Chapter%207%20Sampling%20and%20Sampling%20Distributions

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Chapter 7Sampling and Sampling Distributions

• Simple Random Sampling
• Point Estimation
• Introduction to Sampling Distributions
• Sampling Distribution of
• Sampling Distribution of
• Sampling Methods

n 100
n 30
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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.
• 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.

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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.

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Simple Random Sampling

• Finite Population
• 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.

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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.

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Simple Random Sampling

• Infinite Population
• 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.

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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 as the point estimator of the
  population mean ?.
• s is the point estimator of the population
  standard deviation ?.
• is the point estimator of the population
  proportion p.

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Point Estimation

• When the expected value of a point estimator is
  equal to the population parameter, the point
  estimator is said to be unbiased.

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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
• for sample mean
• for sample standard deviation
• for sample proportion

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Example St. Andrews

• St. Andrews College receives 900 applications
• annually from prospective students. The
  application
• forms contain a variety of information including
  the
• individuals scholastic aptitude test (SAT) score
  and
• whether or not the individual desires on-campus
• housing.

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Example St. Andrews

• 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.

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Example St. Andrews

• 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

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Example St. Andrews

• Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
• Population Standard Deviation

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Example St. Andrews

• Taking a Census of the 900 Applicants
• Applicants Wanting On-Campus Housing
• Population Proportion

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Example St. Andrews

• Take a Sample of 30 Applicants
• 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 the
  textbooks random number table.

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Example St. Andrews

• Take a Sample of 30 Applicants
• Using 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.

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Example St. Andrews

• 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.

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Example St. Andrews

• Sample Data
• Random
• No. Number Applicant SAT Score
  On-Campus
• 1 744 Connie Reyman 1025 Yes
• 2 436 William Fox 950 Yes
• 3 865 Fabian Avante 1090 No
• 4 790 Eric Paxton 1120 Yes
• 5 835 Winona Wheeler 1015 No
• . . . . .
• 30 685 Kevin Cossack 965 No

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Example St. Andrews

• Take a Sample of 30 Applicants
• Using Computer-Generated 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.

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Using Excel to Selecta Simple Random Sample

• Formula Worksheet

Note Rows 10-901 are not shown.
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Using Excel to Selecta Simple Random Sample

• Value Worksheet

Note Rows 10-901 are not shown.
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Using Excel to Selecta Simple Random Sample

• Value Worksheet (Sorted)

Note Rows 10-901 are not shown.
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Example St. Andrews

• Point Estimates
• as Point Estimator of ?
• s as Point Estimator of ?
• as Point Estimator of p

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Example St. Andrews

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

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Sampling Distribution of

• Process of Statistical Inference

Population with mean m ?
A simple random sample of n elements is
selected from the population.
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Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .
• Expected Value of
• E( ) ?
• where
• ? the population mean

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Sampling Distribution of

• Standard Deviation of
• Finite Population Infinite
  Population
• A finite population is treated as being
  infinite if n/N lt .05.
• is the finite correction
  factor.
• is referred to as the standard error of the
  mean.

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Sampling Distribution of

• If we use a large (n gt 30) simple random sample,
  the central limit theorem enables us to conclude
  that the sampling distribution of can be
  approximated by a normal probability
  distribution.
• When the simple random sample is small (n lt 30),
  the sampling distribution of can be
  considered normal only if we assume the
  population has a normal probability distribution.

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Example St. Andrews

• Sampling Distribution of 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
  will be between 980 and 1000?

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Example St. Andrews

• Sampling Distribution of for the SAT Scores
• Using the standard normal probability table with
  
• z 10/14.6 .68, we have area (.2518)(2)
  .5036
• The probability is .5036 that the sample mean
  will be within /-10 of the actual population
  mean.

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample proportion .
• Expected Value of
• where
• p the population proportion

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Sampling Distribution of

• Standard Deviation of
• Finite Population Infinite Population
• is referred to as the standard error of the
  proportion.

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Sampling Distribution of

• The sampling distribution of can be
  approximated by a normal probability distribution
  whenever the sample size is large.
• The sample size is considered large whenever
  these conditions are satisfied
• np gt 5
• and
• n(1 p) gt 5

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Sampling Distribution of

• For values of p near .50, sample sizes as small
  as 10 permit a normal approximation.
• With very small (approaching 0) or large
  (approaching 1) values of p, much larger samples
  are needed.

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Example St. Andrews

• Sampling Distribution of for In-State
  Residents
• The normal probability distribution is an
  acceptable approximation because
• np 30(.72) 21.6 gt 5
• and
• n(1 - p) 30(.28) 8.4 gt 5.

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Example St. Andrews

• Sampling Distribution of for In-State
  Residents

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Example St. Andrews

• Sampling Distribution of 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 on-campus housing that is within plus or
  minus .05 of the actual population proportion?
• In other words, what is the probability that
• will be between .67 and .77?

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Example St. Andrews

• Sampling Distribution of for In-State
  Residents

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Example St. Andrews

• Sampling Distribution of for In-State
  Residents
• 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.

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Example St. Andrews

• Sampling Distribution of for In-State
  Residents

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Sampling Methods

• Stratified Random Sampling
• Cluster Sampling
• Systematic Sampling
• Convenience Sampling
• Judgment Sampling

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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
  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.

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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.

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Cluster Sampling

• 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

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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.

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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

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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.

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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.
• 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.

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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.
• 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.

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End of Chapter 7