- [Instructor] The scatterplot below displays a set of
bivariate data along with its least-squares regression line.
Consider removing the outlier 95 comma one.
So 95 comma one, we're talking about that outlier
right over there.
And calculating a new least-squares regression line.
What effects would removing the outlier have?
Choose all answers that apply.
Like always, pause this video
and see if you could figure it out.
Well let's see, even with this outlier here,
we have an upward sloping regression line.
And so, it looks like our r already
is going to be greater than zero.
And of course, it's going to be less than one.
So our r is going to be greater than zero and less than one.
We know it's not going to be equal one because then
we would go perfectly through all of the dots
and it's clear that this point right over here
is indeed an outlier.
The residual between this point and the line is quite high.
We have a pretty big distance right over here.
It would be a negative residual and so,
this point is definitely bringing down the r
and it's definitely bringing down the slope
of the regression line.
If we were to remove this point, we're more likely
to have a line that looks something like this,
in which case, it looks like we would get a much,
a much much much better fit.
The only reason why the line isn't doing that
is it's trying to get close to this point right over here.
So if we remove this outlier, our r would increase.
So, r would increase
and also the slope of our line would increase.
And slope would increase.
We'd have a better fit to this positively correlated data
and we would no longer have this point dragging
the slope down anymore.
So let's see which choices apply.
The coefficient of determination r squared would increase.
Well if r would increase, then squaring that value
would increase as well.
So I will circle that.
The coefficient, the correlation coefficient r
would get close to zero.
No, in fact, it would get closer to one
because we would have a better fit here.
And so, I will rule that out.
The slope of the least-squares regression line
would increase.
Yes, indeed.
This point, this outlier's pulling it down.
If you take it out, it'll allow the slope to increase.
So I will circle that as well.
Let's do another example.
The scatterplot below displays a set of bivariate data
along with its least-squares regression line.
Same idea.
Consider removing the outlier ten comma negative 18,
so we're talking about that point there,
and calculating a new least-squares regression line.
So what would happen this time?
So as is without removing this outlier,
we have a negative slope for the regression line,
so we're dealing with a negative r.
So we already know that negative one is less than r
which is less than zero without even removing the outlier.
We know it's not going to be negative one.
If it was negative, if r was exactly negative one,
then it would be in downward-sloping line
that went exactly through all of the points.
But if we remove this point, what's going to happen?
Well, this least-squares regression is being
pulled down here by this outlier.
So if you remove this point,
the least-squares regression line could move up
on the left-hand side and so you'll probably
have a line that looks more like that.
And I'm just hand drawing it.
But even what I hand drew looks like a better fit
for the leftover points.
And so, clearly the new line that I drew after removing
the outlier, this has a more negative slope.
So removing the outlier would decrease r,
r would get closer to negative one, it would be
closer to being a perfect negative correlation.
And also, it would decrease the slope.
Decrease the slope.
Which choices match that?
The coefficient of determination r squared would decrease.
So let's be very careful.
R was already negative.
If we decrease it, it's going to become more negative.
If you square something that is more negative,
it's not going to become smaller.
Let's say before you remove the data point,
r was, I'm just gonna make up a value,
let's say it was negative 0.4, and then after
removing the outlier, r becomes more negative
and it's going to be equal to negative 0.5.
We'll if you square this, this would be
positive 0.16 while this would be
positive 0.25.
So if r is already negative and if you
make it more negative, it would not decrease r squared,
it actually would increase r squared.
So I will rule this one out.
The slope of the least-squares regression line
would increase.
No, it's going to decrease.
It's going to be a stronger negative correlation.
Rule that one out.
The y-intercept of the least-squares regression line
would increase.
Yes, by getting rid of this outlier,
you could think of it as the left side of this line
is going to increase.
Or another way to think about it,
the slope of this line is going to decrease,
it's going to become more negative.
We know that the least-squares regression line
will always go through the mean of both variables.
So we're just gonna pivot around the mean of both variables
which would mean that the y-intercept will go higher.
So I will fill that in.