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Chapter 5
Introduction to inferential Statistics:
Sampling and the Sampling Distribution
Copyright © 2012 by Nelson Education Limited.
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In this presentation
you will learn about:
• The basic logic and terminology of
inferential statistics
• Random sampling
• The sampling distribution
Copyright © 2012 by Nelson Education Limited.
5-2
Basic Logic and
Terminology of
Inferential Statistics
– Population and Sample
– Parameter and Statistic
– Representative and EPSEM
Copyright © 2012 by Nelson Education Limited.
5-3
Population and Sample
• Problem: The
populations we
wish to study are
almost always so
large that we are
unable to gather
information from
every case.
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Copyright © 2012 by Nelson Education Limited.
5-4
Population and Sample
(continued)
• Solution: We
choose a sample -a carefully chosen
subset of the
population – and
use information
gathered from the
cases in the sample
to generalize to the
population.
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Copyright © 2012 by Nelson Education Limited.
5-5
Parameter and Statistic
• Statistics are
mathematical
characteristics of
samples.
• Parameters are
mathematical
characteristics of
populations.
• Statistics are used
to estimate
parameters.
PARAMETER
STATISTIC
Copyright © 2012 by Nelson Education Limited.
5-6
Representative and
EPSEM
• Sample must be representative of the population.
– Representative: The sample has the same
characteristics as the population.
• How can we ensure a sample is representative?
– Samples drawn according to the rule of EPSEM (equal
probability of selection method), in which every case in
the population has the same chance of being selected for
the sample are likely to be representative.
Copyright © 2012 by Nelson Education Limited.
5-7
Random Sampling
Techniques
• The most basic EPSEM sampling technique is
simple random sampling (SRS).
• There are variations of this technique, such as
sstratified random sampling and cluster sampling.
– The textbook, however, considers only simple random
sampling since it is the most straightforward application
of EPSEM.
Copyright © 2012 by Nelson Education Limited.
5-8
Simple Random
Sampling (SRS)
• To draw a simple random sample, we
need:
– A list of the population.
– A method for selecting cases from the
population so each case has an equal
probability of being selected (EPSEM).
Copyright © 2012 by Nelson Education Limited.
5-9
Simple Random Sampling
(SRS): An Example
• You want to know what % of students at a
large university work during the semester.
• Draw a sample of 500 from a list of all
students at the university (N = 20,000).
• Assume the list is available from the Registrar.
Copyright © 2012 by Nelson Education Limited.
5-10
Simple Random Sampling
(SRS): An Example (continued)
• How can you draw names so every student
has the same chance of being selected?
– Each student has a unique 6 digit Student ID
Number that ranges from 000000 to 999999.
• Use a Table of Random Numbers (or a computer
program) to select 500 ID numbers with 6 digits each.
• Each time a randomly selected 6-digit number matches
the ID of a student, that student is selected for the
sample.
Copyright © 2012 by Nelson Education Limited.
5-11
Simple Random Sampling (SRS): An Example (continued)
• Below is an excerpt from a Table of Random Numbers:
• To select a random sample from a numbered population list, begin
anywhere in the Table and select groups of digits the same length as the
largest number on your population list.
– Since the population list numbers in our example are in the 100,000s (i.e. Student ID
Numbers range from 000000 to 999999), we select digits in groups of six.
• Every time a number selected from the Table corresponds to the
identification number of a case on the population list, that case is selected
for the sample.
• Continue the process until the desired number of cases (500 in our example)
is selected for the sample.
– Disregard duplicate numbers.
– Ignore cases in which the randomly selected number does not match an identification
number on the population list.
• e.g. ignore cases in which no student ID matches the randomly selected number.
Copyright © 2012 by Nelson Education Limited.
5-12
Simple Random Sampling
(SRS): An Example (continued)
• For example, if we decide to begin from the left of the top
row of the Table, and work across the row, we take the first
set of six digits (i.e., 104801, as boxed in the table below).
– If this number (104801) corresponds to the identification number of a case on the
population list (i.e., a Student's ID Number), that case is selected for the sample.
• Next, we take the second set of six digits (501101).
•
If this number corresponds to a Student's ID Number, that case is selected for the
sample.
• We continue this process until 500 names are selected for the sample.
Copyright © 2012 by Nelson Education Limited.
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Simple Random Sampling
(SRS): An Example (continued)
• Part of the list of 500 selected students might
look like this:
ID #
501101
691791
255958
Student
Andrea Oja
Sasha Albright
Keisha Evans
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Simple Random Sampling
(SRS): An Example (continued)
• After questioning each of these 500
students, you find that 368 (74%) work
during the semester.
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5-15
Applying Logic and
Terminology of Inferential
Statistics
• In the research situation
above, identify each the
following:
– Population
– Sample
– Statistic
– Parameter
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5-16
Applying Logic and
Terminology of Inferential
Statistics (continued)
• Population = All 20,000 students.
• Sample = The 500 students selected
and interviewed.
• Statistic =74% (% of sample that held
a job during the semester).
• Parameter = % of all students in the
population who held a job.
Copyright © 2012 by Nelson Education Limited.
5-17
The Sampling Distribution
• The single most important concept in
inferential statistics.
• Definition: The theoretical,
probabilistic distribution of a statistic
for all possible samples of a given
size (n).
• The sampling distribution is a
theoretical concept.
Copyright © 2012 by Nelson Education Limited.
5-18
The Sampling Distribution
(continued)
• Every application of
inferential statistics involves
3 different distributions:
1. Population: empirical;
unknown
2. Sampling Distribution:
theoretical; known
3. Sample: empirical;
known
• Information from the sample
is linked to the population
via the sampling distribution.
Population
Sampling Distribution
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Sample
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The Sampling Distribution:
Properties
1. Normal in shape.
2. Has a mean equal to the
population mean.
3. Has a standard deviation (called
the standard error) equal to the
population standard deviation, σ,
divided by the square root of n.
Copyright © 2012 by Nelson Education Limited.
5-20
The Sampling Distribution:
First Theorem
• Tells us the shape of the sampling
distribution and defines its mean and
standard deviation.
– If repeated random samples of size n are
drawn from a normal population with mean µ
and standard deviation σ, then the sampling
distribution of sample means will be normal
with a mean µ and a standard deviation
(standard error) of σ/n
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5-21
The Sampling Distribution:
First Theorem (continued)
• Again, if we begin with a trait that is normally
distributed across a population (height,
weight) and take an infinite number of equally
sized random samples from that population,
the sampling distribution of sample means
will be normal.
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5-22
The Sampling Distribution:
Second (Central Limit) Theorem
• If repeated random samples of size n are
drawn from any (e.g., non-normal) shaped
population with mean µ and standard
deviation σ, then as n becomes large the
sampling distribution of sample means will
approach normality, with a mean µ and
standard deviation of σ/n
– Again, for any trait or variable, even those that
are not normally distributed in the population
(income, wealth) as sample size grows larger, the
sampling distribution of sample means will
become normal in shape.
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The Sampling Distribution:
Second (Central Limit) Theorem
(continued)
• The importance of the Central Limit
Theorem is that it removes the constraint of
normality in the population.
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5-24
The Sampling Distribution:
Theorem Proofs
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The Sampling Distribution:
Theorem Proofs (continued)
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The Sampling Distribution:
Theorem Proofs (continued)
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The Sampling Distribution: Theorem Proofs
(continued)
Proof: population mean, 5, equals mean of the sampling
distribution, 5, and population standard deviation divided by
the square root of n, 2.236/2, is equal to standard
deviation of the sampling distribution, 1.581.
Copyright © 2012 by Nelson Education Limited.
5-28
Sampling Distribution: Key Points
1. Definition: Distribution of a statistic for all
possible outcomes of a certain size.
• The statistic for example can be a mean or a
proportion.
2. Shape: Normal
• Since the sampling distribution is normal, we can use
Appendix A to find areas.
3. Central tendency and dispersion: Mean is
same as the population mean; standard
deviation of sampling distribution – the standard
error – is equal to the population standard
deviation divided by the square root of n.
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5-29
Sampling Distribution: Key Points
(continued)
• Role of sampling distribution in inferential
statistics: Links the sample with the population:
Population
Sampling Distribution
Sample
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Three Distributions and their
Symbols
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