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Statistics. Sampling and Sampling Distribution. STATISTICS in PRACTICE. MeadWestvaco Corporation ’ s products include textbook paper, magazine paper, and office products. MeadWestvaco ’ s internal consulting

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Statistics

Statistics

Sampling and Sampling Distribution


Statistics in practice

STATISTICSin PRACTICE

  • MeadWestvaco Corporation’s

    products include textbook

    paper, magazine paper, and

    office products.

  • MeadWestvaco’s internal consulting

    group uses sampling to provide information that enables the company to obtain significant productivity benefits and remain competitive.


Statistics in practice1

STATISTICSin PRACTICE

  • Managers need reliable and accurate information about the timberlands and forests to evaluate the company’s ability to meet its future raw material needs.

  • Data collected from sample plots throughout the forests are the basis for learning about the population of trees owned by the company.


Contents

Contents

  • The Electronics Associates Sampling Problem

  • Simple Random Sampling

  • Point Estimation

  • Introduction to Sampling Distributions

  • Sampling Distribution of p

  • Properties of Point Estimators

  • Other Sampling Methods


Statistics

Statistical Inference

  • 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.


Statistics

Statistical Inference

  • The sample results provide only estimates of

  • the values of the population characteristics.

  • With proper sampling methods, the sample

  • results can provide “good” estimates of the

  • population characteristics.

  • A parameter is a numerical characteristic of a

  • population.


The electronics associates sampling problem

The Electronics Associates Sampling Problem

  • Often the cost of collecting information from a sample is substantially less than from a population,

  • Especially when personal interviews must be conducted to collect the information.


Simple random sampling finite population

Simple Random Sampling:Finite Population

  • Finite populations are often defined by lists such as:

    • Organization membership roster

    • Credit card account numbers

    • Inventory product numbers


Simple random sampling finite population1

Simple Random Sampling:Finite Population

  • A simple random sample of size n 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.


Simple random sampling finite population2

Simple Random Sampling:Finite Population

  • Replacing each sampled element before

  • selecting subsequent elements is called

  • sampling with replacement.

  • Sampling without replacement is the

  • procedure used most often.


Simple random sampling

Simple Random Sampling

  • Random Numbers: the numbers in the table are random, these four-digit numbers are equally likely.


Simple random sampling infinite population

Simple Random Sampling:Infinite Population

  • Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.


Simple random sampling infinite population1

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.


Simple random sampling infinite population2

Simple Random Sampling:Infinite Population

  • In the case of infinite populations, it is

  • impossible to obtain a list of all elements

  • in the population.

  • The random number selection procedure

  • cannot be used for infinite populations.


Statistics

  • We refer to as the point estimator of the

  • population mean .

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.


Statistics

  • is the point estimator of the population

  • proportion p.

Point Estimation

  • sis the point estimator of the population standard

  • deviation .


Point estimation

Point Estimation

  • Example: to estimate the population mean, the population standard deviation and population proportion.


Sampling error

Sampling Error

  • When the expected value of a point estimator

  • is equal to the population parameter, the point

    estimator is said to be unbiased.

  • The absolute value of the difference between an

    unbiased point estimate and the corresponding

    population parameter is called the

    sampling error.


Sampling error1

Sampling Error

  • Sampling error is the result of using a subset

  • of the population (the sample), and not the

  • entire population.

  • Statistical methods can be used to make

  • probability statements about the size of the

  • sampling error.


Statistics

for sample mean

for sample standard deviation

for sample proportion

Sampling Error

  • The sampling errors are:


Example st andrew s

Example: St. Andrew’s

St. Andrew’s College

receives 900 applications

annually from

prospective students.

The application form

contains a variety of information

including the individual’s scholastic aptitude

test (SAT) score and whether or not the

individual desires on-campus housing.


Example st andrew s1

Example: St. Andrew’s

The director of admissions

would like to know the

following information:

  • the average SAT score

    for the 900 applicants,

    and

  • the proportion of applicants that want to live on campus.


Example st andrew s2

Example: St. Andrew’s

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 Excel


Statistics

Conducting a Census

  • If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3.

  • We will assume for the moment that conducting a census is practical in this example.


Conducting a census

Conducting a Census

  • Population Mean SAT Score

  • Population Standard Deviation for SAT Score

  • Population Proportion Wanting On-Campus Housing


Statistics

Simple Random Sampling

  • Now suppose that the necessary data on the

    current year’s applicants were not yet entered

    in the college’s database.

  • Furthermore, the Director of Admissions must

  • obtain estimates of the population parameters of

  • interest for a meeting taking place in a few hours.


Statistics

Simple Random Sampling

  • Now suppose that the necessary data on the current year’s applicants were not yet entered in the college’s database.

  • Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours.


Statistics

Simple Random Sampling

  • The applicants were numbered, from 1 to 900, as their applications arrived.


Simple random sampling using a random number table

Simple Random Sampling:Using a Random Number Table

  • Taking a Sample of 30 Applicants

  • Because 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 the textbook’s random number table , and continue into the fourth column as needed.


Simple random sampling using a random number table1

Simple Random Sampling:Using a Random Number Table

  • Taking a Sample of 30 Applicants

  • 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.


  • Simple random sampling using a random number table2

    Simple Random Sampling:Using a Random Number Table

    • (We will go through all of column 3 and

    • part of column 4 of the random number table,

    • encountering in the process five numbers

    • greater than 900 and one duplicate, 835.)


    Simple random sampling using a random number table3

    Simple Random Sampling:Using a Random Number Table

    • Use of Random Numbers for Sampling

    3-Digit

    Random Number

    Applicant

    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

    836

    No. 836

    . . . and so on


    Simple random sampling using a random number table4

    Simple Random Sampling:Using a Random Number Table

    • Sample Data

    Random

    Number

    SAT

    Score

    Live On-

    Campus

    No.

    Applicant

    1744 Conrad Harris1025 Yes

    2436 Enrique Romero 950 Yes

    3865 Fabian Avante1090 No

    4790 Lucila Cruz1120 Yes

    5835 Chan Chiang 930 No

    . . . . .

    . . . . .

    30 498 Emily Morse 1010 No


    Simple random sampling using a computer

    Simple Random Sampling:Using a Computer

    • Taking a Sample of 30 Applicants

    • Computers can be used to generate random

    • numbers for selecting random samples.

    • For example, Excel’s function

    • = RANDBETWEEN(1,900)

    • can be used to generate random numbers

    • between 1 and 900.

    • Then we choose the 30 applicants

    • corresponding to the 30 smallest random

    • numbers as our sample.


    Point estimation1

    • as Point Estimator of 

    • pas Point Estimator of p

    Point Estimation

    • s as Point Estimator of 


    Point estimation2

    Point Estimation

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


    Statistics

    = Sample mean

    SAT score

    = Sample pro-

    portion wanting

    campus housing

    Summary of Point Estimates

    Obtained from a Simple Random Sample

    Population

    Parameter

    Parameter

    Value

    Point

    Estimator

    Point

    Estimate

    m = Population mean

    SAT score

    990

    997

    80

    s = Sample std.

    deviation for

    SAT score

    75.2

    s = Population std.

    deviation for

    SAT score

    .72

    .68

    p = Population pro-

    portion wanting

    campus housing


    Sampling distribution

    Sampling Distribution

    • Example: Relative Frequency Histogram of Sample

      Mean Values from 500 Simple Random Samples of

      30 each.


    Sampling distribution1

    Sampling Distribution

    • Example: Relative Frequency Histogram of Sample

      Proportion Values from 500 Simple Random

      Samples of 30 each.


    Statistics

    The value of is used to

    make inferences about

    the value of m.

    The sample data

    provide a value for

    the sample mean .

    Sampling Distribution of

    • Process of Statistical Inference

    A simple random sample

    of n elements is selected

    from the population.

    Population

    with mean

    m = ?


    Statistics

    E( ) = 

    Sampling Distribution of

    The sampling distribution of is the probability

    distribution of all possible values of the sample

    mean .

    Expected Value of

    where:

    = the population mean


    Statistics

    Sampling Distribution of

    • Standard Deviation of

    InfinitePopulation

    Finite Population

    • A finite population is treated as being

    • infinite if n/N< .05.


    Statistics

    Sampling Distribution of

    • is the finite correction factor.

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


    Statistics

    Form of the Sampling Distribution

    of

    • If we use a large (n> 30) simple random sample,

    • the central limit theorem enables us to conclude

    • that the sampling distribution of can be

    • approximated by a normal distribution.

    • When the simple random sample is small (n < 30),

    • the sampling distribution of can be considered

    • normal only if we assume the population has a

    • normal distribution.


    Central limit theorem

    Central Limit Theorem

    • Illustration of The Central Limit Theorem


    Relationship between the sample size and the sampling distribution of sample mean

    Relationship Between the Sample Size and the Sampling Distribution of Sample Mean

    • A Comparison of The Sampling Distributions of Sample Mean for Simple Random Samples of n = 30 and n = 100.


    Statistics

    Sampling Distribution offor SAT Scores

    Sampling

    Distribution

    of


    Statistics

    Sampling Distribution offor 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 +/-10 of the actual population mean ?

    In other words, what is the probability that

    will be between 980 and 1000?


    Statistics

    Sampling Distribution offor SAT Scores

    Step 1: Calculate the z-value at the upper

    endpoint of the interval.

    z = (1000 - 990)/14.6= .68

    Step 2: Find the area under the curve to the

    left of the upper endpoint.

    P(z< .68) = .7517


    Statistics

    Sampling Distribution offor SAT Scores

    Cumulative Probabilities for

    the Standard Normal Distribution


    Statistics

    Sampling Distribution offor SAT Scores

    Sampling

    Distribution

    of

    Area = .7517

    990

    1000


    Statistics

    Sampling Distribution offor SAT Scores

    P(z< -.68) = P(z> .68)

    = 1 - P(z< .68)

    = 1 - . 7517

    = .2483

    Step 3: Calculate the z-value at the lower

    endpoint of the interval.

    z = (980 - 990)/14.6= - .68

    Step 4: Find the area under the curve to the

    left of the lower endpoint.


    Statistics

    Sampling Distribution offor SAT Scores

    Sampling

    Distribution

    of

    Area = .2483

    980

    990


    Statistics

    Sampling Distribution offor SAT Scores

    P(-.68 <z< .68) = P(z< .68) - P(z< -.68)

    = .7517 - .2483

    = .5034

    Step 5: Calculate the area under the curve

    between the lower and upper endpoints

    of the interval.

    The probability that the sample mean SAT score will be between 980 and 1000 is:

    P(980 << 1000) = .5034


    Statistics

    Sampling Distribution offor SAT Scores

    Sampling

    Distribution

    of

    Area = .5034

    980

    990

    1000


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    • E( ) = m regardless of the sample size.

    • in our example, E( ) remains at 990.

    • Suppose we select a simple random sample

      of 100 applicants instead of the 30 originally

      considered.


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    • Whenever the sample size is increased, the

      standard error of the mean is decreased.

      With the increase in the sample size to n = 100,

      the standard error of the mean is decreased to:


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    With n = 100,

    With n = 30,


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    • We follow the same steps to solve for

    • P(980 << 1000)

    • when n = 100 as we showed earlier when n = 30.

    • Recall that when n = 30,

    • P(980 << 1000) = .5034.


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    • Because the sampling distribution with n = 100

    • has a smaller standard error, the values of have

    • less variability and tend to be closer to the

    • population mean than the values of with n = 30.

    • Now, with n = 100, P(980 << 1000) = .7888.


    Statistics

    Relationship Between the Sample Size and the Sampling Distribution of

    Sampling

    Distribution

    of

    Area = .7888

    980

    990

    1000


    Sampling distribution2

    Sampling Distribution

    • Example: Relative Frequency Histogram of Sample Proportion Values from 500 Simple Random Samples of 30 each.


    Statistics

    The sample data

    provide a value for the

    sample proportion .

    The value of is used

    to make inferences

    about the value of p.

    Sampling Distribution of p

    • Making Inferences about a Population Proportion

    A simple random sample

    of n elements is selected

    from the population.

    Population

    with proportion

    p = ?


    Statistics

    Sampling Distribution of

    The sampling distribution of p is the probability

    distribution of all possible values of the sample

    proportion p .

    Expected Value of p

    where:

    p = the population proportion


    Statistics

    is referred to as the standard error of the proportion.

    Sampling Distribution of

    Standard Deviation of p

    Infinite Population

    Finite Population


    Statistics

    • The sampling distribution of can be

    • approximatedby a normal distribution whenever

    • the sample size is large.

    The sample size is considered large whenever these conditions are satisfied:

    Form of the Sampling Distribution of

    and

    np> 5

    n(1 – p) > 5


    Statistics

    • For values of p near .50, sample sizes as

    • small as 10 permit a normal approximation.

    • With very small (approaching 0) or very

    • large (approaching 1) values of p, much

    • larger samples are needed.

    Form of the Sampling Distribution of


    Statistics

    Sampling Distribution of

    • Example: St. Andrew’s College

    Recall that 72% of the prospective students

    applying to St. Andrew’s College desire

    on-campus housing.


    Statistics

    Sampling Distribution of

    • Example: St. Andrew’s College

    What is the probability that a simple random

    sample of 30 applicants will provide an estimate

    of the population proportion of applicant

    desiring on-campus housing that is within plus or

    minus .05 of the actual population proportion?


    Statistics

    Sampling Distribution of

    For our example, with n = 30 and p = .72,

    the normal distribution is an acceptable

    approximation because:

    np= 30(.72) = 21.6 > 5

    and

    n(1 - p) = 30(.28) = 8.4 > 5


    Statistics

    Sampling

    Distribution

    of

    Sampling Distribution of


    Statistics

    Sampling Distribution of

    Step 1: Calculate the z-value at the upper

    endpoint of the interval.

    z = (.77 - .72) /.082 = .61

    Step 2: Find the area under the curve to the

    left of the upper endpoint.

    P(z< .61) = .7291


    Statistics

    Sampling Distribution of

    Cumulative Probabilities for

    the Standard Normal Distribution


    Statistics

    Sampling

    Distribution

    of

    Sampling Distribution of

    Area = .7291

    .72

    .77


    Statistics

    Sampling Distribution of

    Step 3: Calculate the z-value at the lower

    endpoint of the interval.

    z = (.67 - .72) /.082 = - .61

    Step 4: Find the area under the curve to the

    left of the lower endpoint.

    P(z< -.61) = P(z> .61)

    = 1 - P(z< .61)

    = 1 - . 7291

    = .2709


    Statistics

    Sampling

    Distribution

    of

    Sampling Distribution of

    Area = .2709

    .67

    .72


    Statistics

    P(.67 << .77) = .4582

    Sampling Distribution of

    Step 5: Calculate the area under the curve between

    the lower and upper endpoints of the interval.

    P(-.61 <z< .61) = P(z< .61) - P(z< -.61)

    = .7291 - .2709

    = .4582

    The probability that the sample proportion of

    applicants wanting on-campus housing will be

    within +/-.05 of the actual population proportion :


    Statistics

    Sampling

    Distribution

    of

    Sampling Distribution of

    Area = .4582

    .67

    .72

    .77


    Point estimators

    Point Estimators

    • Notations:

    • θ = the population parameter of interest.

      For example, population mean, population standard deviation, population proportion, and so on.

    ^


    Point estimators1

    Point Estimators

    • Notations:

    • θ = the sample statistic or point estimator of θ .

      Represents the corresponding sample statistic such as the sample mean, sample standard deviation, and sample proportion.

    • The notation θ is the Greek letter theta.

    • the notation θ is pronounced “theta-hat.”

    ^

    ^


    Statistics

    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.

    • Unbiased

    • Efficiency

    • Consistency


    Statistics

    Properties of Point Estimators

    • Unbised

    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.


    Properties of point estimators

    Properties of Point Estimators

    • Unbised

      The sample statistic θis unbiased estimator of the population parameter θ if

    ^

    ^

    E(θ)=θ

    where

    ^

    E(θ)=the expected value of the sample statistic θ


    Properties of point estimators1

    Properties of Point Estimators

    • Examples of Unbiased and Biased Point Estimators


    Statistics

    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.


    Properties of point estimators2

    Properties of Point Estimators

    • Example: Sampling Distributions of Two

      Unbiased Point Estimators.


    Statistics

    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.


    Other sampling methods

    Other Sampling Methods

    • Stratified Random Sampling(分層隨機抽樣)

    • Cluster Sampling(部落抽樣)

    • Systematic Sampling(系統抽樣)

    • Convenience Sampling(便利抽樣)

    • Judgment Sampling(判斷抽樣)


    Statistics

    Stratified Random Sampling

    • 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

    • withineach stratum are as much alike as possible

    • (i.e. a homogeneous group).


    Stratified random sampling

    Stratified Random Sampling

    • Diagram for Stratified Random Sampling


    Statistics

    Stratified Random Sampling

    • A simple random sample is taken from each

    • stratum.

    • Formulas are available for combining the

    • stratum sample results into one population

    • parameter estimate.


    Statistics

    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,

    • and so on.


    Statistics

    Cluster Sampling

    • The population is first divided into separate

    • groupsof 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.


    Cluster sampling

    Cluster Sampling

    • Diagram for ClusterSampling


    Statistics

    Cluster Sampling

    • Example: A primary application is area

    • sampling, where clusters are city blocks or

    • other well-defined areas.

    • 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.


    Statistics

    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.


    Statistics

    Systematic Sampling

    • This method has the properties of a simple

    • random sample, especially if the list of the

    • population elements is a random ordering.

    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


    Statistics

    Convenience Sampling

    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.

    Example: A professor conducting research might

    use student volunteers to constitute a sample.


    Statistics

    Convenience Sampling

    • Advantage: Sample selection and data collection

    • are relatively easy.

    • Disadvantage: It is impossible to determine how

    • representative of the population the sample is.


    Statistics

    Judgment Sampling

    • 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.

    • Example: A reporter might sample three or four

    • senators, judging them as reflecting the general

    • opinion of the senate.


    Statistics

    Judgment Sampling

    • 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.


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