Al
Coons has an activity regarding this topic that is archived at this location.
Dan
Teague gives a nice explanation of the math involved for r^2 on this archived
post.
r^2
was discussed on the list on this date.
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Do
I have to teach log transformations?
Yes!
Why? (especially when my calculator can do it for me and has
all these fancy commands! Can't I have my students use those buttons?)
It's on the course description! And here's why:
The idea of transforming data to achieve linearity is a
powerful and important idea. It is this idea we are teaching.
Re-expressing data and dealing with it in it's transformed and linear state
is crucial. As is understanding how to back-transform to make an
appropriate prediction.
Dave
Bock discusses transformations for linearity on this archived post.
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What is the difference between
confounding and lurking variables?
Paul
Velleman gives a great post about confounding and lurking variables here.
The
list had a great discussion about confounding and lurking variables on Nov.
13th and 12th. Click here and also go back one day to see the full
discussion.
Josh
Zucker discusses the issue of extraneous variables and gives a list of links.
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Independence vs. mutually exclusive:
If two events are independent, the outcome of one will not
affect the outcome of the other. i.e., Whether or not it rains and
whether or not a coin flips heads or tails.
If two events are mutually exclusive, if one happens the
other event cannot happen. For example, in picking one M&M from a
bag, I can find the probability of drawing green or red. But if I draw
green, I cannot draw red.
As we saw on the '02 MC #23, it is useful to notice that if
two events are mutually exclusive, they affect each other quite
powerfully: if one of them happens, the other CANNOT occur. Thus
they are dependent.
Independence
vs. mutually exclusive has been discussed on the list.
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How much probability do I teach?
'Floyd
Bullard has submitted an awesome post about probability that can be read in
the archives. I would strongly encourage rookies to read this post BEFORE teaching
probability!
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Why doesn't X + X = 2X?
Here's a great explanation from Dave Bock:
For a short answer, try a thought experiment.
Let X represent the outcome when you roll a die. the 2X represents
rolling one die and doubling the result. The possible outcomes are {2,
4, 6, 8, 10, 12}; they are equiprobable.
On the other hand, X+X represents rolling two dice (or one twice). Now
the possible outcomes are {2, 3, 4, 5, 6, 7, ..., 12}. Some are far
less likely than others. Clearly this is a very different situation.
You can actually calculate both variances, but first just think about
the distributions. It should be pretty obvious that X+X is unimodal and
symmetric, peaking around 7 with very low tails while 2X is uniform
across the same range. The two means are the same, but X+X has a
smaller variance than 2X.
When confronting these situations, students must learn to ask
themselves how many random values they are working with. One random
value multiplied by a constant behaves much differently from summing
several different random values.
I urge students to recognize that a random variable in Statistics is
not the same animal as a variable in algebra. In algebra what we call a
"variable" is really just an unspecified constant. With that
understand, no matter what number I use for X I'll always substitute
that same value every time I see an X, so it must be true that X+X+X =
3X.
Then I put my Statistics hat on, declare them "random variables", and
pick up a die. I substitute the results of the first roll for the first
X, roll again for the value of the second X, etc. It's pretty clear now
that this X+X+X = 3X equation that seems so obvious in algebra is false
for random variables in Statistics. (One time the four values I
randomly rolled actually worked! The kids thought that was hilarious.
Their laughter at my bad luck clearly showed they understood the
issue.)
Read
a list discussion thread about adding random variables.
Pete
Flannigan-Hyde as written an article for AP Central about adding random
variable.
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I'm behind! Help!
 |
Note that once you introduce inference, you can teach the
last part of the year very quickly! Especially inference for slope,
which is on the AP test. |
|
For inference for slope, focusing on interpreting the
computer output can save time. |
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Not getting into all the nitty-gritty details about
homogeniety and independence can save time. |
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Following the pacing guide that comes with the textbooks,
can help avoid this problem to begin with, but if you're reading this, it
may be too late! :o) |
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Starting cumulative review while finishing inference can
eliminate the need for lots of days of review. |
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Reviewing regression while teaching inference for slope is a
natural and helpful step for preparing for the exam. |
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How much work to show?
The short answer is that most list contributors recommend
that students show formulas. Both with just variables and then with
the numbers plugged in. It shows that the student understands what is
going on and it eliminates the concern that students would lose points if
they accidentally plugged something into their calculator incorrectly.
A note about the t* for t-intervals. If a student uses
technology for certain procedures (e.g., 1-sample with n = 167 or any
2-sample interval), the t* will not be on the table. It is OK to leave
the formula with all the numbers plugged in and the t* just stays as a
variable. OR a student can use a conservative approach that uses a t*
that is on the table, but then they need to calculate their interval by hand
so their answer matches the df they used.
If students and/or teacher really want to find the t*, they
can use the inverse t function. If students have an 83, they need a
t-inverse program. This program is legal (because it just matches the
84) and can be made by typing this simple little program:
Prompt N
Prompt A
solve(tcdf(X,1E99,N-1)-A,X,1)-
->K
Disp K
A few other points about this:
|
For hypothesis tests and confidence intervals, the AP
rubrics have (thus far!) required name OR formula. So students can
get full credit without the formula. |
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Numerous multiple problems on the '02 exam require formula
understanding:
 |
#8--1 sample t-interval |
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#11--Chi-Sq expected |
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#21--Confidence interval for slope |
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#32--Binomial and geometric formulas |
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#38--Binomial and 1-prop z formulas |
|
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TI-talk is discouraged. Statements like:
normalcdf (1.2, 9999) are just not good communication. While showing
a total by-hand formula is not required, good communication is. For
example, on a binomial problem, students could write:
|
Binomial |
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n = 6 |
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p = 0.87 |
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P(x = 4) = ----- (from calculator) |
|
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It has been frequently recommended on this list that
students show z-score calculations and don't use technology to shortcut
that step! |