HELLO!
I'm Mr. Tarrou.
Ok.
Part 2 of Log Transformations.
We are going to talk about Power.
We just got done showing how if we take the y values, we have a curved scatter plot, if
we take the logged y values and then see that our transformed data is straight then the
original data was exponential.
And please go back and look at that video if you need to.
But now we are going to take part 2 which is looking at... or try to transform curved
data whether it curves up or down Power Rule.
So if it is exponential, if your growth or decay is exponential all you have to do log
the y's to make that scatter plot straight to make another scatter plot of the transformed
data to make it straight.
Now we are going to talk about the Power Model.
So if logging the y does not help make the scatter approximately linear, then you are
also going to have to log the x's.
Now all of this is going to be as if the Power Model actually fits.
Of you log the y's and do not get a linear pattern, then it is not exponential.
If you then also log the x's, so now both the x's and the y's are both logged, if your
scatter plot is still not straight then your original data was not exponential nor was
it power and well we are just stuck.
Whatever that original data, whatever that curved scatter plot is, you are going to be
just stuck talking about that curved pattern.
Talking about the form, direction, and strength and any outliers and just describing what
you see.
But that is going to be it.
You are not going to be able to do any math analysis to it.
Ok.
So let's see what that looks like.
This will not be a real long video.
We have data.
It looks curved.
Again I have it curved up for some reason like the exponential was, but it can curve
down... this way...
Whatever.
It is not straight.
Yes! ok.
So if the problem immediately tells you to check that is a Power Model or find the power
model, then you are going to go straight away and log the x and the y variables.
And hopefully see that your transformed scatter plot is then approximately linear.
I will be talking again how to check for that linearity besides it just looks like it.
If you log the x's the log the y's and you get a linear pattern, then your equation...
your least squares regression line, don't forget when you write it to put log next to
the x and y variables.
Now the calculator is not going to say the log of y hat is equal to a plus b times log
of x.
It is just going to give you the equation like it always does, y=a+bx.
'a' equals blank, 'b' equals blank, 'r' equals blank.
excuse me... and r^2 is blank.
Blank...
blankity blank blank!
But when you write your transformation, or when you write your regression line down after
the transformation please remember to put log next to the x and y variables if indeed
that is what you did.
We will never just, or at least in my class, we will never just be logging the x's.
It is always log the y's and then if it is straight it is exponential, or log both and
then if that works it is power.
If logging x and y makes a linear scatter then the original data is POWER!
Ok, whatever.
[nanananana] Ok.
The second and last screen of this video.
Now my notes are in form of working with TI-83 and TI-84.
When I do my calculator lab which will probably be posted tomorrow if you are watching these
as I get them out to you.
I will show you a lab with a TI-NSPIRE.
I have a new color one so it is very exciting.
And a TI-84.
So you will see how to use both of them.
These notes are in terms of TI-83 and TI-84 use because that is what my students are using
this year before I hopefully get them to move onto INSPIRES next year.
And these notes are as if you are just given data and you are asked to analyze it.
You are not told ahead of time it is exponential and you are not told ahead of time it is power.
You are just given some x and y values, remember the explanatory variable goes on the x axis
and the response variable goes on the y.
And we are just looking at it.
So you put your x's and y's in list 1 and list 2.
Maybe the scatter plot is curved.
You can't work with curved data so we are going to attempt to make it straight.
Well if you have no idea if it is Power or Exponential, the first thing you will do is
log the y's.
And I have got the notation you would use in your TI-83 or TI-84 up here above it.
And you will have the x's, the y's, log the y's and then make the scatter plot.
If it looks like it is straight make a regression line and check with a residual plot that it
is actually straight.
And maybe as soon as you make a scatter plot it is obviously not straight.
Or maybe you have to go as far as making that residual plot before you see that the transformed
data is still curved.
At any rate we already covered exponential growth and decay.
So whatever the case is, whether it is obviously curved or it looks curved after you make the
residual plot, we are going to go back to our data list in the calculator and then log
the x's because logging the y's did not give us a linear pattern.
And again, this is your trial and error idea.
So you have your x's, your y's, you logged the y's to make it straight and it did not
work, and now you are logging the x's to make a straight linear pattern... approximately
straight.
That will be L4 equals the log of L1.
So these are the log of your x's.
And you make the scatter plot again.
Now it looks like it is straight and you made the regression line again, and you made the
residual plot again.
It is very monotonous with a TI-83 and TI-84.
It is almost happens instantaneously with an NSPIRE which you will see in the calculator
lab.
But, you check your residuals and this is the process here.
You are going to make your...
To make a residual plot to verify that your transformed data is indeed linear, you are
going to have to make the scatter plot.
That will be with L4 as the x axis and L3 as the y axis.
Make your regression line, again with L4... if you are following this pattern...
L4 as your x and L3 as your y values.
And then once you make your scatter plot and once you make your regression line, your calculator
is going to make all of your residuals for you.
You are going to, with an 83 or 84, go to the list menu... go find where it says RESIDUAL
and store those residuals into a list.
And if you are starting this from scratch, your empty list will be list five.
Once you make and redo your...
Well you are going to make another scatter plot which will be a residual plot.
Your logged x's will be on the x axis and your residuals will be on the y axis.
You want your residual plot to have a nice horizontal band of points with no clear curvature
to it.
If you have that then your transformed data was indeed linear.
Now remember, log the y's and get a linear pattern the original data was exponential.
Log the x and the y's and get a linear pattern, your original data was then the power model.
Now we have done all of this, we have checked that our transformed data was linear, and
you get this and I have done this in general form.
You have got the log of y hat equals 'a' plus 'b' times the log of x.
This model is of course fitting the transformed data.
Like I talked about in the last video, this is... some teachers teach this, some teachers
don't.
But if we are statisticians and we are analyzing data, most people don't understand logarithms.
You might be struggling with them as well, so clearly you cannot go out and tell someone
these are the results and have them all in terms of logged values.
So how do you take this equation and manipulate it so that it actually models the original
data.
You know that original data that was curved.
You need to get the logs out of that function and you do that by making both sides of the
equation an exponent of ten.
And as I talk about in the last video that power of 10 is going to undo the common log
function.
You can go back and watch that video's end if you like.
But the base ten and log base ten, if we don't write that it is assumed to be base ten, if
they are stacked they will cancel out.
So immediately after making both sides an exponent of 10, the log function cancels out
and you get y hat... equals...
Now it is 10 to the a+b*log(x) power.
So the exponent has two things being added together.
When do you add exponents?
When you are multiplying like bases.
So 10 the a times 10 to the b log of x.
Great.
But this is still not going to cancel out because 10 and log base 10 are not immediately
stacked on top of each other.
So what do you do?
Use one of your properties of logarithms that allows you to take your leading coefficient
and float it up as an exponent.
That is right.
So now you have got y hat equals 10^a times 10 to the log of x to the b power.
So it is a base with an exponent, and that exponent has an exponent.
Well not, but floating the b up, the log function with a base of ten is stacked on a base of
ten.
They are going to cancel out.
Badda Bing, Badda Boom! y hat equals 10^a .... excuse me... 10^a times x^b.
Now this is not a form that you would see in your book.
If you are actually working with your calculator you will know what 'a' is and this will actually
be a decimal of some form.
'b' will be some kind of decimal, probably a decimal exponent.
But this is the model, the Power Model.
See how the base is x.
I am going to repeat that on this side of the notes here.
Our exponent, excuse me... our base is now our variable of x.
Unlike the exponential functions where the x was in the exponent.
And this is modeling or original data.
Our original data followed the Power Model.
So you can go back to the original graph, the original list one and list two... put
this in the Y sub 1 and it should follow that original curved data fairly well.
So, again residual plots will check for linearity.
You will be seeing a calculator lab about that shortly.
Again, this is possibly what the power model will look like in your textbook.
It might use different variables.
But again you have an initial value which is c... times a base which is your variable...
and the exponent now is fixed unlike with the exponential model which is where the exponent
was changing and the base was fixed.
And that is the Power Model and that is the end of my lecture.
BAM!
Go do your homework!
I will say it again.
BAM!
[haha]