http://mathforum.org/kb/plaintext.jspa?messageID=721741
Date: Nov 13, 1997 10:00 AM
Author: George W. Cobb
Subject: Re: Moore's outline of BPS
Dear Jayne:
Of course David Moore himself would be the best person to answer your
questions, but his approach is one I like, and so I'll try to explain why
I like that way of doing things.
At the same time, though, I want to
emphasize two other points that seem relevant:
I believe teaching and
learning always involves finding the right balance between
curricular goals and the interpersonal aspects of what goes on
in the classroom, and that you've offered us an important
example: sometimes its better to teach something you know well
and enjoy teaching than to try something new that you haven't
had time to become comfortable with. Moreover, the binomial can be a
good choice, for reasons I'll mention later.
As to why BPS does things the way it does, I think in terms of the
answers to a couple of questions:
In a course that seeks to
introduce statistical practice, (1) where does randomness come from,
and (2) how is probability actually used?
The short answer is: (1) the randomness comes from random samples
or randomized experiments, and (2) the single most important
probability idea needed for inference is the sampling distribution.
Here's a more elaborate version:
(1) The randomness comes from the use of a chance device in the
process of data production, by choosing random samples from
populations, or assigning treatments to subjects in a randomized
experiment. To oversimplify, but only a little, if your data don't
come from a randomized study, it is often difficult to justify any
role for probability, and so difficult to justify the relevance of
the usual methods of formal inference. In a slogan: "If you don't
randomize, don't generalize." Or, by way of a bad pun, "Unless you
randomize, your inference will never have a chance."
(2) How is probability actually used? When your data come from a
randomized process, you have a recipe for data production that
allows you (in principle) to repeat the data production process
2
again and again, which makes it possible to ask, "What would happen
if I were to repeat this many times?"
If "this" corresponds to
"produce the data and compute a summary statistic," then the
sampling distribution is the answer to the question "What would
happen if I were to produce-the-data-set- and-compute-the-summarystatistic many times?" The sampling distribution gives us a way to
relate observed data to hypotheses and possible parameter values, by
asking
how
likely
a
particular
observed
result
is.
Although
it
is
possible
to
base
inferences
directly
on
the
binomial
distribution, using the normal approximation has advantages:
it avoids
some complications, and it makes it easier for students to see inference
for proportions and inference for means as variations on the same theme.
So for the typical first course, what students need most from probability is to
understand what a sampling distribution is, and to be able to work with the normal
distribution, which is continuous.
What about the binomial?
Because most intro courses don't use it
for inference (except via the normal approximation), the binomial, like
combinations and permutations and formulas for P(AUB), is not something
that
gets
used
later
on,
and
not
something
that
is
essential
for
understanding the main ideas of statistical practice.
All the same,
though,
there's
lovely
mathematics
related
to
the
binomial,
and
a
lot
of
interesting
intellectual
history.
And
teaching
the
binomial
offers
an
especially
nice
chance
to
talk
about
modelling
real-world
situations:
Is xxx a binomial?
Is yyy?
In this context, I always
introduce a mnemonic I learned from Fred Mosteller when I was one of his
TAs for an intro stat course:
To recognize binomial situations in the
real world, check B.I.N.S.:
B
Binary outcomes
I
Independent trials
N
Number n of trials is fixed
S
Same probability of success on each trial.
Students
who
practice
checking
these
four
criteria
for
a
variety
of
situations get to practice thinking about the links between an abstract
model and the messier world we live in -- a kind of practice that
extremely valuable but often hard to provide.
George
George W. Cobb
Mount Holyoke College
South Hadley, MA
413-538-2401
01075
3
On Wed, 12 Nov 1997, Jayne Lutgen wrote:
> Date: Wed, 12 Nov 1997 14:54:51 -0800
> From: Jayne Lutgen markgreg@sdcoe.k12.ca.us
> To: "'apstat-l@etc.bc.ca'" apstat-l@etc.bc.ca
> Subject: Moore's outline of BPS
>
>
> I'm using Moore's BPS and have some questions as to his outline of Chapter 4.
> > Out of the countless stats books I've seen the usual presentation is of
probability distributions and then directly into the binomial distribution as a
special
case.
> > In Dr. Moore's book he introduces the general concept of a probability
distribution and discusses both discrete and continuous variables (most don't
approach the concept of continuous data just yet). Then the big suprise (at least
for me...)
He devotes the next section to sample proportions.
This seems like
such a departure from most other books. What do you think is his motivation for
this?
> > I've looked at other sources and can't seem to find any parallel presentation.
I need to confess I am brand new to this and right now my insecurity is at an all
time
low.
>
> Can anyone out there help me see THE BIG PICTURE?
I skipped the section on
Sample Proportions for now and taught Binomial stuff first cause I didn't want to
try to fake it too much today!!!
>
> Thanks for help!
>
> Jayne Lutgen
>