Download INTRODUCTION - University of Western Ontario

Chapter 4 (cont.)
The Sampling Distribution
What is the sampling distribution?



The sampling distribution is a theoretical
probability distribution that allows us to
determine the probabilities of possible values
associated with a sample statistic such as a
sample mean or a sample proportion.
Every sample statistic has its own sampling
distribution.
Works on the same principles as the normal
probability distribution.
Logic And Terminology

Problem:
The populations we wish to study are almost
always so large that we are unable to gather
information from every case.

Solution:
We choose a random sample -- a carefully
chosen subset of the population – and use
information gathered from the cases in the sample
to generalize to the population.
Remember:




Statistics are mathematical
characteristics of samples.
Parameters are
mathematical characteristics
of populations.
Statistics are used to
estimate parameters.
The Sampling Distribution is
the link between a statistic
and a parameter.
PARAMETER
STATISTIC
The Sampling Distribution




We can use the sampling distribution to
calculate our population parameter based on
our sample statistic.
The single most important concept in
inferential statistics.
It is the distribution of a statistic for all
possible samples of a given size (N).
The sampling distribution is a theoretical
concept based on the principles of the normal
curve.
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 of Y
1. Normal in shape.
2. Has a mean equal to the population mean.
Y  
3. Has a standard deviation (standard error) equal to
the population standard deviation divided by the
square root of N.
Y   / n
Sampling Distribution of a Proportion

Mean of the distribution:

S.E.:

Note that as N increases,
S.E. decreases
Y   / n
  y( y )
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.

When n>30, the distribution is approximately normal.
The Sampling Distribution Summary

The Sampling Distribution is always normal so we can
use Table A (z-table) or 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 (pop. mean).

We do not know the value of the pop. standard
deviation (σ) but the Stnd. Dev. of the S.D. is equal to σ
divided by the square root of N.

We can substitute s for σ
Three Distributions
Shape
Central
Tendency
Sample
Varies
_
X
s
Sampling
Distribution
Normal
μy=μ
σy= σ/√N
Population
Varies
μ
Dispersion
σ