Download The Sampling Distribution Healey Ch. 6

Introduction to Inferential
Statistics
Sampling and the
Sampling Distribution
Outline

The logic and terminology of inferential
statistics

Random sampling

The sampling distribution
Logic And Terminology
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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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Logic And Terminology (cont.)
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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.
Basic Logic And Terminology

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Statistics are
mathematical
characteristics of
samples.
Parameters are
mathematical
characteristics of
populations.
Statistics are used to
estimate parameters.
PARAMETER
STATISTIC
Samples:

Must be representative of the population.


Representative: The sample has the same
characteristics as the population.
How can we ensure samples are
representative?

Samples drawn according to the rule of
EPSEM (every case in the population has the
same chance of being selected for the
sample) are likely to be representative.
Random Sampling Techniques

Simple Random Sampling (SRS)

Systematic Random Sampling

Stratified Random Sampling

Cluster Sampling
Suppose we select a random sample of 500
from a university student body and find that
74% of our sample has worked during the
semester….

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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.
The Sampling Distribution

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We can use the sampling distribution to
calculate our population parameter based on
our sample statistic.
The single most important concept in
inferential statistics.
Definition: The distribution of a statistic for
all possible samples of a given size (N).
The sampling distribution is a theoretical
concept.
The Sampling Distribution


Every application of
inferential statistics
involves 3 different
distributions.
Information from the
sample is linked to the
population via the
sampling distribution.
Population
Sampling Distribution
Sample
The Sampling Distribution: Properties
1. Normal in shape.
2. Has a mean equal to the population mean.
μx=μ
3. Has a standard deviation (standard error)
equal to the population standard deviation
divided by the square root of N.
σx= σ/√N
First Theorem

Tells us the shape of the sampling distribution
and defines its mean and standard deviation.

If we begin with a trait that is normally distributed
across a population (IQ, height) and take an
infinite number of equally sized random samples
from that population, the sampling distribution of
sample means will be normal.
Central Limit Theorem

For any trait or variable, even those that are not
normally distributed in the population, as sample
size grows larger, the sampling distribution of
sample means will become normal in shape.

The importance of the Central Limit Theorem is that
it removes the constraint of normality in the
population.
The Sampling Distribution

The Sampling Distribution is normal so we
can use Appendix A to find areas.

We do not know the value of the population
mean (μ) but the mean of the Sampling
Distribution is the same value as μ.

We do not know the value of the pop. SD (σ)
but the SD of the Sampling Distribution is
equal to σ divided by the square root of N.
Three Distributions
Shape
Central
Tendency
Sample
Varies
_
X
s
Sampling
Distribution
Normal
μx=μ
σx= σ/√N
OR s/√n-1
Population
Varies
μ
Dispersion
σ