Interpreting computer regression data

WEBVTT
Kind: captions
Language: en

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- [Narrator] In other videos,

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we've done linear regressions by hand,

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but we mentioned that most regressions

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are actually done using some
type of computer or calculator.

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And so what we're going
to do in this video,

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is look at an example of
the output that we might see

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from a computer, and to
not be intimidated by it,

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and to see how it gives us

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the equation for the regression line,

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and some of the other data it gives us.

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So here it tells us,
Cheryl Dixon is interested

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to see if students who
consume more caffeine

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tend to study more as well.

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She randomly selects 20
students at her school,

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and records their caffeine
intake in milligrams

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and the number of hours spent studying.

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A scatterplot of the data
showed a linear relationship.

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This is a computer output
from a least-squares

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regression analysis on the data.

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So we have these things
called the predictors,

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coefficient, and then we
have these other things,

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standard error of coefficient, T and P,

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and then all of these things down here,

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how do we make sense of this

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in order to come up with an equation

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for our linear regression?

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So let's just get
straight on our variables.

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Let's just say that we say that Y

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is the thing that we're trying to predict,

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so this is the hours spent studying,

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hours studying.

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And then let's say X is

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what we think explains the hours studying,

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or is one of the things that
explains the hours studying,

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and this is the amount
of caffeine ingested,

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so this is caffeine

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consumed in milligrams.

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And so, our regression
line would have the form

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Y hat, this tells us that
this is a linear regression,

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it's trying to estimate the
actual Y values for given Xs,

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is going to be equal to, MX plus B.

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Now how do we figure out what M and B are,

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based on this computer output?

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So when you look at this table here,

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this first column says predictor,

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and it says constant, and it has caffeine.

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And so all this is saying is,

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when you're trying to predict
the number of hours studying,

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when you're trying to predict Y,

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there's essentially two inputs there.

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There is the constant value,
and there is your variable,

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in this case caffeine, that
you are using to predict

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the amount that you study.

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And so this tells you
the coefficients on each.

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The coefficient on a
constant is the constant.

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You could view this as

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the coefficient on the
X to the zeroth term.

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And so the coefficient on the constant,

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that is the constant,

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two point five four four.

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And then, the coefficient on the caffeine,

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well, we just said that X
is the caffeine consumed,

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so this is that coefficient,

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zero point one six four.

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So just like that, we actually have

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the equation for the regression line,

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that is why these computer
things are useful.

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So, we can just write it out,

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Y hat is equal to

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zero point one six four X plus

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two point five four four,

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two point five four four.

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So that's the regression line,

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what is this other
information they give us?

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Well, I won't give you a
very satisfying answer,

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because all of this is actually

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useful for inferential statistics.

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To think about things like,

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well, what is the probability
that this is chance

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that we got something to fit this well?

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So this right over here is the R squared,

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and if you wanted to
figure out the R from this,

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you would just take the square root here,

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we could say that R is going
to be equal to the square root

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of zero point six zero zero three two,

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depending on how much precision you have.

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But you might say, well how do we know if

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R is the positive square
root, or the negative

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square root of that, R
can take on values between

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negative one and positive one.

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And the answer is, you would
look at the slope here.

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We have a positive slope,
which tells us that

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R is going to be positive.

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If we had a negative slope, then R,

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then we would take the
negative square root.

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Now this right here is
the adjusted R squared,

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and we really don't have
to worry about it too much

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when we're thinking about
just bivariate data,

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we're talking about
caffeine and hours studying

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in this case.

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If we started to have more variables

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that tried to explain the hours studying,

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then we would care about
adjusted R squared,

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but we're not gonna do that just yet.

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Last but not least, this S variable.

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This is the standard
deviation of the residuals,

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which we study in other videos.

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And why is that useful?

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Well that's a measure of
how well the regression line

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fits the data, it's a
measure of, we could say,

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the typical error.

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So big takeaway, computers are useful,

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they'll give you a lot of data,

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and the key thing is how
do you pick out the things

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that you actually need, because
if you know how to do it,

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it can be quite straightforward.


Standard deviation of residuals or Root-mean-square error (RMSD)

WEBVTT
Kind: captions
Language: en

00:00:00.376 --> 00:00:01.983
- [Instructor] What we're
going to do in this video

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is calculate a typical measure

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of how well the actual data points

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agree with a model, in
this case, a linear model

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and there's several names for it.

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We could consider this to
be the standard deviation

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of the residuals

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and that's essentially what
we're going to calculate.

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You could also call it
the root-mean-square error

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and you'll see why it's called this

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because this really describes
how we calculate it.

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So, what we're going to do

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is look at the residuals
for each of these points

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and then we're going to find
the standard deviation of them.

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So, just as a bit of review,

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the ith residual is going to
be equal to the ith Y value

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for a given X

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minus the predicted Y value

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for a given X.

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Now, when I say Y hat right over here,

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this just says what would
the linear regression predict

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for a given X?

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And this is the actual Y for a given X.

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So, for example, and we've
done this in other videos,

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this is all review,

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the residual here when X is equal to one,

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we have Y is equal to one

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but what was predicted by the model

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is 2.5 times one minus two which is .5.

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So, one minus .5,

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so this residual here,

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this residual is equal to one

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minus 0.5 which is equal to 0.5

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and it's a positive 0.5

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and if the actual point is above the model

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you're going to have a positive residual.

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Now, the residual over here

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you also have the actual point
being higher than the model,

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so this is also going to
be a positive residual

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and once again, when X is equal to three,

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the actual Y is six,

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the predicted Y is 2.5 times three,

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which is 7.5 minus two which is 5.5.

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So, you have six minus 5.5,

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so here I'll write residual
is equal to six minus 5.5

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which is equal to 0.5.

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So, once again you have
a positive residual.

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Now, for this point that
sits right on the model,

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the actual is the predicted,

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when X is two, the actual is three

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and what was predicted
by the model is three,

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so the residual here is
equal to the actual is three

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and the predicted is three,

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so it's equal to zero

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and then last but not least,

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you have this data point

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where the residual is
going to be the actual,

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when X is equal to two is two,

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minus the predicted.

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Well, when X is equal to two,

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you have 2.5 times two,

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which is equal to five
minus two is equal to three.

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So, two minus three is
equal to negative one.

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And so, when your actual is
below your regression line,

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you're going to have a negative residual,

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so this is going to be
negative one right over there.

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Now we can calculate
the standard deviation

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of the residuals.

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We're going to take this first residual

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which is 0.5,

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and we're going to square it,

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we're going to add it
to the second residual

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right over here, I'll use
this blue or this teal color,

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that's zero, gonna square that.

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Then we have this third
residual which is negative one,

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so plus negative one squared

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and then finally, we
have that fourth residual

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which is 0.5 squared,

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0.5 squared,

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so once again, we took
each of the residuals,

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which you could view as the distance

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between the points and what
the model would predict,

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we are squaring them,

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when you take a typical
standard deviation,

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you're taking the distance
between a point and the mean.

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Here we're taking the
distance between a point

00:04:00.200 --> 00:04:02.628
and what the model would have predicted

00:04:02.628 --> 00:04:04.818
but we're squaring each of those residuals

00:04:04.818 --> 00:04:07.026
and adding them all up together,

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and just like we do with the
sample standard deviation,

00:04:10.086 --> 00:04:13.505
we are now going to divide by one less

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than the number of residuals
we just squared and added,

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so we have four residuals,

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we're going to divide by four minus one

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which is equal to of course three.

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You could view this part as
a mean of the squared errors

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and now we're gonna take
the square root of it.

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So, let's see, this is going
to be equal to square root

00:04:38.029 --> 00:04:41.946
of this is 0.25, 0.25,

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this is just zero,

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this is going to be positive one,

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and then this 0.5 squared

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is going to be 0.25,

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0.25, all of that over three.

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Now, this numerator is going to be 1.5

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over three, so this is
going to be equal to,

00:05:07.129 --> 00:05:10.005
1.5 is exactly half of three,

00:05:10.005 --> 00:05:11.259
so we could say this is equal

00:05:11.259 --> 00:05:13.842
to the square root of one half,

00:05:14.779 --> 00:05:17.157
this one over the square root of two,

00:05:17.157 --> 00:05:20.240
one divided by the square root of two

00:05:22.994 --> 00:05:24.881
which gets us to,

00:05:24.881 --> 00:05:27.545
so if we round to the nearest thousandths,

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it's roughly 0.707.

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So, approximately 0.707.

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And if you wanted to visualize that,

00:05:38.621 --> 00:05:41.649
one standard deviation of
the residuals below the line

00:05:41.649 --> 00:05:44.151
would look like this,

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and one standard deviation above the line

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for any given X value would
go one standard deviation

00:05:48.989 --> 00:05:50.946
of the residuals above it,

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it would look something like that.

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And this is obviously just
a hand-drawn approximation

00:05:57.048 --> 00:06:00.948
but you do see that this does
seem to be roughly indicative

00:06:00.948 --> 00:06:03.476
of the typical residual.

00:06:03.476 --> 00:06:04.424
Now, it's worth noting,

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sometimes people will say
it's the average residual

00:06:07.126 --> 00:06:10.227
and it depends how you
think about the word average

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because we are squaring the residuals,

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so outliers, things that are
really far from the line,

00:06:16.522 --> 00:06:18.001
when you square it are going to have

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disproportionate impact here.

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If you didn't want to have that behavior

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we could have done
something like find the mean

00:06:24.058 --> 00:06:25.961
of the absolute residuals,

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that actually in some ways
would have been the simple one

00:06:27.874 --> 00:06:31.233
but this is a standard way of
people trying to figure out

00:06:31.233 --> 00:06:35.566
how much a model disagrees
with the actual data,

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and so you can imagine
the lower this number is

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the better the fit of the model.


Threshold for low percentile

WEBVTT
Kind: captions
Language: en

00:00:00.622 --> 00:00:02.891
- [Instructor] The distribution
of average wait times

00:00:02.891 --> 00:00:05.367
in drive-through restaurant
lines in one town

00:00:05.367 --> 00:00:10.016
was approximately normal
with mean of 185 seconds

00:00:10.016 --> 00:00:13.327
and standard deviation of 11 seconds.

00:00:13.327 --> 00:00:16.135
Amelia only likes to use the drive-through

00:00:16.135 --> 00:00:18.226
for restaurants where
the average wait time

00:00:18.226 --> 00:00:22.303
is in the bottom 10% for that town.

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What is the maximum average wait time

00:00:25.064 --> 00:00:27.101
for restaurants where Amelia likes

00:00:27.101 --> 00:00:29.343
to use the drive-through?

00:00:29.343 --> 00:00:31.753
Round to the nearest whole second.

00:00:31.753 --> 00:00:34.309
Like always, if you feel
like you can tackle this,

00:00:34.309 --> 00:00:37.789
pause this video and try to do so.

00:00:37.789 --> 00:00:38.985
I'm assuming you've paused it.

00:00:38.985 --> 00:00:40.247
Now let's work through this together.

00:00:40.247 --> 00:00:41.816
So let's think about what's going on.

00:00:41.816 --> 00:00:43.697
They're telling us that the distribution

00:00:43.697 --> 00:00:47.295
of average wait times
is approximately normal,

00:00:47.295 --> 00:00:51.107
so let's get a visualization
of a normal distribution.

00:00:51.107 --> 00:00:52.507
And they tell us several things

00:00:52.507 --> 00:00:54.000
about this normal distribution.

00:00:54.000 --> 00:00:58.319
They tell us that the mean is 185 seconds,

00:00:58.319 --> 00:00:59.616
so that's 185 there,

00:00:59.616 --> 00:01:02.216
the standard deviation is 11 seconds.

00:01:02.216 --> 00:01:06.383
So, for example, this is gonna
be 11 more than the mean,

00:01:07.437 --> 00:01:10.562
so this would be 196 seconds,

00:01:10.562 --> 00:01:12.862
this would be another 11.

00:01:12.862 --> 00:01:16.131
Each of these dotted lines are
one standard deviation more.

00:01:16.131 --> 00:01:17.881
So this would be 207.

00:01:19.863 --> 00:01:22.884
This would be 11 seconds
less than the mean,

00:01:22.884 --> 00:01:26.634
so this would be 174,
and so on and so forth.

00:01:27.839 --> 00:01:31.107
And we want to find the
maximum average wait time

00:01:31.107 --> 00:01:32.974
for restaurants where Amelia likes

00:01:32.974 --> 00:01:34.186
to use the drive-through.

00:01:34.186 --> 00:01:35.646
Well, what are those restaurants?

00:01:35.646 --> 00:01:37.232
That's where the average wait time is

00:01:37.232 --> 00:01:40.504
in the bottom 10% for that town.

00:01:40.504 --> 00:01:42.043
So how do we think about it?

00:01:42.043 --> 00:01:44.189
Well, there's going to be some value,

00:01:44.189 --> 00:01:48.090
let me mark it off right
over here in this red color,

00:01:48.090 --> 00:01:50.474
so we're going to have
some threshold value

00:01:50.474 --> 00:01:54.641
right over here where this is
anything that level or lower

00:01:56.882 --> 00:02:00.353
is going to be in the bottom 10%.

00:02:00.353 --> 00:02:01.880
Or another way to think about it is,

00:02:01.880 --> 00:02:03.924
this is the largest wait time

00:02:03.924 --> 00:02:07.417
for which you are still in the bottom 10%.

00:02:07.417 --> 00:02:10.287
And so this area right over here

00:02:10.287 --> 00:02:12.841
is going to be 10% of the total,

00:02:12.841 --> 00:02:14.924
or it's going to be 0.10.

00:02:16.136 --> 00:02:18.392
So the way we can tackle this is

00:02:18.392 --> 00:02:20.521
we can get up a z-table and figure out

00:02:20.521 --> 00:02:24.438
what z-score gives us a
proportion of only 0.10

00:02:26.016 --> 00:02:29.164
being less than that z-score,

00:02:29.164 --> 00:02:30.978
and then using that z-score,

00:02:30.978 --> 00:02:33.348
we can figure out this value,

00:02:33.348 --> 00:02:35.137
the actual wait time.

00:02:35.137 --> 00:02:36.975
So let's get our z-table out,

00:02:36.975 --> 00:02:40.330
and since we know that
this is below the mean,

00:02:40.330 --> 00:02:42.159
the mean would be the 50th percentile,

00:02:42.159 --> 00:02:44.661
we know we're gonna
have a negative z-score.

00:02:44.661 --> 00:02:46.378
So I'm gonna take out
the part of the table

00:02:46.378 --> 00:02:50.286
that has the negative z-scores on it.

00:02:50.286 --> 00:02:53.823
And remember, we're looking for 10%,

00:02:53.823 --> 00:02:56.251
but we don't want to go beyond 10%.

00:02:56.251 --> 00:02:58.286
We want to be sure that that value is

00:02:58.286 --> 00:03:00.009
within the 10th percentile,

00:03:00.009 --> 00:03:03.550
that any higher will be
out of the 10th percentile.

00:03:03.550 --> 00:03:06.441
So let's see, when we have
theses really negative z's,

00:03:06.441 --> 00:03:08.759
so far it only gets, it
only doesn't even get

00:03:08.759 --> 00:03:11.226
to the first percentile yet.

00:03:11.226 --> 00:03:13.376
So let's scroll down a little bit

00:03:13.376 --> 00:03:15.552
and let's remember as
we do so that this is

00:03:15.552 --> 00:03:19.241
zero in the hundreds place,
one, two, three, four, five,

00:03:19.241 --> 00:03:23.408
six, seven, eight, nine, so
let's remember those columns.

00:03:25.703 --> 00:03:30.665
So let's see, if we are at
a z-score of negative 1.28,

00:03:30.665 --> 00:03:32.224
remember this is the hundreds

00:03:32.224 --> 00:03:35.949
is zero, one, two, three,
four, five, six, seven, eight,

00:03:35.949 --> 00:03:40.116
so this right over here is
a z-score of negative 1.28

00:03:41.938 --> 00:03:46.105
and that's a little bit
crossing the 10th percentile.

00:03:47.066 --> 00:03:50.828
But if we get a little bit
more negative than that,

00:03:50.828 --> 00:03:52.720
we are in the 10th percentile.

00:03:52.720 --> 00:03:56.887
So this is negative 1.29,
and this does seem to be

00:03:58.331 --> 00:04:01.463
the highest z-score for which we are

00:04:01.463 --> 00:04:03.713
within the 10th percentile.

00:04:04.836 --> 00:04:07.503
So negative 1.29 is our z-score.

00:04:10.898 --> 00:04:13.731
So this is z equals negative 1.29,

00:04:16.487 --> 00:04:19.279
and if we want to figure out
the actual value for that,

00:04:19.279 --> 00:04:23.246
we would start with
the mean, which is 185,

00:04:23.246 --> 00:04:24.858
and then we would say,

00:04:24.858 --> 00:04:27.588
"Well, we want to go
1.29 standard deviations

00:04:27.588 --> 00:04:29.152
"below the mean."

00:04:29.152 --> 00:04:31.338
The negative says we're
going below the mean.

00:04:31.338 --> 00:04:35.505
So we could say minus 1.29
times the standard deviation

00:04:38.414 --> 00:04:40.309
and they tell us up here

00:04:40.309 --> 00:04:43.517
the standard deviation is 11 seconds.

00:04:43.517 --> 00:04:46.350
So it's going to be 1.29 times 11.

00:04:47.793 --> 00:04:50.460
And this is going to be equal to

00:04:51.735 --> 00:04:54.402
1.29 times 11 is equal to 14.19.

00:04:58.631 --> 00:05:00.660
And then I'll make that negative,

00:05:00.660 --> 00:05:02.378
and then add that to 185.

00:05:02.378 --> 00:05:04.711
Plus 185 is equal to 170.81.

00:05:09.498 --> 00:05:10.331
170.81.

00:05:13.170 --> 00:05:16.245
Now they say round to
the nearest whole second.

00:05:16.245 --> 00:05:17.980
There's a couple of
ways to think about it.

00:05:17.980 --> 00:05:20.770
If you really want to ensure
that you're not gonna cross

00:05:20.770 --> 00:05:22.352
that 10th percentile,

00:05:22.352 --> 00:05:25.794
you might want to round to the
nearest second that is below

00:05:25.794 --> 00:05:27.152
this threshold.

00:05:27.152 --> 00:05:31.889
So you might say that this
is approximately 170 seconds.

00:05:31.889 --> 00:05:34.635
If you were to just round
normally this would go to 171,

00:05:34.635 --> 00:05:36.368
but just by doing that,

00:05:36.368 --> 00:05:38.757
you might have crossed the threshold.

00:05:38.757 --> 00:05:40.996
But in all likelihood for this application

00:05:40.996 --> 00:05:43.001
where someone is concerned about wait time

00:05:43.001 --> 00:05:44.456
at drive-through restaurants,

00:05:44.456 --> 00:05:47.392
that difference in a
second between 170 and 171

00:05:47.392 --> 00:05:50.931
is not going to be mission
critical, so to speak.

00:05:50.931 --> 00:05:55.149
But you could safely say
that 170 or 171 seconds

00:05:55.149 --> 00:05:57.232
will meet Amelia's needs.


Residual plots

WEBVTT
Kind: captions
Language: en

00:00:00.791 --> 00:00:02.116
- [Instructor] What we're
going to do in this video

00:00:02.116 --> 00:00:05.533
is talk about the idea of a residual plot

00:00:06.408 --> 00:00:08.658
for a given regression

00:00:08.658 --> 00:00:11.940
and the data that it's trying to explain.

00:00:11.940 --> 00:00:13.183
So right over here,

00:00:13.183 --> 00:00:16.499
we have a fairly simple
least squares regression.

00:00:16.499 --> 00:00:19.508
We're trying to fit four points.

00:00:19.508 --> 00:00:21.657
And in previous videos,
we actually came up

00:00:21.657 --> 00:00:25.674
with the equation of this
least squares regression line.

00:00:25.674 --> 00:00:27.100
What I'm going to do now

00:00:27.100 --> 00:00:31.617
is plot the residuals
for each of these points.

00:00:31.617 --> 00:00:33.691
So what is a residual?

00:00:33.691 --> 00:00:35.633
Well, just as a reminder,

00:00:35.633 --> 00:00:36.716
your residual

00:00:38.875 --> 00:00:42.208
for a given point is equal to the actual

00:00:44.133 --> 00:00:45.716
minus the expected.

00:00:47.970 --> 00:00:50.458
So how do I make that tangible?

00:00:50.458 --> 00:00:53.324
Well, what's the residual for
this point right over here?

00:00:53.324 --> 00:00:54.596
For this point here,

00:00:54.596 --> 00:00:58.013
the actual y when x equals one is one,

00:00:59.333 --> 00:01:02.142
but the expected, when x equals one

00:01:02.142 --> 00:01:04.149
for this least squares regression line,

00:01:04.149 --> 00:01:06.149
2.5 times one minus two,

00:01:07.412 --> 00:01:09.308
well, that's gonna be .5.

00:01:09.308 --> 00:01:12.554
And so our residual is one minus .5,

00:01:12.554 --> 00:01:14.387
so we have a positive,

00:01:15.566 --> 00:01:18.233
we have a positive 0.5 residual.

00:01:19.912 --> 00:01:22.545
Over for this point,
you have zero residual.

00:01:22.545 --> 00:01:25.374
The actual is the expected.

00:01:25.374 --> 00:01:27.583
For this point right over here,

00:01:27.583 --> 00:01:31.500
the actual, when x
equals two, for y is two,

00:01:32.350 --> 00:01:35.009
but the expected is three.

00:01:35.009 --> 00:01:37.283
So our residual over here,

00:01:37.283 --> 00:01:40.450
once again, the actual is y equals two

00:01:41.491 --> 00:01:42.987
when x equals two.

00:01:42.987 --> 00:01:47.410
The expected, two times
2.5 minus two is three,

00:01:47.410 --> 00:01:50.692
so this is going to be two minus three,

00:01:50.692 --> 00:01:54.479
which equals a residual of negative one.

00:01:54.479 --> 00:01:57.441
And then over here, our
residual are actual.

00:01:57.441 --> 00:02:00.333
When x equals three is six,

00:02:00.333 --> 00:02:03.916
our expected when x equals three is 5.5.

00:02:05.225 --> 00:02:06.642
So six minus 5.5,

00:02:07.841 --> 00:02:09.758
that is a positive 0.5.

00:02:11.386 --> 00:02:14.208
So those are the residuals,
but how do we plot it?

00:02:14.208 --> 00:02:16.708
Well, we would set up or axes.

00:02:17.982 --> 00:02:21.210
Let me do it right over here.

00:02:21.210 --> 00:02:22.626
One,

00:02:22.626 --> 00:02:24.233
two,

00:02:24.233 --> 00:02:25.833
and three.

00:02:25.833 --> 00:02:29.579
And let's see, the maximum
residual here is positive .5

00:02:29.579 --> 00:02:31.734
and then the minium one
here is negative one.

00:02:31.734 --> 00:02:35.264
So let's see, this could be .5, one.

00:02:35.264 --> 00:02:37.025
negative .5, negative one.

00:02:37.025 --> 00:02:38.800
So this is negative one.

00:02:38.800 --> 00:02:40.399
This is positive one here.

00:02:40.399 --> 00:02:44.161
And so when x equals one,
what was the residual?

00:02:44.161 --> 00:02:47.744
Well, the actual was
one, expected was 0.5,

00:02:49.503 --> 00:02:50.994
one minus 0.5

00:02:50.994 --> 00:02:51.827
is 0.5.

00:02:54.021 --> 00:02:56.953
So this right over here, we
can plot right over here.

00:02:56.953 --> 00:02:58.620
The residual is 0.5.

00:03:01.716 --> 00:03:04.704
When x equals two, we
actually have two data points.

00:03:04.704 --> 00:03:06.815
First, I'll do this one.

00:03:06.815 --> 00:03:09.867
When we have the point two comma three,

00:03:09.867 --> 00:03:12.254
the residual there is zero.

00:03:12.254 --> 00:03:15.957
So for one of them, the residual is zero.

00:03:15.957 --> 00:03:19.099
Now for the other one, the
residual is negative one.

00:03:19.099 --> 00:03:21.082
Let me do that in a different color.

00:03:21.082 --> 00:03:23.445
For the other one, the
residual is negative one,

00:03:23.445 --> 00:03:26.834
so we would plot it right over here.

00:03:26.834 --> 00:03:28.533
And then this last point,

00:03:28.533 --> 00:03:30.866
the residual is positive .5.

00:03:32.528 --> 00:03:34.528
So it is just like that.

00:03:35.716 --> 00:03:37.750
And so this thing that
I have just created,

00:03:37.750 --> 00:03:38.962
where we're just seeing,

00:03:38.962 --> 00:03:42.141
for each x where we have
a corresponding point,

00:03:42.141 --> 00:03:45.102
we plot the point above or below the line

00:03:45.102 --> 00:03:46.892
based on the residual.

00:03:46.892 --> 00:03:49.246
This is called a residual plot.

00:03:49.246 --> 00:03:50.891
Now, one question is why do people

00:03:50.891 --> 00:03:52.207
even go through the trouble

00:03:52.207 --> 00:03:55.957
of creating a residual plot like this.

00:03:55.957 --> 00:03:59.733
The answer is, regardless of
whether the regression line

00:03:59.733 --> 00:04:02.570
is upward sloping or downward sloping,

00:04:02.570 --> 00:04:05.699
this gives you a sense
of how good a fit it is

00:04:05.699 --> 00:04:09.232
and whether a line is good at explaining

00:04:09.232 --> 00:04:11.621
the relationship between the variables.

00:04:11.621 --> 00:04:14.092
The general idea is if you see the point

00:04:14.092 --> 00:04:16.383
pretty evenly scattered
or randomly scattered

00:04:16.383 --> 00:04:17.432
above and below this line,

00:04:17.432 --> 00:04:20.450
you don't really discern any trend here,

00:04:20.450 --> 00:04:24.908
then a line is probably a
good model for the data.

00:04:24.908 --> 00:04:26.708
But if you do see some type of trend,

00:04:26.708 --> 00:04:29.105
if the residuals had an
upward trend like this

00:04:29.105 --> 00:04:31.249
or if they were curving
up and then curving down,

00:04:31.249 --> 00:04:32.857
or they had a downward trend,

00:04:32.857 --> 00:04:35.192
then you might say, "Hey,
this line isn't a good fit,

00:04:35.192 --> 00:04:38.925
"and maybe we would have
to do a non-linear model."

00:04:38.925 --> 00:04:41.300
What are some examples
of other residual plots?

00:04:41.300 --> 00:04:44.300
And let's try to analyze them a bit.

00:04:45.342 --> 00:04:47.166
So right here you have a regression line

00:04:47.166 --> 00:04:48.846
and its corresponding residual plot.

00:04:48.846 --> 00:04:50.414
And once again, you see here,

00:04:50.414 --> 00:04:52.683
the residual is slightly positive.

00:04:52.683 --> 00:04:54.999
The actual is slightly above the line,

00:04:54.999 --> 00:04:57.291
and you see it right over
there, it's slightly positive.

00:04:57.291 --> 00:04:59.650
This one's even more
positive, you see it there.

00:04:59.650 --> 00:05:02.024
But like the example we just looked at,

00:05:02.024 --> 00:05:04.890
it looks like these residuals
are pretty evenly scattered

00:05:04.890 --> 00:05:06.066
above and below the line.

00:05:06.066 --> 00:05:08.232
There isn't any discernible trend.

00:05:08.232 --> 00:05:10.590
And so I would say that
a linear model here,

00:05:10.590 --> 00:05:12.558
and in particular, this regression line,

00:05:12.558 --> 00:05:15.058
is a good model for this data.

00:05:17.306 --> 00:05:19.157
But if we see something like this,

00:05:19.157 --> 00:05:22.309
a different picture emerges.

00:05:22.309 --> 00:05:24.807
When I look at just the residual plot,

00:05:24.807 --> 00:05:26.599
it doesn't look like
they're evenly scattered.

00:05:26.599 --> 00:05:28.607
It looks like there's
some type of trend here.

00:05:28.607 --> 00:05:30.574
I'm going down here,

00:05:30.574 --> 00:05:33.141
but then I'm going back up.

00:05:33.141 --> 00:05:35.542
When you see something like
this, where on the residual plot

00:05:35.542 --> 00:05:38.659
you're going below the
x-axis and then above,

00:05:38.659 --> 00:05:40.380
then it might say, "Hey, a linear model

00:05:40.380 --> 00:05:41.684
"might not be appropriate.

00:05:41.684 --> 00:05:44.236
"Maybe some type of non-linear model.

00:05:44.236 --> 00:05:46.633
"Some type of non-linear curve

00:05:46.633 --> 00:05:48.217
"might better fit the data,"

00:05:48.217 --> 00:05:51.742
or the relationship
between the y and the x

00:05:51.742 --> 00:05:52.909
is non-linear.

00:05:53.792 --> 00:05:55.082
Another way you could think about it

00:05:55.082 --> 00:05:57.116
is when you have a lot of residuals

00:05:57.116 --> 00:06:00.433
that are pretty far away from the x-axis

00:06:00.433 --> 00:06:02.783
in the residual plot, you'd also say,

00:06:02.783 --> 00:06:05.107
"This line isn't such a good fit."

00:06:05.107 --> 00:06:07.474
If you calculate the R value here,

00:06:07.474 --> 00:06:09.711
it would only be slightly positive,

00:06:09.711 --> 00:06:11.677
but it would not be close to one.


Calculating the equation of a regression line

WEBVTT
Kind: captions
Language: en

00:00:01.385 --> 00:00:02.900
- [Instructor] In previous videos, we took

00:00:02.900 --> 00:00:06.631
this bivariate data and we
calculated the correlation

00:00:06.631 --> 00:00:09.436
coefficient, and just
as a bit of a review,

00:00:09.436 --> 00:00:11.874
we have the formula here, and it looks

00:00:11.874 --> 00:00:14.475
a bit intimidating, but
in that video we saw

00:00:14.475 --> 00:00:18.642
all it is is an average of
the product of the z scores

00:00:20.119 --> 00:00:22.119
for each of those pairs.

00:00:23.077 --> 00:00:25.509
And as we said, if r is equal to one,

00:00:25.509 --> 00:00:28.016
you have a perfect positive correlation.

00:00:28.016 --> 00:00:30.493
If r is equal to negative
one, you have a perfect

00:00:30.493 --> 00:00:33.528
negative correlation, and
if r is equal to zero,

00:00:33.528 --> 00:00:36.614
you don't have a correlation, but for this

00:00:36.614 --> 00:00:40.781
particular bivariate dataset,
we got an r of 0.946,

00:00:42.734 --> 00:00:46.665
which means we have a fairly
strong positive correlation.

00:00:46.665 --> 00:00:49.351
What we're going to do
on this video is build

00:00:49.351 --> 00:00:52.735
on this notion and actually
come up with the equation

00:00:52.735 --> 00:00:56.402
for the least squares
line that tries to fit

00:00:57.325 --> 00:00:58.408
these points.

00:00:59.881 --> 00:01:02.717
So before I do that, let's just visualize

00:01:02.717 --> 00:01:05.199
some of the statistics that we have here

00:01:05.199 --> 00:01:06.293
for these data points.

00:01:06.293 --> 00:01:08.300
We clearly have the four
data points plotted,

00:01:08.300 --> 00:01:10.798
but let's plot the statistics for x.

00:01:10.798 --> 00:01:13.654
So the sample mean and the
sample standard deviation

00:01:13.654 --> 00:01:16.821
for x are here in red, and actually

00:01:18.298 --> 00:01:20.396
let me box these off
in red so that you know

00:01:20.396 --> 00:01:23.250
that's what is going on
here, so the sample mean

00:01:23.250 --> 00:01:25.649
for x, it's easy to calculate
one plus two plus two

00:01:25.649 --> 00:01:29.291
plus three divided by four,
is eight divided by four,

00:01:29.291 --> 00:01:32.867
which is two, so we have x
equals two right over here.

00:01:32.867 --> 00:01:34.991
And then this is one
sample standard deviation

00:01:34.991 --> 00:01:37.779
above the mean, this is one
sample standard deviation

00:01:37.779 --> 00:01:41.184
below the mean, and then
we could do the same thing

00:01:41.184 --> 00:01:42.851
for the y variables.

00:01:44.313 --> 00:01:47.980
So the mean is three,
and this is one sample

00:01:51.031 --> 00:01:53.311
standard deviation for y above the mean

00:01:53.311 --> 00:01:57.939
and this is one standard
deviation for y below the mean.

00:01:57.939 --> 00:02:00.250
And visualizing these means, especially

00:02:00.250 --> 00:02:03.934
their intersection and also
their standard deviations,

00:02:03.934 --> 00:02:06.520
will help us build an
intuition for the equation

00:02:06.520 --> 00:02:08.687
of the least squares line.

00:02:09.649 --> 00:02:11.973
So generally speaking,
the equation for any line

00:02:11.973 --> 00:02:15.223
is going to be y is equal to mx plus b,

00:02:17.047 --> 00:02:19.906
where this is the slope and
this is the y intercept.

00:02:19.906 --> 00:02:23.535
For the regression line, we'll
put a little hat over it.

00:02:23.535 --> 00:02:25.869
So this, you would literally say y hat,

00:02:25.869 --> 00:02:27.960
this tells you that this
is a regression line

00:02:27.960 --> 00:02:30.462
that we're trying to fit to these points.

00:02:30.462 --> 00:02:34.203
First, what is going to be the slope.

00:02:34.203 --> 00:02:38.120
Well the slope is going
to be r times the ratio

00:02:39.071 --> 00:02:42.126
between the sample standard
deviation in the y direction

00:02:42.126 --> 00:02:46.407
over the sample standard
deviation in the x direction.

00:02:46.407 --> 00:02:48.025
This might not seem intuitive at first,

00:02:48.025 --> 00:02:49.677
but we'll talk about it in a few seconds

00:02:49.677 --> 00:02:51.334
and hopefully it'll make a lot more sense,

00:02:51.334 --> 00:02:53.424
but the next thing we
need to know is alright,

00:02:53.424 --> 00:02:55.474
if we can calculate our
slope, how do we calculate

00:02:55.474 --> 00:02:57.335
our y intercept?

00:02:57.335 --> 00:02:59.789
Well like you first
learned in Algebra one,

00:02:59.789 --> 00:03:02.384
you can calculate the y
intercept if you already know

00:03:02.384 --> 00:03:05.314
the slope by saying well
what point is definitely

00:03:05.314 --> 00:03:07.371
going to be on my line?

00:03:07.371 --> 00:03:10.157
And for a least squares regression line,

00:03:10.157 --> 00:03:11.875
you're definitely going to have the point

00:03:11.875 --> 00:03:16.078
sample mean of x comma sample mean of y.

00:03:16.078 --> 00:03:20.290
So you're definitely going
to go through that point.

00:03:20.290 --> 00:03:23.510
So before I even calculate
for this particular example

00:03:23.510 --> 00:03:26.378
where in previous videos
we calculated the r

00:03:26.378 --> 00:03:30.173
to be 0.946 or roughly equal to that,

00:03:30.173 --> 00:03:32.959
let's just think about what's going on.

00:03:32.959 --> 00:03:35.620
So our least squares line
is definitely going to go

00:03:35.620 --> 00:03:37.203
through that point.

00:03:38.574 --> 00:03:42.741
Now if r were one, if we had a
perfect positive correlation,

00:03:43.597 --> 00:03:46.967
then our slope would be
the standard deviation

00:03:46.967 --> 00:03:50.346
of y over the standard deviation of x.

00:03:50.346 --> 00:03:52.450
So if you were to start at
this point and if you were

00:03:52.450 --> 00:03:56.714
to run your standard
deviation of x and rise

00:03:56.714 --> 00:03:59.283
your standard deviation of y,

00:03:59.283 --> 00:04:00.995
well with a perfect positive correlation,

00:04:00.995 --> 00:04:03.578
your line would look like this.

00:04:06.419 --> 00:04:08.458
And that makes a lot of sense.

00:04:08.458 --> 00:04:10.089
Because you're looking at your spread of y

00:04:10.089 --> 00:04:13.935
over your spread of x,
if r were equal to one,

00:04:13.935 --> 00:04:17.128
this would be your slope,
standard deviation of y

00:04:17.128 --> 00:04:19.478
over standard deviation of x.

00:04:19.478 --> 00:04:21.095
That has parallels to when you first learn

00:04:21.095 --> 00:04:22.202
about slope.

00:04:22.202 --> 00:04:25.042
Change in y over change
in x, you're seeing

00:04:25.042 --> 00:04:28.060
you could say the average spread in y

00:04:28.060 --> 00:04:30.268
over the average spread in x.

00:04:30.268 --> 00:04:32.705
And this would be the case when r is one,

00:04:32.705 --> 00:04:34.114
so let me write that down.

00:04:34.114 --> 00:04:37.633
This would be the case
if r is equal to one.

00:04:37.633 --> 00:04:40.654
What if r were equal to negative one?

00:04:40.654 --> 00:04:43.482
It would look like this.

00:04:43.482 --> 00:04:44.981
That would be our line if we had

00:04:44.981 --> 00:04:48.133
a perfect negative correlation.

00:04:48.133 --> 00:04:50.104
Now what if r were zero?

00:04:50.104 --> 00:04:53.348
Then your slope would be
zero and then your line

00:04:53.348 --> 00:04:58.327
would just be this line, y
is equal to the mean of y,

00:04:58.327 --> 00:05:02.229
so you would just go through
that right over there.

00:05:02.229 --> 00:05:04.417
But now let's think about this scenario.

00:05:04.417 --> 00:05:08.084
In this scenario, our
r is 0.946, so we have

00:05:10.330 --> 00:05:14.245
a fairly strong correlation,
this is pretty close to one,

00:05:14.245 --> 00:05:18.147
and so if you were to
take 0.946 and multiply it

00:05:18.147 --> 00:05:21.559
by this ratio, if you
were to move forward in x

00:05:21.559 --> 00:05:25.392
by the standard deviation
in x, for this case,

00:05:26.261 --> 00:05:28.897
how much would you move up in y?

00:05:28.897 --> 00:05:33.064
Well you would move up r times
the standard deviation of y.

00:05:34.139 --> 00:05:36.051
And as we said if r is one,
you would get all the way

00:05:36.051 --> 00:05:38.974
up to this perfect
correlation line, but here

00:05:38.974 --> 00:05:41.848
it's a 0.946, so you would get up

00:05:41.848 --> 00:05:45.379
about 95% of the way to that.

00:05:45.379 --> 00:05:46.962
And so our line without even looking

00:05:46.962 --> 00:05:51.129
at the equation is going to
look something like this,

00:05:52.859 --> 00:05:56.400
which we can see is a pretty
good fit for those points.

00:05:56.400 --> 00:05:58.590
I'm not proving it here in this video.

00:05:58.590 --> 00:06:00.989
But now that we have an
intuition for these things,

00:06:00.989 --> 00:06:02.380
hopefully you'll appreciate
this isn't just coming

00:06:02.380 --> 00:06:04.486
out of nowhere into some strange formula,

00:06:04.486 --> 00:06:07.335
it actually makes intuitive
sense, let's calculate it

00:06:07.335 --> 00:06:10.760
for this particular set of data.

00:06:10.760 --> 00:06:14.927
M is going to be equal to
r, 0.946, times the sample

00:06:16.945 --> 00:06:20.862
standard deviation of y,
2.160, over the sample

00:06:23.027 --> 00:06:25.610
standard deviation of x, 0.816.

00:06:27.959 --> 00:06:29.729
We can get our calculator
out to calculate that,

00:06:29.729 --> 00:06:32.146
so we have 0.946 times 2.160,

00:06:40.443 --> 00:06:43.526
divided by 0.816, it gets us to 2.50,

00:06:49.498 --> 00:06:51.314
let's just round to the nearest hundredth

00:06:51.314 --> 00:06:54.780
for simplicity here, so
this is approximately equal

00:06:54.780 --> 00:06:55.613
to 2.50.

00:06:59.972 --> 00:07:02.005
And so how do we figure
out the y intercept?

00:07:02.005 --> 00:07:04.459
Well remember, we go through this point,

00:07:04.459 --> 00:07:08.459
so we're going to have
2.50 times our x mean,

00:07:10.182 --> 00:07:13.099
so our x mean is two, times two,

00:07:14.087 --> 00:07:15.758
remember this right
over here is our x mean,

00:07:15.758 --> 00:07:18.675
plus b, plus b is going to be equal

00:07:20.859 --> 00:07:25.233
to our y mean, our y mean we
see right over here is three,

00:07:25.233 --> 00:07:26.976
and so what do we get?

00:07:26.976 --> 00:07:30.059
We get three is equal to five plus b.

00:07:35.693 --> 00:07:38.305
And so what is b, well
if you subtract five

00:07:38.305 --> 00:07:42.302
from both sides, you get b
is equal to negative two.

00:07:42.302 --> 00:07:44.439
And so there you have it.

00:07:44.439 --> 00:07:47.085
The equation for our
regression line, we deserve

00:07:47.085 --> 00:07:49.689
a little bit of a drum
roll here, we would say

00:07:49.689 --> 00:07:53.363
y hat, the hat tells us
that this is the equation

00:07:53.363 --> 00:07:57.280
for a regression line,
is equal to 2.50 times x

00:08:02.682 --> 00:08:05.932
minus two, minus two, and we are done.


Calculating correlation coefficient r

WEBVTT
Kind: captions
Language: en

00:00:00.405 --> 00:00:02.501
- [Instructor] What we're
going to do in this video

00:00:02.501 --> 00:00:04.869
is calculate by hand

00:00:04.869 --> 00:00:09.740
the correlation coefficient
for a set of bi-variated data.

00:00:09.740 --> 00:00:11.437
Now, when I say bi-variate

00:00:11.437 --> 00:00:14.712
it's just a fancy way of
saying for each X data point,

00:00:14.712 --> 00:00:16.996
there's a corresponding Y data point.

00:00:16.996 --> 00:00:19.461
Now, before I calculate the
correlation coefficient,

00:00:19.461 --> 00:00:20.640
let's just make sure we understand

00:00:20.640 --> 00:00:23.751
some of these other statistics
that they've given us.

00:00:23.751 --> 00:00:25.924
So, we assume that these are samples

00:00:25.924 --> 00:00:28.200
of the X and the corresponding Y

00:00:28.200 --> 00:00:30.052
from our broader population.

00:00:30.052 --> 00:00:32.428
And so, we have the sample mean for X

00:00:32.428 --> 00:00:34.984
and the sample standard deviation for X.

00:00:34.984 --> 00:00:37.307
The sample mean for X
is quite straightforward

00:00:37.307 --> 00:00:40.508
to calculate, it would
just be one plus two

00:00:40.508 --> 00:00:42.925
plus two plus three over four

00:00:44.729 --> 00:00:46.450
and this is eight over four

00:00:46.450 --> 00:00:49.689
which is indeed equal to two.

00:00:49.689 --> 00:00:53.010
The sample standard deviation for X,

00:00:53.010 --> 00:00:54.350
we've also seen this before,

00:00:54.350 --> 00:00:55.925
this should be a little bit review,

00:00:55.925 --> 00:00:56.807
it's gonna be the square root

00:00:56.807 --> 00:00:59.976
of the distance from each of these points

00:00:59.976 --> 00:01:02.728
to the sample mean squared.

00:01:02.728 --> 00:01:05.429
So, one minus two squared

00:01:05.429 --> 00:01:08.117
plus two minus two squared

00:01:08.117 --> 00:01:10.941
plus two minus two squared

00:01:10.941 --> 00:01:13.358
plus three minus two squared,

00:01:14.443 --> 00:01:16.189
all of that over, since
we're talking about

00:01:16.189 --> 00:01:18.131
sample standard deviation,

00:01:18.131 --> 00:01:19.940
we have four data points,

00:01:19.940 --> 00:01:24.006
so one less than four is
all of that over three.

00:01:24.006 --> 00:01:26.568
Now, this actually simplifies quite nicely

00:01:26.568 --> 00:01:29.906
because this is zero, this is zero,

00:01:29.906 --> 00:01:32.485
this is one, this is one

00:01:32.485 --> 00:01:34.174
and so you essentially get the square root

00:01:34.174 --> 00:01:38.341
of 2/3 which is if you approximate 0.816.

00:01:41.133 --> 00:01:42.317
So, that's that.

00:01:42.317 --> 00:01:44.226
And the same thing is true for Y.

00:01:44.226 --> 00:01:45.395
The sample mean for Y,

00:01:45.395 --> 00:01:48.169
if you just add up one plus two plus three

00:01:48.169 --> 00:01:52.054
plus six over four, four data points,

00:01:52.054 --> 00:01:56.025
this is 12 over four which
is indeed equal to three

00:01:56.025 --> 00:01:58.246
and then the sample standard deviation

00:01:58.246 --> 00:02:00.297
for Y you would calculate
the exact same way

00:02:00.297 --> 00:02:01.272
we did it for X

00:02:01.272 --> 00:02:03.272
and you would get 2.160.

00:02:05.022 --> 00:02:07.300
Now, with all of that out of the way,

00:02:07.300 --> 00:02:09.270
let's think about how we calculate

00:02:09.270 --> 00:02:11.603
the correlation coefficient.

00:02:12.601 --> 00:02:15.237
Now, right over here is a representation

00:02:15.237 --> 00:02:19.098
for the formula for the
correlation coefficient

00:02:19.098 --> 00:02:21.954
and at first it might
seem a little intimating

00:02:21.954 --> 00:02:24.560
until you realize a few things.

00:02:24.560 --> 00:02:28.703
All this is saying is for
each corresponding X and Y,

00:02:28.703 --> 00:02:30.620
find the Z score for X,

00:02:31.575 --> 00:02:34.173
so we could call this Z sub X

00:02:34.173 --> 00:02:35.937
for that particular X,

00:02:35.937 --> 00:02:38.041
so Z sub X sub I

00:02:38.041 --> 00:02:39.845
and we could say this is the Z score

00:02:39.845 --> 00:02:40.854
for that particular Y.

00:02:40.854 --> 00:02:45.594
Z sub Y sub I is one way that
you could think about it.

00:02:45.594 --> 00:02:47.437
Look, this is just saying
for each data point,

00:02:47.437 --> 00:02:50.399
find the difference
between it and its mean

00:02:50.399 --> 00:02:55.025
and then divide by the
sample standard deviation.

00:02:55.025 --> 00:02:57.450
And so, that's how many
sample standard deviations

00:02:57.450 --> 00:03:00.191
is it away from its mean,

00:03:00.191 --> 00:03:03.043
and so that's the Z score
for that X data point

00:03:03.043 --> 00:03:06.879
and this is the Z score for
the corresponding Y data point.

00:03:06.879 --> 00:03:08.849
How many sample standard
deviations is it away

00:03:08.849 --> 00:03:11.375
from the sample mean?

00:03:11.375 --> 00:03:13.921
In the real world you
won't have only four pairs

00:03:13.921 --> 00:03:15.587
and it'll be very hard to do it by hand

00:03:15.587 --> 00:03:18.926
and we typically use software
computer tools to do it

00:03:18.926 --> 00:03:20.895
but it's really valuable to do it by hand

00:03:20.895 --> 00:03:24.514
to get an intuitive understanding
of what's going on here.

00:03:24.514 --> 00:03:27.590
So, in this particular situation,

00:03:27.590 --> 00:03:31.180
R is going to be equal
to one over N minus one.

00:03:31.180 --> 00:03:34.345
We have four pairs, so it's gonna be 1/3

00:03:34.345 --> 00:03:37.009
and it's gonna be times
a sum of the products

00:03:37.009 --> 00:03:38.658
of the Z scores.

00:03:38.658 --> 00:03:41.418
So, this first pair right over here,

00:03:41.418 --> 00:03:44.218
so the Z score for this one

00:03:44.218 --> 00:03:47.846
is going to be one
minus how far it is away

00:03:47.846 --> 00:03:50.412
from the X sample mean,

00:03:50.412 --> 00:03:54.579
divided by the X sample
standard deviation, 0.816,

00:03:57.151 --> 00:03:58.401
that times one,

00:04:01.281 --> 00:04:02.961
now we're looking at the Y variable,

00:04:02.961 --> 00:04:06.805
the Y Z score, so it's one minus three,

00:04:06.805 --> 00:04:10.972
one minus three over the Y
sample standard deviation, 2.160

00:04:15.155 --> 00:04:16.701
and we're just going keep doing that.

00:04:16.701 --> 00:04:17.596
I'll do it like this.

00:04:17.596 --> 00:04:21.857
So, the next one it's
going to be two minus two

00:04:21.857 --> 00:04:25.524
over 0.816, this is
where I got the two from

00:04:27.327 --> 00:04:30.179
and I'm subtracting from
that the sample mean

00:04:30.179 --> 00:04:34.482
right over here, times, now
we're looking at this two,

00:04:34.482 --> 00:04:36.649
two minus three over 2.160

00:04:41.977 --> 00:04:45.644
plus I'm happy there's
only four pairs here,

00:04:46.899 --> 00:04:49.383
two minus two again, two minus two

00:04:49.383 --> 00:04:53.550
over 0.816 times now we're
gonna have three minus three,

00:04:56.914 --> 00:04:59.247
three minus three over 2.160

00:05:01.276 --> 00:05:05.525
and then the last pair you're
going to have three minus two,

00:05:05.525 --> 00:05:06.603
three minus two

00:05:06.603 --> 00:05:09.353
over 0.816 times six minus three,

00:05:12.872 --> 00:05:15.122
six minus three over 2.160.

00:05:17.485 --> 00:05:19.457
So, before I get a calculator out,

00:05:19.457 --> 00:05:22.006
let's see if there's some
simplifications I can do.

00:05:22.006 --> 00:05:24.147
Two minus two, that's gonna be zero,

00:05:24.147 --> 00:05:25.657
zero times anything is zero,

00:05:25.657 --> 00:05:27.309
so this whole thing is zero,

00:05:27.309 --> 00:05:28.218
two minus two is zero,

00:05:28.218 --> 00:05:29.176
three minus three is zero,

00:05:29.176 --> 00:05:31.186
this is actually gonna be zero times zero,

00:05:31.186 --> 00:05:33.381
so that whole thing is zero.

00:05:33.381 --> 00:05:35.569
Let's see this is going
to be one minus two

00:05:35.569 --> 00:05:37.341
which is negative one,

00:05:37.341 --> 00:05:40.345
one minus three is negative two,

00:05:40.345 --> 00:05:43.756
so this is going to be R is equal to 1/3

00:05:43.756 --> 00:05:47.697
times negative times negative is positive

00:05:47.697 --> 00:05:51.114
and so this is going to be two over 0.816

00:05:52.391 --> 00:05:56.474
times 2.160 and then plus
three minus two is one,

00:05:58.621 --> 00:06:00.842
six minus three is three,

00:06:00.842 --> 00:06:03.925
so plus three over 0.816 times 2.160.

00:06:07.330 --> 00:06:09.131
Well, these are the same denominator,

00:06:09.131 --> 00:06:11.961
so actually I could rewrite
if I have two over this thing

00:06:11.961 --> 00:06:13.685
plus three over this thing,

00:06:13.685 --> 00:06:15.620
that's gonna be five over this thing,

00:06:15.620 --> 00:06:17.843
so I could rewrite this whole thing,

00:06:17.843 --> 00:06:20.093
five over 0.816 times 2.160

00:06:24.616 --> 00:06:26.274
and now I can just get a calculator out

00:06:26.274 --> 00:06:28.085
to actually calculate this,

00:06:28.085 --> 00:06:30.688
so we have one divided by three

00:06:30.688 --> 00:06:33.938
times five divided by 0.816 times 2.16,

00:06:41.764 --> 00:06:42.885
the zero won't make a difference

00:06:42.885 --> 00:06:44.283
but I'll just write it down,

00:06:44.283 --> 00:06:46.927
and then I will close that parentheses

00:06:46.927 --> 00:06:49.421
and let's see what we get.

00:06:49.421 --> 00:06:51.299
We get an R of, and since everything else

00:06:51.299 --> 00:06:53.124
goes to the thousandth place,

00:06:53.124 --> 00:06:54.630
I'll just round to the thousandths place,

00:06:54.630 --> 00:06:55.797
an R of 0.946.

00:06:59.351 --> 00:07:01.768
So, R is approximately 0.946.

00:07:04.843 --> 00:07:07.544
So, what does this tell us?

00:07:07.544 --> 00:07:10.392
The correlation coefficient is a measure

00:07:10.392 --> 00:07:15.090
of how well a line can
describe the relationship

00:07:15.090 --> 00:07:16.423
between X and Y.

00:07:17.792 --> 00:07:20.221
R is always going to be greater than

00:07:20.221 --> 00:07:21.924
or equal to negative one

00:07:21.924 --> 00:07:24.155
and less than or equal to one.

00:07:24.155 --> 00:07:26.112
If R is positive one,

00:07:26.112 --> 00:07:28.593
it means that an upwards sloping line

00:07:28.593 --> 00:07:31.448
can completely describe the relationship.

00:07:31.448 --> 00:07:33.976
If R is negative one,

00:07:33.976 --> 00:07:36.430
it means a downwards sloping line

00:07:36.430 --> 00:07:39.014
can completely describe the relationship.

00:07:39.014 --> 00:07:40.846
R anywhere in between says well,

00:07:40.846 --> 00:07:42.494
it won't be as good.

00:07:42.494 --> 00:07:45.303
If R is zero that means
that a line isn't describing

00:07:45.303 --> 00:07:48.050
the relationships well at all.

00:07:48.050 --> 00:07:50.081
Now in our situation here,

00:07:50.081 --> 00:07:51.557
not to use a pun,

00:07:51.557 --> 00:07:53.739
in our situation here,

00:07:53.739 --> 00:07:56.072
our R is pretty close to one

00:07:57.272 --> 00:08:01.312
which means that a line
can get pretty close

00:08:01.312 --> 00:08:02.964
to describing the relationship

00:08:02.964 --> 00:08:05.131
between our Xs and our Ys.

00:08:06.497 --> 00:08:09.975
So, for example, I'm just
going to try to hand draw

00:08:09.975 --> 00:08:10.892
a line here

00:08:12.021 --> 00:08:13.877
and it does turn out that
our least squares line

00:08:13.877 --> 00:08:15.878
will always go through the mean

00:08:15.878 --> 00:08:17.517
of the X and the Y,

00:08:17.517 --> 00:08:19.245
so the mean of the X is two,

00:08:19.245 --> 00:08:20.443
mean of the Y is three,

00:08:20.443 --> 00:08:23.837
we'll study that in more
depth in future videos

00:08:23.837 --> 00:08:25.222
but let's see, this
actually does look like

00:08:25.222 --> 00:08:27.183
a pretty good line.

00:08:27.183 --> 00:08:29.677
So, let me just draw it right over there.

00:08:29.677 --> 00:08:32.151
You see that I actually can draw a line

00:08:32.151 --> 00:08:34.049
that gets pretty close to describing it.

00:08:34.049 --> 00:08:34.965
It isn't perfect.

00:08:34.965 --> 00:08:36.703
If it went through every point

00:08:36.703 --> 00:08:39.465
then I would have an R of one

00:08:39.465 --> 00:08:42.514
but it gets pretty close to describing

00:08:42.514 --> 00:08:44.313
what is going on.

00:08:44.313 --> 00:08:45.403
Now, the next thing I wanna do

00:08:45.403 --> 00:08:47.481
is focus on the intuition.

00:08:47.481 --> 00:08:49.930
What was actually going on
here with these Z scores

00:08:49.930 --> 00:08:54.060
and how does taking products
of corresponding Z scores

00:08:54.060 --> 00:08:56.616
get us this property
that I just talked about

00:08:56.616 --> 00:09:00.783
where an R of one will be
strong, positive correlation,

00:09:01.691 --> 00:09:02.685
R of negative one

00:09:02.685 --> 00:09:05.486
would be strong, negative correlation?

00:09:05.486 --> 00:09:08.141
Well, let's draw the sample means here.

00:09:08.141 --> 00:09:10.784
So, the X sample mean is two,

00:09:10.784 --> 00:09:12.956
this is our X axis here,

00:09:12.956 --> 00:09:14.623
this is X equals two

00:09:15.708 --> 00:09:18.291
and our Y sample mean is three.

00:09:19.498 --> 00:09:22.984
This is the line Y is equal to three.

00:09:22.984 --> 00:09:26.569
Now, we can also draw
the standard deviations.

00:09:26.569 --> 00:09:28.342
This is, let's see, the standard deviation

00:09:28.342 --> 00:09:31.926
for X is 0.816 so I'll
be approximating it,

00:09:31.926 --> 00:09:34.759
so if I go .816 less than our mean

00:09:35.803 --> 00:09:39.136
it'll get us at some place around there,

00:09:40.925 --> 00:09:43.095
so that's one standard
deviation below the mean,

00:09:43.095 --> 00:09:44.801
one standard deviation above the mean

00:09:44.801 --> 00:09:48.134
would put us some place right over here,

00:09:50.022 --> 00:09:51.939
and if I do the same thing in Y,

00:09:51.939 --> 00:09:55.546
one standard deviation
above the mean, 2.160

00:09:55.546 --> 00:09:59.370
so that'll be 5.160 so it would put us

00:09:59.370 --> 00:10:01.287
some place around there

00:10:02.269 --> 00:10:04.874
and one standard deviation below the mean,

00:10:04.874 --> 00:10:07.799
so let's see we're gonna
go, if we took away two,

00:10:07.799 --> 00:10:10.095
we would go to one

00:10:10.095 --> 00:10:12.516
and then we're gonna go take another .160,

00:10:12.516 --> 00:10:16.349
so it's gonna be some
place right around here.

00:10:17.478 --> 00:10:20.007
So, for example, for this first pair,

00:10:20.007 --> 00:10:21.174
one comma one.

00:10:22.154 --> 00:10:23.303
What were we doing?

00:10:23.303 --> 00:10:26.288
Well, we said alright, how
many standard deviations

00:10:26.288 --> 00:10:28.937
is this below the mean?

00:10:28.937 --> 00:10:33.238
And that turned out to be
negative one over 0.816,

00:10:33.238 --> 00:10:34.858
that's what we have right over here,

00:10:34.858 --> 00:10:37.117
that's what this would have calculated,

00:10:37.117 --> 00:10:38.941
and then how many standard deviations

00:10:38.941 --> 00:10:40.858
for in the Y direction,

00:10:42.064 --> 00:10:45.314
and that is our negative two over 2.160

00:10:46.616 --> 00:10:49.891
but notice, since both
of them were negative

00:10:49.891 --> 00:10:52.519
it contributed to the R,

00:10:52.519 --> 00:10:55.046
this would become a positive value

00:10:55.046 --> 00:10:56.534
and so, one way to think about it,

00:10:56.534 --> 00:11:00.240
it might be helping us
get closer to the one.

00:11:00.240 --> 00:11:03.424
If both of them have a negative Z score

00:11:03.424 --> 00:11:05.869
that means that there's
a positive correlation

00:11:05.869 --> 00:11:07.006
between the variables.

00:11:07.006 --> 00:11:08.175
When one is below the mean,

00:11:08.175 --> 00:11:09.324
the other is you could say,

00:11:09.324 --> 00:11:11.947
similarly below the mean.

00:11:11.947 --> 00:11:13.981
Now, if we go to the next data point,

00:11:13.981 --> 00:11:17.731
two comma two right over
here, what happened?

00:11:18.985 --> 00:11:22.599
Well, the X variable was right on the mean

00:11:22.599 --> 00:11:25.971
and because of that that
entire term became zero.

00:11:25.971 --> 00:11:27.888
The X Z score was zero.

00:11:28.928 --> 00:11:30.518
And so, that would have taken away

00:11:30.518 --> 00:11:32.875
a little bit from our
correlation coefficient.

00:11:32.875 --> 00:11:34.900
The reason why it would take away

00:11:34.900 --> 00:11:36.328
even though it's not negative,

00:11:36.328 --> 00:11:38.543
you're not contributing to the sum

00:11:38.543 --> 00:11:42.245
but you're going to be dividing
by a slightly higher value

00:11:42.245 --> 00:11:44.340
by including that extra pair.

00:11:44.340 --> 00:11:47.312
If you had a data point where
let's say X was below the mean

00:11:47.312 --> 00:11:49.028
and Y was above the mean,

00:11:49.028 --> 00:11:51.187
something like this,

00:11:51.187 --> 00:11:53.321
if this was one of the points,

00:11:53.321 --> 00:11:54.833
this term would have been negative

00:11:54.833 --> 00:11:56.852
because the Y Z score
would have been positive

00:11:56.852 --> 00:11:59.853
and the X Z score would have been negative

00:11:59.853 --> 00:12:02.078
and so, when you put it in the sum

00:12:02.078 --> 00:12:03.471
it would have actually taken away

00:12:03.471 --> 00:12:04.684
from the sum and so,

00:12:04.684 --> 00:12:07.333
it would have made the R score even lower.

00:12:07.333 --> 00:12:09.535
Similarly something like this

00:12:09.535 --> 00:12:12.546
would have made the R score even lower

00:12:12.546 --> 00:12:15.334
because you would have
a positive Z score for X

00:12:15.334 --> 00:12:17.185
and a negative Z score for Y

00:12:17.185 --> 00:12:19.337
and so a product of a
positive and a negative

00:12:19.337 --> 00:12:21.004
would be a negative.


Finding z-score for a percentile

WEBVTT
Kind: captions
Language: en

00:00:00.497 --> 00:00:02.811
- [Instructor] The distribution
of resting pulse rates

00:00:02.811 --> 00:00:05.600
of all students at Santa Maria High School

00:00:05.600 --> 00:00:09.969
was approximately normal with
mean of 80 beats per minute

00:00:09.969 --> 00:00:13.492
and standard deviation
of nine beats per minute.

00:00:13.492 --> 00:00:15.742
The school nurse plans
to provide additional

00:00:15.742 --> 00:00:18.737
screening to students
whose resting pulse rates

00:00:18.737 --> 00:00:22.892
are in the top 30% of the
students who were tested.

00:00:22.892 --> 00:00:26.574
What is the minimum resting
pulse rate at that school

00:00:26.574 --> 00:00:30.348
for students who will
receive additional screening?

00:00:30.348 --> 00:00:32.642
Round to the nearest whole number.

00:00:32.642 --> 00:00:34.287
If you feel like you
know how to tackle this,

00:00:34.287 --> 00:00:38.102
I encourage you to pause this
video and try to work it out.

00:00:38.102 --> 00:00:41.242
All right, now let's
work this out together.

00:00:41.242 --> 00:00:43.444
They're telling us that the distribution

00:00:43.444 --> 00:00:47.079
of resting pulse rates
are approximately normal.

00:00:47.079 --> 00:00:49.948
So we could use a normal distribution.

00:00:49.948 --> 00:00:51.187
And they tell us several things

00:00:51.187 --> 00:00:53.407
about this normal distribution.

00:00:53.407 --> 00:00:56.567
They tell us that the mean
is 80 beats per minute.

00:00:56.567 --> 00:00:59.281
So that is the mean right over there.

00:00:59.281 --> 00:01:00.661
And they tell us that
the standard deviation

00:01:00.661 --> 00:01:02.934
is nine beats per minute.

00:01:02.934 --> 00:01:05.027
So on this normal distribution,

00:01:05.027 --> 00:01:07.830
we have one standard
deviation above the mean,

00:01:07.830 --> 00:01:09.594
two standard deviations above the mean,

00:01:09.594 --> 00:01:12.070
so this distance right over here is nine.

00:01:12.070 --> 00:01:13.718
So this would be 89.

00:01:13.718 --> 00:01:16.083
This one right over here would be 98.

00:01:16.083 --> 00:01:18.010
And you could also go standard
deviations below the mean,

00:01:18.010 --> 00:01:21.975
this right over here would
be 71, this would be 62,

00:01:21.975 --> 00:01:25.527
but what we're concerned
about is the top 30%

00:01:25.527 --> 00:01:28.867
because that is who is going to be tested.

00:01:28.867 --> 00:01:32.370
So there's gonna be some
value here, some threshold.

00:01:32.370 --> 00:01:35.623
Let's say it is right over here,

00:01:35.623 --> 00:01:38.123
that if you are at that score,

00:01:39.489 --> 00:01:41.729
you have reached the minimum threshold

00:01:41.729 --> 00:01:43.379
to get an additional screening.

00:01:43.379 --> 00:01:46.430
You are in the top 30%.

00:01:46.430 --> 00:01:49.633
So that means that this
area right over here

00:01:49.633 --> 00:01:52.216
is going to be 30% or 0.3.

00:01:56.454 --> 00:01:59.164
So what we can do, we can use a z-table

00:01:59.164 --> 00:02:02.395
to say for what z-score is 70%

00:02:02.395 --> 00:02:05.081
of the distribution less than that.

00:02:05.081 --> 00:02:06.941
And then we can take that z-score

00:02:06.941 --> 00:02:09.729
and use the mean and
the standard deviation

00:02:09.729 --> 00:02:13.143
to come up with an actual value.

00:02:13.143 --> 00:02:15.396
In previous examples, we
started with the z-score

00:02:15.396 --> 00:02:17.258
and were looking for the percentage.

00:02:17.258 --> 00:02:19.946
This time we're looking
for the percentage.

00:02:19.946 --> 00:02:22.687
We want it to be at least 70%

00:02:22.687 --> 00:02:25.649
and then come up with the
corresponding z-score.

00:02:25.649 --> 00:02:27.765
So let's see, immediately
when we look at this,

00:02:27.765 --> 00:02:30.358
and we are to the right of the mean,

00:02:30.358 --> 00:02:32.197
and so we're gonna have
a positive z-score.

00:02:32.197 --> 00:02:34.126
So we're starting at 50% here.

00:02:34.126 --> 00:02:38.293
We definitely want to
get to the 67%, 68, 69,

00:02:39.260 --> 00:02:41.448
we're getting close and on our table

00:02:41.448 --> 00:02:44.402
this is the lowest z-score

00:02:44.402 --> 00:02:48.893
that gets us across that 70% threshold.

00:02:48.893 --> 00:02:50.143
It's at 0.7019.

00:02:53.221 --> 00:02:55.663
So it definitely crosses the threshold.

00:02:55.663 --> 00:02:58.413
And so that is a z-score of 0.53.

00:03:00.817 --> 00:03:02.400
0.52 is too little.

00:03:04.448 --> 00:03:06.865
So we need a z-score of 0.53.

00:03:08.744 --> 00:03:10.577
Let's write that down.

00:03:11.612 --> 00:03:13.529
0.53, right over there,

00:03:15.992 --> 00:03:17.198
and we just now have to figure out

00:03:17.198 --> 00:03:20.935
what value gives us a z-score of 0.53.

00:03:20.935 --> 00:03:22.881
Well, this just means 0.53

00:03:22.881 --> 00:03:25.950
standard deviations above the mean.

00:03:25.950 --> 00:03:26.992
So to get the value,

00:03:26.992 --> 00:03:30.161
we would take our mean and we would add

00:03:30.161 --> 00:03:32.161
0.53 standard deviation.

00:03:34.163 --> 00:03:35.746
So 0.53 times nine.

00:03:37.934 --> 00:03:40.017
And this will get us 0.53

00:03:43.167 --> 00:03:45.417
times nine is equal to 4.77

00:03:46.906 --> 00:03:49.073
plus 80 is equal to 84.77.

00:03:53.102 --> 00:03:54.935
84.77 and they want us

00:03:56.361 --> 00:03:58.818
to round to the nearest whole number.

00:03:58.818 --> 00:04:02.501
So we will just round
to 85 beats per minute.

00:04:02.501 --> 00:04:03.859
So that's the threshold.

00:04:03.859 --> 00:04:05.989
If you have that resting heartbeat,

00:04:05.989 --> 00:04:07.897
then the school nurse is going

00:04:07.897 --> 00:04:09.585
to give you some additional screening.

00:04:09.585 --> 00:04:13.752
You are in the top 30% of
students who are tested.


Standard normal table for proportion between values

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.192
- [Instructor] A set of laptop prices

00:00:02.192 --> 00:00:05.977
are normally distributed
with a mean of $750

00:00:05.977 --> 00:00:09.019
and a standard deviation of $60.

00:00:09.019 --> 00:00:11.736
What proportion of laptop prices

00:00:11.736 --> 00:00:13.069
are between $624

00:00:14.406 --> 00:00:15.239
and $768?

00:00:18.098 --> 00:00:20.350
So let's think about what they are asking.

00:00:20.350 --> 00:00:24.274
So we have a normal
distribution for the prices.

00:00:24.274 --> 00:00:26.852
So it would look something like this.

00:00:26.852 --> 00:00:29.035
This is just my hand-drawn sketch

00:00:29.035 --> 00:00:30.939
of a normal distribution.

00:00:30.939 --> 00:00:32.285
So it would look something like this.

00:00:32.285 --> 00:00:33.748
It should be symmetric,

00:00:33.748 --> 00:00:36.767
so I'm making it as symmetric
as I can hand draw it.

00:00:36.767 --> 00:00:38.740
And we have the mean

00:00:38.740 --> 00:00:40.134
right in the center.

00:00:40.134 --> 00:00:41.829
So the mean

00:00:41.829 --> 00:00:43.152
would be right there.

00:00:43.152 --> 00:00:44.569
And that is $750.

00:00:46.008 --> 00:00:49.793
They also tell us that we have
a standard deviation of $60.

00:00:49.793 --> 00:00:52.649
So that means one standard
deviation above the mean

00:00:52.649 --> 00:00:55.459
would be roughly right over here,

00:00:55.459 --> 00:00:58.176
and that would be 750 plus 60.

00:00:58.176 --> 00:01:00.428
So that would be $810.

00:01:00.428 --> 00:01:02.030
One standard deviation below the mean

00:01:02.030 --> 00:01:04.375
would put us right about there,

00:01:04.375 --> 00:01:07.510
and that would be 750 minus $60,

00:01:07.510 --> 00:01:09.177
which would be $690.

00:01:11.295 --> 00:01:13.501
And then they tell us,

00:01:13.501 --> 00:01:15.242
what proportion of laptop prices

00:01:15.242 --> 00:01:16.575
are between $624

00:01:18.493 --> 00:01:19.326
and $768.

00:01:22.115 --> 00:01:23.698
So the lower bound,

00:01:24.971 --> 00:01:25.804
$624,

00:01:27.618 --> 00:01:29.848
that's going to actually be more than

00:01:29.848 --> 00:01:31.310
another standard deviation less,

00:01:31.310 --> 00:01:34.143
so that's going to be right around here.

00:01:34.143 --> 00:01:34.976
So that is

00:01:36.117 --> 00:01:36.950
$624.

00:01:38.810 --> 00:01:40.477
And 768 would put us

00:01:41.783 --> 00:01:42.950
right at about

00:01:44.244 --> 00:01:45.591
right at about there,

00:01:45.591 --> 00:01:47.982
and once again this is
just a hand-drawn sketch.

00:01:47.982 --> 00:01:49.315
But that is 768.

00:01:50.281 --> 00:01:53.323
And so what proportion are
between those two values?

00:01:53.323 --> 00:01:55.459
So we wanna find essentially the area

00:01:55.459 --> 00:01:59.708
under this distribution
between these two values.

00:01:59.708 --> 00:02:01.682
The way we are going to approach it,

00:02:01.682 --> 00:02:04.770
we're going to figure out the z-score

00:02:04.770 --> 00:02:07.301
for 768, it's going to be positive

00:02:07.301 --> 00:02:09.043
because it's above the mean,

00:02:09.043 --> 00:02:11.481
and then we're going to use
a z-table to figure out,

00:02:11.481 --> 00:02:13.898
what proportion is below 768.

00:02:14.732 --> 00:02:16.752
So essentially we're going to figure out

00:02:16.752 --> 00:02:17.750
this entire area.

00:02:17.750 --> 00:02:19.492
We're even going to figure out the stuff

00:02:19.492 --> 00:02:20.699
that's below 624.

00:02:20.699 --> 00:02:22.441
That's what that z-table will give us.

00:02:22.441 --> 00:02:25.645
Then we'll figure out the z-score for 624.

00:02:25.645 --> 00:02:29.476
That will be negative 2 point something,

00:02:29.476 --> 00:02:31.079
and we will use the z-table again

00:02:31.079 --> 00:02:35.328
to figure out the proportion
that is less than that.

00:02:35.328 --> 00:02:38.045
And so then we can subtract this red area

00:02:38.045 --> 00:02:40.506
from the proportion that is less than 768

00:02:40.506 --> 00:02:43.617
to get this area in between.

00:02:43.617 --> 00:02:45.637
So let's do that.

00:02:45.637 --> 00:02:49.747
Let's figure out first
the z-score for 768.

00:02:49.747 --> 00:02:52.511
And then we'll do it for 624.

00:02:52.511 --> 00:02:56.261
The z-score for 768,
I'll write it like that,

00:02:57.805 --> 00:02:59.523
is going to be

00:02:59.523 --> 00:03:00.606
768 minus 750

00:03:03.842 --> 00:03:07.255
over the standard deviation, over 60.

00:03:07.255 --> 00:03:10.460
So this is going to be equal to 18

00:03:10.460 --> 00:03:11.293
over 60,

00:03:13.107 --> 00:03:16.107
which is the same thing as six over,

00:03:17.193 --> 00:03:18.517
let's see if we divide the numerator

00:03:18.517 --> 00:03:20.305
and the denominator by three,

00:03:20.305 --> 00:03:24.299
6/20, and this is the same thing

00:03:24.299 --> 00:03:25.132
as 0.30.

00:03:28.641 --> 00:03:32.286
So that is the z-score
for this upper bound.

00:03:32.286 --> 00:03:35.537
Let's figure out what
proportion is less than that.

00:03:35.537 --> 00:03:37.650
For that, we take out a z-table,

00:03:37.650 --> 00:03:39.577
get our z-table.

00:03:39.577 --> 00:03:42.244
And let's see we wanna get 0.30.

00:03:43.548 --> 00:03:46.404
And so this is 0.3, this first column,

00:03:46.404 --> 00:03:47.704
and we've done this in other videos,

00:03:47.704 --> 00:03:50.769
this goes up until the tenths place

00:03:50.769 --> 00:03:52.348
for our z-score,

00:03:52.348 --> 00:03:54.299
and then if we wanna go
to our hundredths place,

00:03:54.299 --> 00:03:57.039
that's what these other columns give us.

00:03:57.039 --> 00:03:58.456
But we're at 0.3,

00:03:59.547 --> 00:04:01.892
so we're going to be in

00:04:01.892 --> 00:04:02.725
this row,

00:04:05.027 --> 00:04:07.743
and our hundredths place
is right over here,

00:04:07.743 --> 00:04:10.321
it's a zero, so this is the proportion

00:04:10.321 --> 00:04:12.238
that is less than $768.

00:04:15.545 --> 00:04:16.378
So .6179.

00:04:18.494 --> 00:04:19.827
So 0.61, 0.6179.

00:04:26.203 --> 00:04:28.293
So now let's do the same exercise

00:04:28.293 --> 00:04:32.210
but do it for the proportion
that's below $624.

00:04:34.957 --> 00:04:35.874
The z-score

00:04:37.349 --> 00:04:39.932
for 624 is going to be equal to

00:04:41.087 --> 00:04:43.254
624 minus the mean of 750,

00:04:45.685 --> 00:04:47.352
all of that over 60.

00:04:48.657 --> 00:04:50.422
And so what is that going to be?

00:04:50.422 --> 00:04:52.209
I'll get my calculator out for this one,

00:04:52.209 --> 00:04:54.322
don't wanna make a careless error.

00:04:54.322 --> 00:04:55.405
624 minus 750

00:05:00.174 --> 00:05:01.962
is equal to,

00:05:01.962 --> 00:05:03.795
and then divide by 60,

00:05:05.793 --> 00:05:08.115
is equal to negative 2.1.

00:05:08.115 --> 00:05:11.343
So that lower bound is
2.1 standard deviations

00:05:11.343 --> 00:05:12.852
below the mean,

00:05:12.852 --> 00:05:17.310
or you could say it has a
z-score of negative 2.1.

00:05:17.310 --> 00:05:19.888
Is equal to negative 2.1.

00:05:19.888 --> 00:05:22.744
And so to figure out the
proportion that is less than that,

00:05:22.744 --> 00:05:24.950
this red area right over here,

00:05:24.950 --> 00:05:26.552
we go back to our z-table.

00:05:26.552 --> 00:05:29.361
And we actually go to the
first part of the z-table.

00:05:29.361 --> 00:05:32.032
So same idea but this starts at negative,

00:05:32.032 --> 00:05:34.307
a z-score of negative 3.4,

00:05:34.307 --> 00:05:37.187
3.4 standard deviations below the mean.

00:05:37.187 --> 00:05:38.556
And just like we saw before,

00:05:38.556 --> 00:05:41.622
this is our zero
hundredths, one hundredth,

00:05:41.622 --> 00:05:43.897
two hundredths, so on and so forth.

00:05:43.897 --> 00:05:47.334
And we wanna go to negative 2.1.

00:05:47.334 --> 00:05:51.235
We could say negative
2.10 just to be precise.

00:05:51.235 --> 00:05:52.721
So this is going to get us,

00:05:52.721 --> 00:05:54.927
let's see negative 2.1,

00:05:54.927 --> 00:05:55.925
there we go.

00:05:55.925 --> 00:05:57.008
And so we are

00:05:58.317 --> 00:06:01.400
negative 2.1, and it's negative 2.10,

00:06:02.403 --> 00:06:04.098
so we have zero hundredths,

00:06:04.098 --> 00:06:06.838
so we're gonna be right here on our table.

00:06:06.838 --> 00:06:10.693
So we see the proportion
that is less than 624

00:06:10.693 --> 00:06:12.276
is .0179 or 0.0179.

00:06:15.825 --> 00:06:16.658
So 0.0179.

00:06:20.677 --> 00:06:22.117
And so if we wanna
figure out the proportion

00:06:22.117 --> 00:06:23.743
that's in between the two,

00:06:23.743 --> 00:06:25.763
we just subtract this red area

00:06:25.763 --> 00:06:28.340
from this entire area,

00:06:28.340 --> 00:06:30.871
the entire proportion that's less than 768

00:06:30.871 --> 00:06:33.193
to get what's in between.

00:06:33.193 --> 00:06:34.026
0.6179,

00:06:36.235 --> 00:06:37.628
once again I know I keep repeating it,

00:06:37.628 --> 00:06:40.647
that's this entire area right over here,

00:06:40.647 --> 00:06:42.667
and we're gonna subtract
out what we have in red,

00:06:42.667 --> 00:06:43.750
minus 0.0179,

00:06:46.545 --> 00:06:48.472
so we're gonna subtract this out,

00:06:48.472 --> 00:06:49.389
to get 0.6.

00:06:52.001 --> 00:06:55.275
So if we wanna give our
answer to four decimal places,

00:06:55.275 --> 00:06:56.858
it would be 0.6000,

00:06:58.410 --> 00:06:59.664
or another way to think about it is

00:06:59.664 --> 00:07:01.247
exactly 60% is

00:07:02.543 --> 00:07:04.210
between 624 and 768.


Standard normal table for proportion above

WEBVTT
Kind: captions
Language: en

00:00:00.834 --> 00:00:02.844
- [Tutor] A set of philosophy exam scores

00:00:02.844 --> 00:00:06.083
are normally distributed
with a mean of 40 points

00:00:06.083 --> 00:00:09.169
and a standard deviation of three points.

00:00:09.169 --> 00:00:13.002
Ludwig got a score of
47.5 points on the exam.

00:00:14.934 --> 00:00:17.981
What proportion of exam scores are higher

00:00:17.981 --> 00:00:20.097
than Ludwig's score?

00:00:20.097 --> 00:00:23.199
Give your answer correct
to four decimal places.

00:00:23.199 --> 00:00:26.368
So, let's just visualize
what's going on here.

00:00:26.368 --> 00:00:29.334
So, the scores are normally distributed.

00:00:29.334 --> 00:00:31.667
So, it would look like this.

00:00:33.770 --> 00:00:37.235
So, the distribution would
look something like that,

00:00:37.235 --> 00:00:39.769
trying to make that
pretty symmetric looking.

00:00:39.769 --> 00:00:41.602
The mean is 40 points,

00:00:42.588 --> 00:00:45.941
so that would be 40
points right over there.

00:00:45.941 --> 00:00:48.649
Standard deviation is three points,

00:00:48.649 --> 00:00:52.158
so this could be one standard
deviation above the mean,

00:00:52.158 --> 00:00:55.111
that would be one standard
deviation below the mean.

00:00:55.111 --> 00:00:57.167
And once again, this is just very rough.

00:00:57.167 --> 00:01:02.119
And so, this would be 43, this
would be 37 right over here.

00:01:02.119 --> 00:01:06.286
And they say Ludwig got a score
of 47.5 points on the exam.

00:01:07.760 --> 00:01:11.927
So, Ludwig's score is going
to be someplace around here.

00:01:13.093 --> 00:01:15.926
So, Ludwig got a 47.5 on the exam.

00:01:18.203 --> 00:01:20.067
And they're saying, what
proportion of exam scores

00:01:20.067 --> 00:01:22.627
are higher than Ludwig's score?

00:01:22.627 --> 00:01:24.333
So, what we need to do is figure out

00:01:24.333 --> 00:01:27.817
what is the area under the
normal distribution curve

00:01:27.817 --> 00:01:29.400
that is above 47.5.

00:01:31.570 --> 00:01:33.256
So, the way we will tackle this

00:01:33.256 --> 00:01:36.951
is we will figure out
the z score for 47.5.

00:01:36.951 --> 00:01:40.608
How many standard deviations
above the mean is that?

00:01:40.608 --> 00:01:42.479
Then, we will look at a z table

00:01:42.479 --> 00:01:44.811
to figure out what
proportion is below that

00:01:44.811 --> 00:01:46.441
because that's what z tables give us.

00:01:46.441 --> 00:01:48.166
They give us the proportion that is below

00:01:48.166 --> 00:01:49.666
a certain z score.

00:01:50.897 --> 00:01:53.269
And then, we can take one minus that

00:01:53.269 --> 00:01:55.507
to figure out the
proportion that is above.

00:01:55.507 --> 00:01:58.586
Remember, the entire area
under the curve is one,

00:01:58.586 --> 00:02:00.529
so if we can figure out this orange area

00:02:00.529 --> 00:02:03.692
and take one minus that,
we're gonna get the red area.

00:02:03.692 --> 00:02:04.786
So, let's do that.

00:02:04.786 --> 00:02:08.958
So, first of all, let's figure
out the z score for 47.5.

00:02:08.958 --> 00:02:10.266
So, let's see.

00:02:10.266 --> 00:02:11.766
We would take 47.5

00:02:15.507 --> 00:02:17.550
and we would subtract the mean.

00:02:17.550 --> 00:02:18.616
So, this is his score.

00:02:18.616 --> 00:02:20.941
We'll subtract the mean, minus 40.

00:02:20.941 --> 00:02:23.501
We know what that's gonna be, that's 7.5.

00:02:23.501 --> 00:02:25.278
That's how much more above the mean.

00:02:25.278 --> 00:02:27.302
But how many standard deviations is that?

00:02:27.302 --> 00:02:28.917
Well, each standard deviation is three,

00:02:28.917 --> 00:02:31.373
so what's 7.5 divided by three?

00:02:31.373 --> 00:02:34.993
This just means the previous
answer divided by three.

00:02:34.993 --> 00:02:39.076
So, he is 2.5 standard
deviations above the mean.

00:02:40.011 --> 00:02:41.918
So, the z score here,

00:02:41.918 --> 00:02:44.501
z score here is a positive 2.5.

00:02:46.167 --> 00:02:48.065
If he was below the mean,
it would be a negative.

00:02:48.065 --> 00:02:49.570
So now, we can look at a z table

00:02:49.570 --> 00:02:52.367
to figure out what
proportion is less than 2.5

00:02:52.367 --> 00:02:54.474
standard deviations above the mean.

00:02:54.474 --> 00:02:55.508
So, that'll give us that orange

00:02:55.508 --> 00:02:57.867
and then we'll subtract that from one.

00:02:57.867 --> 00:03:00.768
So, let's get our z table.

00:03:00.768 --> 00:03:02.084
So, here we go.

00:03:02.084 --> 00:03:04.703
And we've already done
this in previous videos,

00:03:04.703 --> 00:03:07.160
but what's going on
here is this left column

00:03:07.160 --> 00:03:10.797
gives us our z score
up to the tenths place.

00:03:10.797 --> 00:03:13.583
And then these other columns
give us the hundredths place.

00:03:13.583 --> 00:03:16.016
So, what we want to do is find 2.5

00:03:16.016 --> 00:03:17.818
right over here on the left,

00:03:17.818 --> 00:03:19.976
and it's actually gonna be 2.50.

00:03:19.976 --> 00:03:22.921
There's zero hundredths here.

00:03:22.921 --> 00:03:25.525
So, we want to look up 2.50.

00:03:25.525 --> 00:03:27.608
Let me scroll my z table.

00:03:28.546 --> 00:03:30.963
So, I'm gonna go down to 2.5.

00:03:32.256 --> 00:03:34.589
Alright, I think I am there.

00:03:34.589 --> 00:03:37.589
So, what I have here, so I have 2.5,

00:03:38.569 --> 00:03:41.754
so I am going to be in this row.

00:03:41.754 --> 00:03:43.541
And it's now scrolled off,

00:03:43.541 --> 00:03:45.482
but this first column we saw,

00:03:45.482 --> 00:03:48.565
this is the hundredths place
and this zero hundredths.

00:03:48.565 --> 00:03:51.648
And so, 2.50 puts us right over here.

00:03:52.594 --> 00:03:53.894
Now, you might be tempted to say,

00:03:53.894 --> 00:03:57.122
okay, that's the proportion
that scores higher than Ludwig,

00:03:57.122 --> 00:03:58.028
but you'd be wrong.

00:03:58.028 --> 00:04:01.060
This is the proportion that
scores lower than Ludwig.

00:04:01.060 --> 00:04:04.645
So, what we wanna do is
take one minus this value.

00:04:04.645 --> 00:04:07.895
So, let me get my calculator out again.

00:04:08.749 --> 00:04:12.153
So, what I'm going to
do is I'm going to take

00:04:12.153 --> 00:04:13.403
one minus this.

00:04:15.199 --> 00:04:16.532
One minus 0.9938

00:04:20.843 --> 00:04:22.385
is equal to, now this is,

00:04:22.385 --> 00:04:25.476
so this is the proportion
that scores less than Ludwig.

00:04:25.476 --> 00:04:27.095
One minus that is gonna be the proportion

00:04:27.095 --> 00:04:28.593
that scores more than him.

00:04:28.593 --> 00:04:30.428
The reason why we have to do
this is because the z table

00:04:30.428 --> 00:04:34.061
gives us the proportion
less than a certain z score.

00:04:34.061 --> 00:04:37.561
So, this gives us right over here, 0.0062.

00:04:38.577 --> 00:04:39.773
So, that's the proportion.

00:04:39.773 --> 00:04:40.818
If you thought of it in percent,

00:04:40.818 --> 00:04:45.674
it would be 0.62% scores
higher than Ludwig.

00:04:45.674 --> 00:04:47.836
Now, that makes sense 'cause Ludwig scored

00:04:47.836 --> 00:04:49.174
over two standard deviations,

00:04:49.174 --> 00:04:52.669
two and a half standard
deviations above the mean.

00:04:52.669 --> 00:04:54.752
So, our answer is 0.0062.

00:04:56.741 --> 00:04:59.324
So, this is going to be 0.0062.

00:05:01.463 --> 00:05:03.031
That's the proportion of exam scores

00:05:03.031 --> 00:05:05.281
higher than Ludwig's score.


Standard normal table for proportion below

WEBVTT
Kind: captions
Language: en

00:00:00.280 --> 00:00:02.544
- [Instructor] A set of middle
school students' heights

00:00:02.544 --> 00:00:07.526
are normally distributed with
a mean of 150 centimesters

00:00:07.526 --> 00:00:11.502
and a standard deviation
of 20 centimeters.

00:00:11.502 --> 00:00:13.714
Darnell is a middle school school student

00:00:13.714 --> 00:00:16.631
with a height of 161.4 centimeters.

00:00:18.547 --> 00:00:20.899
What proportion of student heights

00:00:20.899 --> 00:00:23.566
are lower than Darnell's height?

00:00:24.460 --> 00:00:26.391
So, let's think about
what they are asking.

00:00:26.391 --> 00:00:30.300
So, they're saying that heights
are normally distributed.

00:00:30.300 --> 00:00:34.595
So, it would have a shape that
looks something like that.

00:00:34.595 --> 00:00:36.652
That's my hand-drawn version of it.

00:00:36.652 --> 00:00:40.819
There's a mean of 150
centimeters, so right over here.

00:00:42.196 --> 00:00:44.696
That would be 150 centimeters.

00:00:45.922 --> 00:00:46.945
They tell us that there's

00:00:46.945 --> 00:00:49.724
a standard deviation of 20 centimeters

00:00:49.724 --> 00:00:54.199
and Darnell has a height
of 161.4 centimeters.

00:00:54.199 --> 00:00:57.462
So, Darnell is above the mean,

00:00:57.462 --> 00:00:59.940
so let's say he is right over here,

00:00:59.940 --> 00:01:03.322
and I'm not drawing it
exactly, but you get the idea.

00:01:03.322 --> 00:01:05.489
That is 161.4 centimeters.

00:01:06.789 --> 00:01:09.430
And we wanna figure out what
proportion of students' heights

00:01:09.430 --> 00:01:11.763
are lower than Darnell's height.

00:01:11.763 --> 00:01:14.400
So, we wanna figure out what is the area

00:01:14.400 --> 00:01:17.296
under the normal curve right over here.

00:01:17.296 --> 00:01:18.658
That will give us the proportion

00:01:18.658 --> 00:01:21.371
that are lower than Darnell's height.

00:01:21.371 --> 00:01:23.927
And I'll give you a
hint on how to do this.

00:01:23.927 --> 00:01:26.215
We need to think about how
many standard deviations

00:01:26.215 --> 00:01:28.954
above the mean is Darnell,
and we can do that

00:01:28.954 --> 00:01:32.359
because they tell us what
the standard deviation is

00:01:32.359 --> 00:01:33.999
and we know the difference

00:01:33.999 --> 00:01:37.273
between Darnell's height
and the mean height,

00:01:37.273 --> 00:01:39.757
and then once we know how
many standard deviations

00:01:39.757 --> 00:01:43.529
he is above the mean, that's our z score,

00:01:43.529 --> 00:01:46.479
we can look at a z table
to tell us what proportion

00:01:46.479 --> 00:01:51.269
is less than that amount
in a normal distribution.

00:01:51.269 --> 00:01:53.387
So let's do that.

00:01:53.387 --> 00:01:57.684
So, I have my TI-84 Emulator
right over here and let's see.

00:01:57.684 --> 00:02:00.684
Darnell is 161.4 centimeters, 161.4.

00:02:05.028 --> 00:02:08.695
Now, the mean is 150,
minus 150 is equal to,

00:02:11.001 --> 00:02:14.148
we coulda done that in our
head, 11.4 centimeters.

00:02:14.148 --> 00:02:17.482
Now, how many standard deviations
is that above the mean?

00:02:17.482 --> 00:02:19.220
Well, they tell us that
the standard deviation

00:02:19.220 --> 00:02:22.478
in this case for this
distribution is 20 centimeters,

00:02:22.478 --> 00:02:25.543
so we'll take 11.4 divided by 20.

00:02:25.543 --> 00:02:28.085
So, we will just take our previous answer,

00:02:28.085 --> 00:02:29.473
so this just means our previous answer

00:02:29.473 --> 00:02:33.473
divided by 20 centimeters
and that gets us 0.57.

00:02:36.903 --> 00:02:39.486
So we can say that this is 0.57

00:02:42.600 --> 00:02:46.517
standard deviations,
deviations above the mean.

00:02:50.260 --> 00:02:51.809
Now, why is that useful?

00:02:51.809 --> 00:02:55.077
Well, you could take this
z score right over here

00:02:55.077 --> 00:02:57.485
and look at a z table to
figure out what proportion

00:02:57.485 --> 00:03:01.652
is less than 0.57 standard
deviations above the mean.

00:03:03.509 --> 00:03:06.384
So, let's get a z table over here.

00:03:06.384 --> 00:03:07.391
So, what we're going to do is

00:03:07.391 --> 00:03:10.601
we're gonna look up this
z score on this table,

00:03:10.601 --> 00:03:13.193
and the way that you do
it, this first column,

00:03:13.193 --> 00:03:17.360
each rows tells us our z score
up until the tenths place

00:03:18.492 --> 00:03:20.426
and then each of these columns after that

00:03:20.426 --> 00:03:22.260
tell us which hundredths we're in.

00:03:22.260 --> 00:03:26.093
So, 0.57, the tenths
place is right over here,

00:03:28.958 --> 00:03:31.708
so we're going to be in this row.

00:03:32.745 --> 00:03:36.391
And then our hundredths
place is this seven,

00:03:36.391 --> 00:03:38.175
so we'll look right over here.

00:03:38.175 --> 00:03:42.157
So, 0.57, this tells us the proportion

00:03:42.157 --> 00:03:46.324
that is lower than 0.57 standard
deviations above the mean,

00:03:48.556 --> 00:03:53.348
and so it is 0.7157 or
another way to think about

00:03:53.348 --> 00:03:56.557
is if the heights are
truly normally distributed,

00:03:56.557 --> 00:03:58.807
71.57% of the students

00:04:00.186 --> 00:04:03.625
would have a height less than Darnell's.

00:04:03.625 --> 00:04:05.264
But the answer to this question,

00:04:05.264 --> 00:04:06.705
what proportion of students' heights

00:04:06.705 --> 00:04:08.914
are lower than Darnell's height?

00:04:08.914 --> 00:04:11.164
Well, that would be 0.7157,

00:04:13.356 --> 00:04:15.755
and they want our answer
to four decimal places,

00:04:15.755 --> 00:04:18.672
which is exactly what we have done.