QMS 6351 Statistics and Research Methods Chapter 7 Sampling and Sampling Distributions

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QMS 6351Statistics and Research Methods
Chapter 7Sampling andSampling Distributions

• Prof. Vera Adamchik

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Chapter 7 Outline

• Simple random sampling
• Point estimation
• Introduction to sampling distributions
• Sampling distribution of
• Sampling distribution of
• Other sampling methods

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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 in a study.
• A sample is a subset of the population.

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• A parameter is a numerical characteristic of a
  population.
• A sample statistic is a numerical characteristic
  of a sample.
• We will use a sample statistic in order to judge
  tentatively or approximately the value of the
  population parameter.

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• The sample results provide only estimates (that
  is, rough and approximate values) of the values
  of the population characteristics.
• The reason is simply that the sample contains
  only a portion of the population.
• With proper sampling methods, the sample results
  will provide good estimates of the population
  characteristics.

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Simple random sampling procedure
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Selecting a sample

• Sampling from a finite population. Finite
  populations are often defined by lists such as
  organization membership roster, class roster,
  inventory product numbers, etc.
• Sampling from an infinite population (a process).
  The population is usually considered infinite if
  it involves an ongoing process that makes listing
  or counting every element impossible. For
  example, parts being manufactured on a production
  line, customers entering a store, etc.

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Sampling from a 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.

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Example St. Andrews College
St. Andrews College received 900 applications
for admission in the upcoming year from
prospective students. The applicants were
numbered, from 1 to 900, as their applications
arrived. The Director of Admissions would like to
select a simple random sample of 30 applicants.
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Sampling from a finite population using Excel

• RAND() Excel generates a random number between 0
  and 1
• RAND()N Excel generates a random number greater
  than or equal to 0 but less than or equal N
• INT(RAND()900)

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Sampling from an 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.

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Sampling from an 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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Point estimation
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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.
• A point estimate is a statistic computed from
  a sample that gives a single value for the
  population parameter.
• An estimator is a rule or strategy for using
  the data to estimate the parameter.

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Terminology of point estimation

• We refer to
• as the point estimator of the population mean
  ?.
• We refer to
• as the point estimator of the population
  standard deviation ?.

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Terminology of point estimation

• We refer to
• as the point estimator of the population
  proportion p.
• The actual numerical value obtained for
• in a particular sample is called the point
  estimate of the parameter.

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Example St. Andrews College

• Recall that St. Andrews College received 900
  applications from prospective students. The
  application form contains a variety of
  information including the individuals scholastic
  aptitude test (SAT) score and whether or not the
  individual desires on-campus housing.
• At a meeting in a few hours, the Director of
  Admissions would like to announce the average SAT
  score and the proportion of applicants that want
  to live on campus, for the population of 900
  applicants.

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Example St. Andrews College

• However, the necessary data on the applicants
  have not yet been entered in the colleges
  computerized database. So, the Director decides
  to estimate the values of the population
  parameters of interest based on sample
  statistics. The sample of 30 applicants selected
  earlier with Excels RAND() function will be used.

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Point estimation using Excel
Excel Value Worksheet
Note Rows 10-31 are not shown.
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Point estimates
Note Different random numbers would
have identified a different sample which would
have resulted in different point estimates.
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Population parameters
Once all the data for the 900 applicants were
entered in the colleges database, the values of
the population parameters of interest were
calculated.
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Summary of point estimates obtained from a simple
random sample
Population Parameter
Point Estimator
Point Estimate
Parameter Value
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
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Making inferences about a population mean
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• Making inferences about a population mean

A simple random sample of n elements is
selected from the population.
Population with mean m ?
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Population vs sampling distribution

• The population distribution is the probability
  distribution derived from the information on all
  elements of a population.
• The probability distribution of a sample
  statistic ( ) is called its
  sampling distribution.

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Sampling distribution of

• The sampling distribution of the sample mean
  ( ) is the probability distribution of all
  possible values of .
• We need to know
• Expected value of
• Standard deviation of
• Form of the sampling distribution of

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Mean of the sampling distribution of

• The mean of the sampling distribution of
  is equal to the mean of the population. Thus,

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Standard deviation of the sampling distribution
of
(1) Infinite population (N is unknown) (2)
Finite population and n/N 0.05
Finite population and n/N ? 0.05
is referred to as the standard error of
the mean.
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Two important observations

• 1. The spread of the sampling distribution of
  is smaller than the spread of the
  corresponding population distribution. In other
  words, .
• 2. The standard deviation of the sampling
  distribution of decreases as the sample
  size increases.

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Form of the sampling distribution of

• 1. The population has a normal distribution.
• If the population from which the samples are
  drawn is normally distributed, then the sampling
  distribution of the sample mean will also be
  normally distributed for any sample size.

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Form of the sampling distribution of

• 2. The population is not normally distributed
  but the sample size is large (n 30).
• According to the Central Limit Theorem, for a
  large sample size (n 30), the sampling
  distribution of the sample mean is approximately
  normal, irrespective of the shape of the
  population distribution.
• In cases where the population is highly skewed
  or outliers are present, samples of size 50 or
  more may be needed.

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Form of the sampling distribution of

• 3. The sample size is small (n lt 30) and the
  population is not normally distributed.
• Use special statistical procedures.

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Example St. Andrews College
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Example St. Andrews College

• What is the probability that the sample mean
  will be between 980 and 1000? In other words,
  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 ?

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Example St. Andrews College
Area .5034
1000
980
990
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Example St. Andrews College

• The probability of 0.5034 means that, for a large
  number of samples of size 30 selected from the
  population, we can expect that in 50.34 of all
  cases the sample mean will be within /-10 of the
  actual population mean (that is, 980-1000) and in
  49.66 of all cases the sample mean will be
  further than /-10 of the actual population mean
  (that is, below 980 or above 1000).

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Relationship between the sample size and the
sampling distribution of

• Example St. Andrews College
• Suppose we select a simple random sample of 100
  applicants instead of the 30 originally
  considered.

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• regardless of the sample
  size. In our example, E( ) remains at 990.
• 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 from 14.6 to

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Relationship between the sample size and the
sampling distribution of
Example St. Andrews College
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Example St. Andrews College

• Recall that when n 30, P(980 lt lt 1000)
  .5034.
• Now, with n 100, P(980 lt lt 1000) .7888.
• 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.

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Example St. Andrews College
Area .7888
1000
980
990
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Example St. Andrews College

• The probability of 0.7888 means that, for a large
  number of samples of size100 selected from the
  population, we can expect that in 78.88 of all
  cases the sample mean will be within /-10 of the
  actual population mean (that is, 980-1000) and in
  21.12 of all cases the sample mean will be
  further than /-10 of the actual population mean
  (that is, below 980 or above 1000).

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Making inferences about a population proportion
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Making inferences about a population proportion

A simple random sample of n elements is
selected from the population.
Population with proportion p ?
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Sampling distribution of

• The sampling distribution of the sample
  proportion ( ) is the probability
  distribution of all possible values of .
• We need to know
• Expected value of
• Standard deviation of
• Form of the sampling distribution of

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Mean of the sampling distribution of

• The mean of the sampling distribution of
  is equal to the population proportion. Thus,

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Standard deviation of the sampling distribution
of
(1) Infinite population (N is unknown) (2)
Finite population and n/N 0.05
Finite population and n/N ? 0.05
is referred to as the standard error of
the proportion.
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Form of the 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 the
  following two conditions are satisfied

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Form of the 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 very large
  (approaching 1) values of p, much larger samples
  are needed.

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Example St. Andrews College

• For our example, with n 30 and
• p .72, the normal 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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Sampling distribution of
Example St. Andrews College
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Example St. Andrews College

• Recall that 72 of the prospective students
  applying to St. Andrews College desire on-campus
  housing.
• 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?

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Example St. Andrews College
Area .4582
.77
.67
.72
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Other sampling methods
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Other 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. a homogeneous group).
• A simple random sample is taken from each
  stratum.

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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.

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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.

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