Statistical Inference

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Lesson 1

• Statistical Inference
• Random Sampling

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Sampling and Sampling Distributions

• Simple Random Sampling
• Point Estimation
• Introduction to Sampling Distributions
• Sampling Distribution of
• Sampling Distribution of
• Properties of Point Estimators
• Other 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 SamplingFinite Population

• Finite populations are often defined by lists
  such as
• Organization membership roster
• Credit card account numbers
• Inventory product numbers

• 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.

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Simple Random SamplingFinite Population

• 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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Simple Random SamplingInfinite 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.

• 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 SamplingInfinite 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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Random Sampling

• The basis for statistical inference about a
  population based on a sample
• Example Build restaurant in neighborhood?
• Population the collection of items you want to
  understand
• N items. Example all people in neighborhood
• Sample a smaller collection of population units
• n items. Example 100 neighborhood residents who
  agree to be interviewed
• Which 100 residents?
• How to select?

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Random Sample

• A random sample must satisfy
• 1. Each population unit must have an equal chance
  of being selected
• This helps assure representation, because all
  units in the population are equally accessible
• 2. Units must be selected independently of one
  another
• This guarantees that each item to be selected
  will bring new, independent information
• Properties
• Sample is representative of population (on
  average)

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Selecting a Random Sample

• Use Table of Random Digits
• Establish the frame (population units from 1 to
  N)
• Decide starting place in table of random digits
• Read random digits in groups
• e.g., if N 5,281, then use groups of 4 digits
  (N has 4 digits)
• Include number group
• if it is from 1 to N, and has not yet been chosen
• Shuffle the Population (Spreadsheet)
• Arrange the population items in a column from 1
  to N
• Put random numbers in an adjacent column
  RAND()
• Sort population items in order by random numbers

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Table of Random Digits

• For example
• Starting in row 21, column 3
• We find 52794, then 01466

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Example Sample Selection

• Select random sample using random number table
• Of size n 4 from a population of size N 861,
  starting at row 21, column 3 of the Table of
  Random Digits
• Start with random digits
• 52794 01466 85938 14565 79993
• Group by 3 (because N 861 has 3 digits)
• 527 940 146 685 938 145 657
• Omit 000, and also 862, 863, , 999
• Omit duplicates, until n 4 are obtained in the
  sample
• 527 146 685 145
• The random sample includes the following
  population units (numbered by frame)
• 527, 146, 685, 145

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Terminology of Sampling

• Representative Sample
• Same percentages in sample as in population
• Example sample is representative if same
  percent
• work, are young/old, are single/married, etc
• Biased Sample
• Not representative of population in an important
  way
• Example sample is biased if too many retired
  people

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Statistic and Parameter

• Sample Statistic
• Any number computed from sample data
• A random variable. Known
• Example Average weekly food expenditures for 100
  sampled residents
• Random? Yes! Due to randomness of sample
  selection
• Population Parameter
• Any number computed for the entire population
• A fixed number. Unknown
• Example mean weekly food expenditures for all
  77,386 residents
• Do we ever know this? NO!
• But we estimate it (with error)

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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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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 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.

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Sampling Error

• The sampling errors are

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Example UCF BUSINESS STUDENTS

• 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 two alternatives for
  obtaining
• the desired information.
• Conducting a census of the entire 900 applicants
• Selecting a sample of 30 applicants,

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Example UCF

• Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
• Population Standard Deviation
• Applicants Wanting On-Campus Housing
• Population Proportion

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Example UCF

• Take a Sample of 30Applicants Using a Random
  Number Table
• 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.

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Example UCF

• 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 UCF

• 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 UCF

• 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 UCF

• Point Estimates
• as Point Estimator of ?
• s as Point Estimator of ?
• as Point Estimator of p
• Note Different random numbers would have
• identified a different sample which would have
  resulted in different point estimates.

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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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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
• For this class ALWAYS INFINITE

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Sampling Distribution of

• Standard Deviation of
• FOR THIS CLASS ALWAYS INFINITE POPULATION
• is referred to as the standard error of the mean.

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Example

• Let us assume a population of N5
• We can have 5 possible sample of size 4(order is
  not important).

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Continued
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Central Limit Theorem

• In selecting a random sample of size n from a
  population, the sampling distribution of the
  sample mean can be approximated by a
  normal probability distribution as the sample
  size becomes large
• The sampling distribution of can be
  approximated by a normal probability distribution
  whenever the sample size is large. The
  large-sample condition can be assumed for simple
  random samples of size 30 or more
• Whenever the population has a normal probability
  distribution, the sampling distribution of
  has a normal probability distribution for any
  sample size
• Please go to www.ruf.rice.edu./lane/rvls.html

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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 UCF

• Sampling Distribution of for the SAT Scores

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Example UCF BUSINESS STUDENTS

• 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 UCF

• 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

Sampling distribution of
Area .2518
Area .2518
1000
980
990
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• Suppose we select a simple random sample of
  100
• applicants instead of the 30 originally
  considered.

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Area .7888
1000
980
990
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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 Infinite
  Population ONLY
• is referred to as the standard error of the
  proportion.

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np gt 5
n(1 p) gt 5
and
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Example UCF

• Sampling Distribution of for In-State
  Residents
• Recall that 72 of the prospective students
  applying to UCF 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 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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• 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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Example UCF

• Sampling Distribution of for In-State
  Residents

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ExampleUCF

• 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.

Sampling distribution of
Area .2291
Area .2291
0.77
0.67
0.72
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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

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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.

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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.

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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.

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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).
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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.
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.
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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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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.
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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.

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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 LESSON 1