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

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

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the slope of a regression
line based on sample data,

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how we can use that in
order to make inference

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about the slope of the true
population regression line.

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

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what are the conditions for inference

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when we're dealing with regression lines?

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And these are going to be, in some ways,

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similar to the conditions for inference

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that we thought about when we
were doing hypothesis testing

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and confidence intervals for
means and for proportions,

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

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So to help us remember these conditions,

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

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

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

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And this is valuable because, remember,

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we're thinking about linear regression.

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

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

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in the population between
your x and y variables

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actually is a linear relationship,

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so actual

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linear

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relationship,

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relationship

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between,

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between x

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and y.

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

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

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an exam, like an AP exam, for example.

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

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Oftentimes, it'll say assume all

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of these conditions are met.

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They just want you to maybe
know about these conditions.

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But this is something to think about.

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If the underlying
relationship is nonlinear,

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well, then maybe some of your

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inferences might not be as robust.

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Now, the next one is
one we have seen before

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when we're talking about general
conditions for inference,

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and this is the independence,

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independence condition.

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

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Either individual observations

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are independent of each other.

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So you could be sampling with replacement.

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Or you could be thinking
about your 10% rule,

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that we have done when we thought about

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the independence condition
for proportions and for means,

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where we would need to feel confident

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

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of the size of the population.

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Now, the next one is the normal condition,

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which we have talked about
when we were doing inference

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for proportions and for means.

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Although, it means something a
little bit more sophisticated

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when we're dealing with a regression.

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The normal condition, and, once again,

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

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But let me actually
draw a regression line,

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but do it with a little perspective,

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and I'm gonna add a third dimension.

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

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

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And the true population
regression line looks like this.

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And so the normal condition tells us

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that, for any given x
in the true population,

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

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is normal.

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So let me see if I can
draw a normal distribution

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for the y's,

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given that x.

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So that would be that
normal distribution there.

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

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you would expect a normal
distribution as well,

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so just like,

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just like this.

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So if we're given x,

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the distribution of y's should be normal.

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Once again, many times you'll just be

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

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at least in an introductory
statistics class,

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

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Now, the next condition
is related to that,

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and this is the idea of
having equal variance,

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equal variance.

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And that's just saying that each

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of these normal distributions should have

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the same spread for a given x.

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And so you could say equal variance,

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or you could even think about them having

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the equal standard deviation.

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

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all of sudden, you had
a much lower variance,

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made it look like this,

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then you would no longer meet
your conditions for inference.

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

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this is the random condition.

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And this is that the data comes from

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a well-designed random sample or

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some type of randomized experiment.

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And this condition we have
seen in every type of condition

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for inference that we
have looked at so far.

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So I'll leave you there.

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It's good to know.

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It will show up on some exams.

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But many times, when it
comes to problem solving,

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in an introductory statistics
class, they will tell you,

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

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Or what are the conditions for inference?

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

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for example, the normal or
the equal variance condition.

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That might be a bit much

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for an introductory statistics class.