Predicting with Linear Models

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Predicting with Linear Models

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Let's do a couple of problems where we estimate or predict
answers to questions using linear models.
It sounds very fancy.
But what a linear model is is just using a line to describe
some trend in data.
With our linear models we will do a little bit of
interpolation and a little bit of extrapolation.
You might have already heard these words before in your
everyday life.
Interpolation means trying to estimate what happened between
two data points.
We're going to do a couple of examples of
that in this video.
Extrapolation means we'll see what the last few data points
were, and see what that trend would look like, and then keep
on continuing that trend and see what might happen if that
trend were to continue.
I'll show you some examples of that.
So let's do some example questions here.
So I have this chart here.
Median age, remember median is the middle.
So if I have the numbers 3, 7, and 9, the median is the 7.
It's the middle age.
So this is the median age of males and females at first
marriage by year.
So if we just look at this data right here, if we look at
1900, the average man was getting married, not the
average, the median male.
So the middle man was getting married at looks like at
around the age a little over 25, almost 26, while the
median woman was getting married looks at around 22 or
23 years old.
As we went through the century, those ages got lower,
and lower, and lower, until we got to about 1960.
This problem is interesting irrespective
of the actual problem.
It's just interesting to see this trend that at the
beginning of the century, people got married reasonably
similar to when they're getting married right now.
But there was a minimum point.
Around the 1960s, people got married at a younger age.
This was true for males and females.
You can also see back here the age difference between the
median male and the median female.
That's just gotten smaller and smaller over time.
Now it's really small.
So these are just all interesting
things from this chart.
This is the actual data from that chart.
So actually I didn't even have to estimate it here.
In 1900, the median age of males getting married was
25.9, that's that right there, and the median age of women
getting married was 21.9, that's that over there.
So this is just a scatter plot of this data.
We've just plotted each of these points over here, these
in blue, these in red.
Let's do the questions.
Use the data from example 1-- this is example 1 up here-- to
estimate the age at marriage for females in 1946.
Fit a line by hand to the data before 1970.
So what they want to do, is they want me to look at all of
the data before 1970.
So let's see, this is 1970 right here.
That's 1970 right there.
They want to fit a line by hand to the data before 1970.
What they mean is-- and I'm just going to eyeball -- but
they want me to draw a line that gets as close to all of
this data before 1970 as possible.
So it can't go through all of these points.
Because all these points don't sit exactly on one line.
But let me try to draw a line, or a linear model, for the
data from, what is this, 1890 all the way to 1970.
So I'm going to try my best to draw a line here that gets
close to all of these points.
It won't be able to go through all of them.
So all the way to 1970, maybe a line would look
something like that.
Let me draw a better one than that.
I'll draw it in red because we're dealing
with the red data.
So I want to go all the way to 1970.
It would have been nice to have a line drawing tool.
Maybe the line might look something like that.
There are actual mathematical ways to figure out the best
fit line on this.
We won't go into that right now.
That's called linear regression.
But I just eyeballed it.
That line looks like a pretty good fit for all of that data.
None of the data is too far away from that line.
So that's what they mean by fit a line by hand to the data
before 1970.
This is the data before 1970.
If we assume that this is a good linear model for that
data, we can use it to estimate the age at marriage
for females in 1946.
Let's see, this is 1950.
This would be 1945.
1945 would be right over here.
So 1946 would be right over there.
So if we wanted to estimate that, if I were to figure out
where that hits the vertical axis, it seems to hit the
vertical axis at around-- I don't know, my best to guess
is, I should have zoomed in more-- it hits
it right about there.
This is 22.5.
So this looks like maybe 20.5.
Because this would be maybe 20.5 years.
So I'll call it twenty 20.5 years.
Obviously I can't do it exact, but you get the idea.
I am interpolating.
Here I'm using the model to do.
I won't use necessarily the word interpolation yet.
We're going to do a much more direct form of interpolation
later on in this video.
But here I fit a line, and then I just use that line as a
model to estimate what was the median age of women in 1946.
Let's do part 2.
Use the data from example 1 to estimate the age of marriage
for females in 1984.
Fit a line by hand to the data from 1970 on in order to
estimate this accurately.
So now they want us to fit a line from the data on.
So let me make a line.
So fit a line by hand from the data 1970 on.
So let me draw a line that gets close to all of these
points, that gets close to describing
all of these points.
Maybe it looks like something like that.
Not all of them sit exactly on the line, but they're all
pretty close.
If we assume that this is a good a linear model-- I
haven't done this mathematically, I'm eyeballing
it-- but let's see what it would be in 1984.
This is 1990.
So 1984 is going to be right around there.
Actually it looks amazingly similar to the median age of
men in 1970.
So we can actually get the exact number.
It's about 23.2.
So this looks to be about 23.2 years.
Part 3, use the data from example 1 to estimate the age
at marriage for males in 1995.
Use linear interpolation.
Now we're going to do more direct form of interpolation.
We're not going to use a linear model or we're not
going to fit a line.
We're going to use linear interpolation between 1990 and
2000 data points.
We're looking at men now.
So I'll do it in blue.
Between 1990, that is 1990 and that is 2000 for men.
So if I were to draw just a line there, we could assume
that the trend would have looked something like that
between 1990 and 2000.
They want us to interpolate what happened in 1995.
So 1995 is that point right there.
So this is pure interpolation.
We're using the data point in 1990 and the
data point in 2000.
We're drawing a line between them and assuming that 1995
would have been right in between them.
We're interpolating between the 1990 data
and the 2000 data.
So if you were to try to estimate where that is, it
looks like it's a little bit over 26.
Actually if we say that it's right in between these two
values right there, it'll be 26.45 years.
If it's exactly in between, 1995 is exactly
5 years from each.
So 26.45 looks pretty good.
So let's do another example So here's a percentage of women
smokers by year.
I have to say these charts are interesting in and of
themselves.
You'll notice in 1990 there were a good number of women
who were pregnant who smoked.
Almost 18% or 19% of pregnant women smoked in 1990.
Now that's down.
Well this is 2004 so hopefully this trend is continuing to go
down eventually to 0%.
But it went down to 10%.
So there still are a percentage of pregnant women
who unfortunately smoke.
Let's do these problems. Use the data from example 2--
that's this right here-- to estimate the percentage of
pregnant smokers in 1997.
Use linear interpolation between the
1996 and 2000 data.
We're going to use linear interpolation.
So let me draw a line.
You could always do this much more exact if you have better
tools or a spreadsheet.
But I'm going to draw a line, and I want to get 1997.
This is 1997.
It's going to be right over there.
If we use linear interpolation, we are
interpolating between that data point
and that data point.
It's a linear interpolation.
We assume It's a linear trend between the two.
So this looks like it's-- let me go all the way over here--
looks a little bit more than 12, maybe
13% is my best estimate.
Now part 5.
Use the data from example 2-- same example right here-- to
estimate the percentage of pregnant smokers in 2006.
There isn't any 2006.
This chart ends at 2004.
So we have to use linear extrapolation with the final
two data points.
So this here you're going to see the difference between
interpolation and extrapolation.
Interpolation, you are estimating between two data
points you already knew.
Extrapolation, we're going to take the final two data points
and we're going to continue that line.
We're going to extrapolate that line.
So if you were to just continue this line between the
final two data points, it might look
something like that.
So if you had 2006 out here-- these are going up by two, so
2004, 2006-- we extrapolate this line.
The percentage of women smoking might be, well it
looks a little bit under 10%.
So maybe 9.5%.
Once again, it's just an estimate.
But we're just extrapolating the trend.
We're assuming the trend of the last
two data points continues.
Then using that, it would be around 9.5% in 2006.
Now maybe, that's not definitely the only prediction
you could make.
You could extrapolate based on all of the data and get a line
that looks something-- let me try to do it a little bit
better than that-- that looks something like that.
Then this line would predict a lower smoking rate.
So it depends on what data you use, what you're using to do
your linear model with, and how you're extrapolating.
But anyway, hopefully you found this useful.