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## Estimating lines of best fit

# Line of best fit: smoking in 1945

CCSS.Math: , ,

## Video transcript

The graph below shows the
percentage of American adults who smoke over time. Assuming the trend
shown in the data has been consistent
since 1945, use the graph to estimate the percentage
of American adults who smoked in 1945. So let's see what's
going on here. The horizontal axis here,
they say years since 1965. So at this point
right over here, this is 0 years since 1965. So this really represents 1965. And we see it
looks like around-- let's see, if I
were to eyeball it, it looks like it's around 42%
of Americans, just looking at this graph. I know that's not
an exact number. Roughly 41% or 42% of
Americans smoked in 1965 based on this graph. And then five years
later, this would be 1970. 10 years later,
that would be 1975. And they don't sample
the data, or we don't have data from
every given year. This is just from some of the
years that we happen to have. But what is clear,
it looks like we have a negative linear
relationship right over here, that it would not be
difficult to fit a line. So let me try to do that. So I'm just going to
eyeball it and try to fit a line to this data. So our line might look
something like that. So it looks like a pretty strong
negative linear relationship. When I say it's a negative
linear relationship, we see that as time
increases, the percentage of smokers in the
US is decreasing. So that's what makes it
a negative relationship. Now, what are they asking? They want to estimate the
percentage of American adults who smoked in 1945. Well, 1945 would be
to the left of 0. So we could even
think of it as if 1945 is 20 years before 1965. So let me see if
I can draw that. So 20 years before 1965. Let's see, this would be 5
years before 1965, 10 years, 15 years, 20 years before 1965. So I could even put that as
negative 20 right over here. Negative 20 years since 1965
you could view as 20 years before 1965. So that would represent
1945 right over there. And one thing that we
could do is very roughly just try to extend this negative
linear relationship backwards. And they allow us
to do that by saying assuming the trend shown on
the data has been consistent. So the trend has
been consistent. This line represents the trend. So let's just keep
going backwards, keep going backwards at the same
rate, so something like that. I want to make
sure that it looks like it's the same
rate right over here. And you could just
try to eyeball it. You could say, well, let's
see, 20 years ago, 1945. If I were to extend
that line backwards, it looks like there
were about 52% of the population was smoking. It seems like we're about
52% right over here. Another way to
think about it would be to actually try to
calculate the rate of decline. And let's say we do it
over every 20 years, because that will be
useful because we're going 20 years back. So if we go 20 years from
this point, so this is 1965, you go 20 years in the future. So that is 10 years, and
then that is 20 years. So my change in the
horizontal is 20 years. What's the change
in the vertical? Well, it looks like we have a
decrease of a little bit more than 10%. It looks like it's
11% or 12% decrease. So I'll just say
minus 11% right there. And let's see if
that's consistent. If we were to go
another 20 years. So if we go another 20 years,
it looks like once again we've gone down by about 10%. So that looks like roughly 10%. If we're following the
line, it should actually be the same number. So let me write it this way. It's approximately down 10%. So that little
squiggly line, I'm just saying approximately
negative 10% every 20 years. So if you go back 20 years, you
should increase your percentage by 20%. So this should go up by-- or
you should increase your percent by 10%, I should say. So if we started at
41% or 42%, once again, this was what we saw when
we just eyeballed it, you should get to 51% or 52%. So my estimate of the
percentage of American adults who smoked in 1945
would be 51% or 52%.