorpoint estimator of the population of the population standard deviation standard deviation ..

is the is the point estimatorpoint estimator of the population of the population proportion proportion pp..

xx

pp

11 11 Slide Slide

Sampling ErrorSampling Error

The absolute difference between an unbiased The absolute difference between an unbiased point estimate and the corresponding point estimate and the corresponding population parameter is called the population parameter is called the sampling sampling errorerror..

Sampling error is the result of using a subset of Sampling error is the result of using a subset of the population (the sample), and not the entire the population (the sample), and not the entire population to develop estimates.population to develop estimates.

The sampling errors are:The sampling errors are:

for sample meanfor sample mean

||ss - - for sample standard for sample standard deviationdeviation

for sample proportionfor sample proportion

|| x || x

|| pp || pp

12 12 Slide Slide

Example: St. Andrew’sExample: St. Andrew’s

St. Andrew’s University receives 900 St. Andrew’s University receives 900 applicationsapplications

annually from prospective students. The annually from prospective students. The applicationapplication

forms contain a variety of information including forms contain a variety of information including thethe

individual’s scholastic aptitude test (SAT) score individual’s scholastic aptitude test (SAT) score andand

whether or not the individual desires on-campus whether or not the individual desires on-campus

housing.housing.

13 13 Slide Slide


The director of admissions would like to know theThe director of admissions would like to know thefollowing information:following information:

• the average SAT score for the applicants, andthe average SAT score for the applicants, and• the proportion of applicants that want to live on the proportion of applicants that want to live on

campus.campus.We will now look at three alternatives for obtainingWe will now look at three alternatives for obtaining

the desired information.the desired information.• Conducting a census of the entire 900 applicantsConducting a census of the entire 900 applicants• Selecting a sample of 30 applicants, using a Selecting a sample of 30 applicants, using a

random number tablerandom number table• Selecting a sample of 30 applicants, using Selecting a sample of 30 applicants, using

computer-generated random numberscomputer-generated random numbers

14 14 Slide Slide

Taking a Census of the 900 ApplicantsTaking a Census of the 900 Applicants

• SAT ScoresSAT Scores• Population MeanPopulation Mean

• Population Standard DeviationPopulation Standard Deviation

• Applicants Wanting On-Campus HousingApplicants Wanting On-Campus Housing• Population ProportionPopulation Proportion

ix 990900

ix 990900

ix

2( )80

900

ix

2( )80

900

p648

.72900

p648

.72900


15 15 Slide Slide


Take a Sample of 30Applicants Using a Random Take a Sample of 30Applicants Using a Random Number TableNumber Table

Since the finite population has 900 elements, Since the finite population has 900 elements, we will need 3-digit random numbers to randomly we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900.select applicants numbered from 1 to 900.

We will use the We will use the lastlast three digits of the 5-digit three digits of the 5-digit random numbers in the random numbers in the thirdthird column of a random column of a random number table. The numbers we draw will be the number table. The numbers we draw will be the numbers of the applicants we will sample unlessnumbers of the applicants we will sample unless

• the random number is greater than 900 orthe random number is greater than 900 or• the random number has already been used.the random number has already been used.

We will continue to draw random numbers until weWe will continue to draw random numbers until wehave selected 30 applicants for our sample.have selected 30 applicants for our sample.

16 16 Slide Slide


Use of Random Numbers for SamplingUse of Random Numbers for Sampling

3-Digit3-Digit ApplicantApplicant

Random NumberRandom Number Included in Included in SampleSample

744744 No. 744 No. 744 436436 No. 436 No. 436 865865 No. 865 No. 865 790790 No. 790 No. 790 835835 No. 835 No. 835 902902 Number exceeds Number exceeds

900900 190190 No. 190 No. 190 436436 Number already Number already

usedused etc.etc. etc. etc.

17 17 Slide Slide

Sample DataSample Data

RandomRandom

No.No. NumberNumber ApplicantApplicant SAT ScoreSAT Score On-CampusOn-Campus

11 744 744 Connie Reyman Connie Reyman 1025 1025 Yes Yes

22 436 436 William Fox William Fox 950 950 Yes Yes

33 865 865 Fabian Avante Fabian Avante 1090 1090 No No

44 790 790 Eric Paxton Eric Paxton 1120 1120 Yes Yes

55 835 835 Winona Wheeler Winona Wheeler 1015 1015 No No

.. . . . . . . . .

3030 685 685 Kevin Cossack Kevin Cossack 965 965 No No


18 18 Slide Slide


Take a Sample of 30 Applicants Using Take a Sample of 30 Applicants Using Computer-Generated Random NumbersComputer-Generated Random Numbers

• Excel provides a function for generating Excel provides a function for generating random numbers in its worksheet.random numbers in its worksheet.

• 900 random numbers are generated, one 900 random numbers are generated, one for each applicant in the population.for each applicant in the population.

• Then we choose the 30 applicants Then we choose the 30 applicants corresponding to the 30 smallest random corresponding to the 30 smallest random numbers as our sample.numbers as our sample.

• Each of the 900 applicants have the same Each of the 900 applicants have the same probability of being included.probability of being included.

19 19 Slide Slide

Using Excel to SelectUsing Excel to Selecta Simple Random Samplea Simple Random Sample

Formula WorksheetFormula WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes =RAND()3 2 1025 No =RAND()4 3 952 Yes =RAND()5 4 1090 Yes =RAND()6 5 1127 Yes =RAND()7 6 1015 No =RAND()8 7 965 Yes =RAND()9 8 1161 No =RAND()

Note: Rows 10-901 are not shown.Note: Rows 10-901 are not shown.

20 20 Slide Slide


Value WorksheetValue WorksheetA B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 1 1008 Yes 0.413273 2 1025 No 0.795144 3 952 Yes 0.662375 4 1090 Yes 0.002346 5 1127 Yes 0.712057 6 1015 No 0.180378 7 965 Yes 0.716079 8 1161 No 0.90512


21 21 Slide Slide


Value Worksheet (Sorted)Value Worksheet (Sorted)

A B C D

1Applicant Number

SAT Score

On-Campus Housing

Random Number

2 12 1107 No 0.000273 773 1043 Yes 0.001924 408 991 Yes 0.003035 58 1008 No 0.004816 116 1127 Yes 0.005387 185 982 Yes 0.005838 510 1163 Yes 0.006499 394 1008 No 0.00667


22 22 Slide Slide

Point EstimatesPoint Estimates• as Point Estimator of as Point Estimator of

• ss as Point Estimator of as Point Estimator of

• as Point Estimator of as Point Estimator of pp

Note:Note: Different random numbers would have Different random numbers would have identified a different sample which would have identified a different sample which would have resulted in different point estimates.resulted in different point estimates.

xx

pp

ixx29,910

99730 30

ixx29,910

99730 30

ix x

s2( ) 163,996

75.229 29

ix x

s2( ) 163,996

75.229 29

p 20 30 .68 p 20 30 .68


23 23 Slide Slide

Sampling Distribution ofSampling Distribution of

Process of Statistical InferenceProcess of Statistical Inference

Population Population with meanwith mean

= ?= ?

Population Population with meanwith mean

= ?= ?

A simple random sampleA simple random sampleof of nn elements is selected elements is selected

from the population.from the population.

xx

The sample data The sample data provide a value forprovide a value for

the sample meanthe sample mean . .

The sample data The sample data provide a value forprovide a value for

the sample meanthe sample mean . .xx

The value of is used toThe value of is used tomake inferences aboutmake inferences about

the value of the value of ..

The value of is used toThe value of is used tomake inferences aboutmake inferences about

the value of the value of ..

xx

24 24 Slide Slide

The The sampling distribution of sampling distribution of is the is the probability distribution of all possible values of probability distribution of all possible values of the sample the sample

mean .mean . Expected Value ofExpected Value of

EE( ) = ( ) =

where: where:

= the population mean = the population mean

Sampling Distribution of Sampling Distribution of xx

xx

xx

xx

xx

25 25 Slide Slide

Standard Deviation ofStandard Deviation of

Finite PopulationFinite Population Infinite Infinite Population Population

• A finite population is treated as being A finite population is treated as being infinite if infinite if nn//NN << .05. .05.

• is the finite correction is the finite correction factor.factor.

• is referred to as the is referred to as the standard error of the standard error of the meanmean..

xx

x n

N nN

( )1

x n

N nN

( )1

x n

x n

( ) / ( )N n N 1( ) / ( )N n N 1

x x


26 26 Slide Slide

If we use a large (If we use a large (nn >> 30) simple random 30) simple random sample, the sample, the central limit theoremcentral limit theorem enables us enables us to conclude that the sampling distribution of to conclude that the sampling distribution of can be approximated by a normal probability can be approximated by a normal probability distribution.distribution.

When the simple random sample is small (When the simple random sample is small (nn < < 30), the sampling distribution of can be 30), the sampling distribution of can be considered normal only if we assume the considered normal only if we assume the population has a normal probability population has a normal probability distribution.distribution.

xx

xx


27 27 Slide Slide

Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scoresxx


xn

8014.6

30

xn

8014.6

30

E x( ) 990E x( ) 990

xx

28 28 Slide Slide

Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scores

What is the probability that a simple What is the probability that a simple random sample of 30 applicants will provide random sample of 30 applicants will provide an estimate of the population mean SAT score an estimate of the population mean SAT score that is within plus or minus 10 of the actual that is within plus or minus 10 of the actual population mean population mean ? ?

In other words, what is the probability In other words, what is the probability that will be between 980 and 1000?that will be between 980 and 1000?

xx


xx

29 29 Slide Slide

Sampling Distribution of for the SAT ScoresSampling Distribution of for the SAT Scores

Using the standard normal probability table with Using the standard normal probability table with

zz = 10/14.6= .68, we have area = (.2518)(2) = 10/14.6= .68, we have area = (.2518)(2) = .5036= .5036

xx

Sampling distribution of

Sampling distribution of xx

10001000980980 990990

Area = .2518Area = .2518Area = .2518Area = .2518


xx

30 30 Slide Slide

The The sampling distribution of sampling distribution of is the is the probability distribution of all possible values of probability distribution of all possible values of the sample proportion the sample proportion

Expected Value ofExpected Value of

where:where:

pp = the population proportion = the population proportion

Sampling Distribution of Sampling Distribution of pp

pp

pp

pp

E p p( ) E p p( )

31 31 Slide Slide

Sampling Distribution of Sampling Distribution of pp

pp

pp pn

N nN

( )11

pp pn

N nN

( )11

pp pn

( )1 pp pn

( )1

p p

Standard Deviation of Standard Deviation of

Finite PopulationFinite Population Infinite Population Infinite Population

• is referred to as the is referred to as the standard error of the standard error of the proportionproportion..

32 32 Slide Slide

Sampling Distribution of for In-State ResidentsSampling Distribution of for In-State Residents

The normal probability distribution is an The normal probability distribution is an acceptable approximation since acceptable approximation since npnp = 30(.72) = = 30(.72) = 21.6 21.6 >> 5 and 5 and

nn(1 - (1 - pp) = 30(.28) = 8.4 ) = 30(.28) = 8.4 >> 5. 5.

pp

p

.72(1 .72).082

30

p

.72(1 .72).082

30


( ) .72E p ( ) .72E p

33 33 Slide Slide

Sampling Distribution of for In-State Sampling Distribution of for In-State ResidentsResidents

What is the probability that a simple What is the probability that a simple random sample of 30 applicants will provide random sample of 30 applicants will provide an estimate of the population proportion of an estimate of the population proportion of applicants desiring on-campus housing that is applicants desiring on-campus housing that is within plus or minus .05 of the actual within plus or minus .05 of the actual population proportion?population proportion?

In other words, what is the probability In other words, what is the probability that that

will be between .67 and .77?will be between .67 and .77?

pp


pp

34 34 Slide Slide

Sampling Distribution of for In-State ResidentsSampling Distribution of for In-State Residents

For For zz = .05/.082 = .61, the area = (.2291)(2) = .05/.082 = .61, the area = (.2291)(2) = .4582.= .4582.The probability is .4582 that the sample The probability is .4582 that the sample proportion will be within +/-.05 of the actual proportion will be within +/-.05 of the actual population proportion.population proportion.



0.770.770.670.67 0.720.72

Area = .2291Area = .2291Area = .2291Area = .2291

pp

pp


pp

35 35 Slide Slide

Properties of Point EstimatorsProperties of Point Estimators

Before using a sample statistic as a point Before using a sample statistic as a point estimator, statisticians check to see whether estimator, statisticians check to see whether the sample statistic has the following the sample statistic has the following properties associated with good point properties associated with good point estimators.estimators.

• UnbiasednessUnbiasedness

• EfficiencyEfficiency

• ConsistencyConsistency

36 36 Slide Slide


UnbiasednessUnbiasedness

If the expected value of the sample If the expected value of the sample statistic is equal to the population parameter statistic is equal to the population parameter being estimated, the sample statistic is said to being estimated, the sample statistic is said to be an be an unbiased estimatorunbiased estimator of the population of the population parameter.parameter.

37 37 Slide Slide


EfficiencyEfficiency

Given the choice of two unbiased Given the choice of two unbiased estimators of the same population parameter, estimators of the same population parameter, we would prefer to use the point estimator we would prefer to use the point estimator with the smaller standard deviation, since it with the smaller standard deviation, since it tends to provide estimates closer to the tends to provide estimates closer to the population parameter.population parameter.

The point estimator with the smaller The point estimator with the smaller standard deviation is said to have greater standard deviation is said to have greater relative efficiencyrelative efficiency than the other. than the other.

38 38 Slide Slide


ConsistencyConsistency

A point estimator is A point estimator is consistentconsistent if the if the values of the point estimator tend to become values of the point estimator tend to become closer to the population parameter as the closer to the population parameter as the sample size becomes larger.sample size becomes larger.

39 39 Slide Slide

Other Sampling MethodsOther Sampling Methods

Stratified Random SamplingStratified Random Sampling Cluster SamplingCluster Sampling Systematic SamplingSystematic Sampling Convenience SamplingConvenience Sampling Judgment SamplingJudgment Sampling

40 40 Slide Slide

Stratified Random SamplingStratified Random Sampling

The population is first divided into groups of The population is first divided into groups of elements called elements called stratastrata..

Each element in the population belongs to one Each element in the population belongs to one and only one stratum.and only one stratum.

Best results are obtained when the elements Best results are obtained when the elements within each stratum are as much alike as within each stratum are as much alike as possible (i.e. possible (i.e. homogeneous grouphomogeneous group).).

A simple random sample is taken from each A simple random sample is taken from each stratum.stratum.

Formulas are available for combining the Formulas are available for combining the stratum sample results into one population stratum sample results into one population parameter estimate.parameter estimate.

41 41 Slide Slide

分層隨機抽樣 (Stratified Random Sampling)

○○○○ ○○

XXXXX

○○○○○○ ○○○○○ ○ ○

XXXXXXXXX

X

抽樣

第一層

第二層

第三層

42 42 Slide Slide

Stratified Random SamplingStratified Random Sampling

AdvantageAdvantage: If strata are homogeneous, this : If strata are homogeneous, this method is as “precise” as simple random method is as “precise” as simple random sampling but with a smaller total sample size.sampling but with a smaller total sample size.

ExampleExample: The basis for forming the strata : The basis for forming the strata might be department, location, age, industry might be department, location, age, industry type, etc.type, etc.

43 43 Slide Slide

Cluster SamplingCluster Sampling

The population is first divided into separate The population is first divided into separate groups of elements called groups of elements called clustersclusters..

Ideally, each cluster is a representative small-Ideally, each cluster is a representative small-scale version of the population (i.e. scale version of the population (i.e. heterogeneous group).heterogeneous group).

A simple random sample of the clusters is then A simple random sample of the clusters is then taken.taken.

All elements within each sampled (chosen) All elements within each sampled (chosen) cluster form the sample.cluster form the sample.

… … continuedcontinued

44 44 Slide Slide

集體隨機抽樣 (Cluster Random Sampling)

A

B

C

D

E

F

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

○○○○○○○×××

△△△△△

抽出A 、 D

45 45 Slide Slide

Cluster SamplingCluster Sampling

AdvantageAdvantage: The close proximity of elements : The close proximity of elements can be cost effective (I.e. many sample can be cost effective (I.e. many sample observations can be obtained in a short time).observations can be obtained in a short time).

DisadvantageDisadvantage: This method generally requires : This method generally requires a larger total sample size than simple or a larger total sample size than simple or stratified random sampling.stratified random sampling.

ExampleExample: A primary application is area : A primary application is area sampling, where clusters are city blocks or sampling, where clusters are city blocks or other well-defined areas.other well-defined areas.

46 46 Slide Slide

Systematic SamplingSystematic Sampling

If a sample size of If a sample size of nn is desired from a is desired from a population containing population containing NN elements, we might elements, we might sample one element for every sample one element for every nn//NN elements in elements in the population.the population.

We randomly select one of the first We randomly select one of the first nn//NN elements from the population list.elements from the population list.

We then select every We then select every nn//NNth element that th element that follows in the population list.follows in the population list.

This method has the properties of a simple This method has the properties of a simple random sample, especially if the list of the random sample, especially if the list of the population elements is a random ordering.population elements is a random ordering.

… … continuedcontinued

47 47 Slide Slide

∣ ○●○○∣ ○●○○∣ ○●○○∣ ○●○○∣ ○●○○∣ 母體

↓

●●●●● 樣本

系統抽樣 (Systematic Sampling)

48 48 Slide Slide

Systematic SamplingSystematic Sampling

AdvantageAdvantage: The sample usually will be easier : The sample usually will be easier to identify than it would be if simple random to identify than it would be if simple random sampling were used.sampling were used.

ExampleExample: Selecting every 100: Selecting every 100thth listing in a listing in a telephone book after the first randomly telephone book after the first randomly selected listing.selected listing.

49 49 Slide Slide

Convenience Sampling (Convenience Sampling ( 便利抽樣便利抽樣 ))

It is a It is a nonprobability sampling techniquenonprobability sampling technique. Items . Items are included in the sample without known are included in the sample without known probabilities of being selected.probabilities of being selected.

The sample is identified primarily by The sample is identified primarily by convenienceconvenience..

AdvantageAdvantage: Sample selection and data collection : Sample selection and data collection are relatively easy.are relatively easy.

DisadvantageDisadvantage: It is impossible to determine how : It is impossible to determine how representative of the population the sample is. representative of the population the sample is.

ExampleExample: A professor conducting research might : A professor conducting research might use student volunteers to constitute a sample. use student volunteers to constitute a sample.

50 50 Slide Slide

Judgment Sampling (Judgment Sampling ( 立意抽樣立意抽樣 ))

The person most knowledgeable on the subject The person most knowledgeable on the subject of the study selects elements of the population of the study selects elements of the population that he or she feels are most representative of that he or she feels are most representative of the population.the population.

It is a It is a nonprobability sampling techniquenonprobability sampling technique.. AdvantageAdvantage: It is a relatively easy way of : It is a relatively easy way of

selecting a sample.selecting a sample. DisadvantageDisadvantage: The quality of the sample results : The quality of the sample results

depends on the judgment of the person depends on the judgment of the person selecting the sample.selecting the sample.

ExampleExample: A reporter might sample three or four : A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate.opinion of the senate.

51 51 Slide Slide

End of Chapter 7End of Chapter 7

1 1 slide 統計學 fall 2003 授課教師：統計系余清祥 日期： 2003 年 11 月 18 日...

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1 1 slide 統計學 fall 2003 授課教師：統計系余清祥日期： 2003 年 11 月 18 日...