Quadratic Regression

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Quadratic Regression: Using a calculator to perform a quadratic regression

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I have some data here.
It says a golf ball is hit down a fairway.
The following table shows the height of the ball with
respect to time.
The ball is hit at an angle of 70 degrees with the horizontal
with a speed of 40 meters per second.
And then they give us a bunch of, essentially data samples.
At time 0, we're at 0 meters.
At time 0.5 seconds, we're at 17.2 meters.
So they give us a bunch of data points.
And what I want to do is use these data, or use these data
points, to essentially find a quadratic function that fits
it best. Or I guess to view it the other way, I want to model
this data or I want to model this phenomenon, I want to
model the golf ball being hit down the fairway, or the
height of the golf ball using a quadratic equation, or
quadratic function.
And we're going to do it using a graphing calculator.
And actually, you know, if you do a little bit of physics,
you can figure out what the actual equation
would be in a vacuum.
But this is actual data taken from a golf ball hit in air.
And that changes things a bunch, it makes the physics
really complicated.
So we'll actually look at the measurements and then try to
figure out what the actual function is.
And actually, the calculator is going to do most of the
work for us.
I won't go into the details of what the
calculators algorithm is.
I just want to show you how to do it.
It's safe to say that this is going to be well approximated
by some type of quadratic function or a parabola.
So let's try to fit a function to it.
So what we want to do is we want to hit stat on our
calculator.
You hit stat, then you hit edit.
That's what we use to input our information.
We can click enter twice.
This just says these are the two variable names for where
we input our data.
I'm just hitting enter twice.
And now I can start entering the data.
And I'll use my keyboard to do this.
So the first data point is 0.
Enter.
The x is 0, y is 0.
Press enter again.
The next data point, 0.5 and 17.2.
I'm just looking at this right here, 0.5 and 17.2.
Next data point.
We have x is 1.5.
When x is 1.5, y is 42.9.
When x is 2, y is 51.6.
When x is 2.5, y is 57.7.
When x is 3, y is 61.2.
I just keep pressing enter every time.
When x is 3.5, y is 62.3 meters in the air.
When x is 4, y is 61 meters in the air.
And when x is 4.5, y is 57.2.
So I've entered all of the data.
And then the next thing I want to do, I'm going
to plot all of it.
But before I plot all of this data, I want to make sure that
I have the proper range on my display.
So I'm going to go to graph, hit range.
And see.
My x minimum, I want it to be 0.
That's the lowest value here.
And my x maximum, I could set it actually,
a little bit lower.
Well, I set my x maximum, that's fine.
I'll go up to 7 seconds.
I'll make the scale, that's the measurements on the
x-axis, as 1.
y minimum is 0.
y maximum, 70 looks pretty good.
It seems like it'll cover all of these numbers, and some of
these get pretty close.
They're in the 60's, so 70's pretty good.
And I'll make the y scale 5.
I don't have to make any changes here, but you could
make changes if your calculator
isn't set to this already.
And so we can go back to stat.
And then what we want to do is have the calculator calculate
a quadratic function that matches these points as well
as possible.
So we'll go to calc.
I'm assuming that means for calculate.
This is saying, where is the data?
It's the same variables we used to input the data, so
I'll just click enter.
And enter twice.
And now here are the different types of
regressions we can do.
We can do a linear regression.
We can do an exponential regression, power regression.
You click more and you'll see a power 2 regression.
This means a second degree regression.
So here, we can literally click-- we
just select this option.
That means a quadratic or this would be a third degree
regression, a fourth degree regression.
So we're just going to do a quadratic regression.
And it figured out the coefficients of the quadratic
that best matches this data.
So I'm going to write it down.
So it's negative 5.2.
It's negative 5.2.
Scroll over a bit.
35.99999.
And then 0.292514.
So let me write these down over here.
So let me take it off the screen so I can
keep looking at it.
And so we have-- it's telling us, the calculator has told
us, that if the quadratic is equal to ax squared plus bx
plus c, it's telling us that the first coefficient, a, is
negative 5.20128.
I'll probably round that off a little bit.
It's telling us b is equal to 35.9-- let me scroll over my
calculator a bit-- 993.
I'll probably round these.
9934.
And it's telling my c values that it calculates for the
best fit quadratic is 0.2925.
Which is interesting.
Before I did this problem, I actually figured out what this
quadratic would be in a vacuum.
I just used kind of some of the physics principles.
And then you have this number in a vacuum.
If you didn't have air resistance, this number would
be negative 4.9.
This number I think turned out to be something like 37.
And this number turned out to-- well, this was 0, because
you start at the ground.
But this is interesting.
This is actual data and actual data is always going to be a
little bit different, or sometimes a lot different than
the theoretical ideal, if you had no air resistance.
So that's what we're getting right here.
And so we can now use this.
This is telling me, my graphing calculator's telling
me, that the quadratic I should use-- you know, y is a
function of x-- is negative 5.2x squared
plus 35.99x plus 0.29.
I rounded them off a little bit.
So now we can use this information, or maybe I
should say of t.
I could have made this-- well, x is equal to time.
I could have made it t instead, but I
think you get the idea.
And now we can use this information to figure out what
the ball was probably doing at other points in time.
So, for example, if we wanted to know what was
happening at time 5.
What's y of, let's say 5.2 seconds into it?
So what happens a little bit further off?
We're extrapolating, we're going beyond what the data
that we have right here.
So y of 5.2.
We can do it two ways.
We can look at it graphically.
So now that we've calculated this regression, here are the
coefficients again.
As you see here, I'm just scrolling to the left.
It gave me the coefficients.
Negative 5.2, 35.99, and then it gave me the 0.2925.
So now that it calculated, we could actually plot all the
points and draw the regression.
I want to pick this draw option up here.
So I can either exit and then I pick draw.
And first I'll just draw the data.
So that's scattered.
It'll make a scattered plot of all of the data.
I select it there.
And notice, it drew the data points as they were given.
That's the data points right there.
5.2 is going to be out here some place.
And now I can draw the actual regression.
This is draw the regression using these
coefficients right here.
So draw regression.
And you see it actually fits the data very,
very, very, very well.
So let's use this function that seems to fit the data
very, very, very well to actually come up with how high
was the ball at 5.2 seconds?
So let's exit from here.
And let's just calculate it.
It is negative 5.2 times 5.2 squared.
Just by coincidence, I picked a time that equals that first
coefficient.
Therefore, it's the positive version of it.
So 5.2 squared plus 35.99 times 5.2-- remember, we're
dealing with when time is equal to 5.2, or where x is
equal to 5.2-- plus 0.29.
So we're just going to evaluate this function at time
or at x is equal to 5.2.
And we get 46.83 feet.
So it's equal to 46.83, or I should say 0.83 meters.
So I hope you enjoyed that.
I just wanted to show you a quick thing of how a graphing
calculator is useful.
Maybe in the future, I'll make a more advanced video on the
actual algorithm or the actual process that the calculator
used to figure out this best curve.

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