C:\Documents and Settings\AYounger\My Documents\Teaching\Math Stats\Lecture Notes\Week 7 Central
Limit Theorem.docx
1
Week 7
Sampling Distributions & the Central Limit Theorem (WMS Ch 7)
1
INTRODUCTION
Chapter 5 was an important turning point.
Chapter 2 introduced the ideas of random events and of a
mathematical “probability measure”.
Chapters 3 through 5 investigated a wide variety of
“theoretical distributions” appropriate to different sorts of random experiment.
Starting with Chapter 7, we now begin to focus more closely on “statistics”.
These are various
functions of the observed values of random variables found in our sample data.
Eventually, we use
them to infer certain things about the “population” from which they were drawn.
So, from here on, you should feel that there is a more obvious “practical” side to the theory than may
have been obvious hitherto.
There is also a good deal more opportunity to carry out simulations and
analyses using R (or whatever other favourite software you may prefer).
On the other hand, we now
must be very careful to understand the differences between sample and population distributions.
Chapter 7 is also important because it introduces the Central Limit Theorem – an idea that is central
to applied statistics and econometrics as you go forward.
We have skipped Chapter 6 in the interest of time.
But in the following notes you will find
explanations of a few things from Chapter 6 that we really need.
2
SAMPLING AND POPULATION DISTRIBUTIONS
The basic situation is illustrated:
The basic idea:
•
Draw a random sample of n
observations from the
population
•
Use those observations to
calculate an estimator
for the
population statistic
•
Our concern is with the error
of the estimate
•
If we can find the probability
distribution of the estimator
,
we can calculate the
probability of error
Key provisos:
•
Sampling is random
•
Pop N large relative to sample n
•
We treat the observations as independent and identically distributed (iid) random variables.
Population
Mean:
µ
St Dev:
σ
Sample
Mean:
Y
μ
St Dev:
Y
σ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentC:\Documents and Settings\AYounger\My Documents\Teaching\Math Stats\Lecture Notes\Week 7 Central
Limit Theorem.docx
2
(This seems like a shift in terminology.
Consider the example of height.
Hitherto, we would have
said that “height” is an RV defined on a sample space of university students, for example.
Now we
want to talk of each observed height y
i
is itself an RV Y
i
.
I’m not completely convinced that this
consistent with what we have done up until now…but let’s just accept it for now and proceed…)
3
APPLICATION TO A SAMPLING MEAN
Continue with the example of the height of university students.
In principle, there is a “true”
population mean
1
µ
that we could calculate with a 100% coverage survey.
But instead, for reasons
of time and cost, we decide just to examine a sample of n Torontoarea
2
students.
We therefore have
a set of observations {y
1
, y
2
…y
n
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '10
 arthuryounger
 Normal Distribution, Variance, Probability theory, tai yi, Settings\AYounger\My Documents\Teaching\Math Stats\Lecture, Documents\Teaching\Math Stats\Lecture Notes\Week

Click to edit the document details