Conditions for inference on slope | More on regression | AP Statistics | Khan Academy

- [Instructor] In a previous
video, we began to think about

how we can use a regression
line and, in particular,

the slope of a regression
line based on sample data,

how we can use that in
order to make inference

about the slope of the true
population regression line.

In this video, what we're
going to think about,

what are the conditions for inference

when we're dealing with regression lines?

And these are going to be, in some ways,

similar to the conditions for inference

that we thought about when we
were doing hypothesis testing

and confidence intervals for
means and for proportions,

but there's also going to
be a few new conditions.

So to help us remember these conditions,

you might want to think about
the LINER acronym, L-I-N-E-R.

And if it isn't obvious to
you, this almost is linear.

Liner, if it had an A, it would be linear.

And this is valuable because, remember,

we're thinking about linear regression.

So the L right over here
actually does stand for linear.

And here, the condition is, is
that the actual relationship

in the population between
your x and y variables

actually is a linear relationship,

so actual

linear

relationship,

relationship

between,

between x

and y.

Now, in a lot of cases, you
might just have to assume

that this is going to be
the case when you see it on

an exam, like an AP exam, for example.

They might say, hey, assume
this condition is met.

Oftentimes, it'll say assume all

of these conditions are met.

They just want you to maybe
know about these conditions.

But this is something to think about.

If the underlying
relationship is nonlinear,

well, then maybe some of your

inferences might not be as robust.

Now, the next one is
one we have seen before

when we're talking about general
conditions for inference,

and this is the independence,

independence condition.

And there's a couple of
ways to think about it.

Either individual observations

are independent of each other.

So you could be sampling with replacement.

Or you could be thinking
about your 10% rule,

that we have done when we thought about

the independence condition
for proportions and for means,

where we would need to feel confident

that the size of our
sample is no more than 10%

of the size of the population.

Now, the next one is the normal condition,

which we have talked about
when we were doing inference

for proportions and for means.

Although, it means something a
little bit more sophisticated

when we're dealing with a regression.

The normal condition, and, once again,

many times people just
say assume it's been met.

But let me actually
draw a regression line,

but do it with a little perspective,

and I'm gonna add a third dimension.

Let's say that's the x-axis,

and let's say this is the y-axis.

And the true population
regression line looks like this.

And so the normal condition tells us

that, for any given x
in the true population,

the distribution of y's that
you would expect is normal,

is normal.

So let me see if I can
draw a normal distribution

for the y's,

given that x.

So that would be that
normal distribution there.

And then let's say, for
this x right over here,

you would expect a normal
distribution as well,

so just like,

just like this.

So if we're given x,

the distribution of y's should be normal.

Once again, many times you'll just be

told to assume that that has
been met because it might,

at least in an introductory
statistics class,

be a little bit hard to
figure this out on your own.

Now, the next condition
is related to that,

and this is the idea of
having equal variance,

equal variance.

And that's just saying that each

of these normal distributions should have

the same spread for a given x.

And so you could say equal variance,

or you could even think about them having

the equal standard deviation.

So, for example, if, for a
given x, let's say for this x,

all of sudden, you had
a much lower variance,

made it look like this,

then you would no longer meet
your conditions for inference.

Last, but not least, and this
is one we've seen many times,

this is the random condition.

And this is that the data comes from

a well-designed random sample or

some type of randomized experiment.

And this condition we have
seen in every type of condition

for inference that we
have looked at so far.

So I'll leave you there.

It's good to know.

It will show up on some exams.

But many times, when it
comes to problem solving,

in an introductory statistics
class, they will tell you,

hey, just assume the conditions
for inference have been met.

Or what are the conditions for inference?

But they're not going to
actually make you prove,

for example, the normal or
the equal variance condition.

That might be a bit much

for an introductory statistics class.