Introduction to inference about slope in linear regression

WEBVTT
Kind: captions
Language: en

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- [Instructor] In this
video, we're going to talk

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about regression lines,
but it's not gonna be

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the first time we're talking
about regression lines.

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And so if the idea of a
regression is foreign to you,

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I encourage you to watch the
introductory videos on it.

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Here we're gonna think about
how we can make inferences

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from a regression line.

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And so if the idea of statistical
inference is new to you

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or hypothesis testing, once again,

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watch those videos as well.

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But let's say we think
there's a positive association

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between shoe size

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and height.

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And so what we might
want to do is we could,

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here on the horizontal
axis, that is shoe size.

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Our sizes could go size
one, two, three, four, five,

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six, seven, eight, nine, 10, 11, 12,

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and it could keep going up from there.

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And then, on this height,

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or on this axis, our y-axis,

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this would be height,

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so one foot, two feet,
three feet, four feet,

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five feet, six feet, seven feet.

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And then you could,

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to see if there's an association,
you might take a sample.

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Let's say you take a random sample

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of 20 people from the population.

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And in future videos, we'll
talk about the conditions

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necessary for making
appropriate inferences.

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Let's say those 20 people
are these 20 data points.

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So there's a young child, and
maybe there's a grown adult,

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with bigger feet and who's taller.

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And then three, four, five,
six, seven, eight, nine,

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10, 11, 12,

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13, 14, 15,

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16, 17,

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18, 19, 20,

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and so you have these 20 data points.

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And then what you're likely to do

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is input them into a computer.

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You could do it by hand,

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but we have computers now
to do that for us usually.

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And the computer could try
to fit a regression line.

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And there's many techniques for doing it,

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but one typical technique is
to try to overall minimize

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the squared distance between
these points and that line.

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And this regression line
will have an equation,

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as any line would have.

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And we tend to show that as saying y hat,

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this hat tells us that
this is a regression line,

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is equal to the y-intercept,

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a plus the slope times our x variable.

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So this right over here would be a.

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Now, to be clear, if
you took another sample,

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you might get different results here.

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In fact, let's call this y
sub one for our first sample,

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a sub one, b sub one,
and this is a sub one.

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If you were to take
another sample of 20 folks,

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so let's do that.

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Maybe you get one, two, three,

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four, five, six, seven, eight,

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nine, 10, 11, 12,

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13, 14, 15, 16,

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17, 18, 19, 20

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and then you tried to fit a line to that,

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that line might look something like this.

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It might have a slightly
different y-intercept

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and a slightly different slope.

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So we could call that,
for the second sample,

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y sub two or y hat sub
two is equal to a sub two

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plus b sub two times x.

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And so every time you take a sample,

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you are likely to get different results

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for these values, which
are essentially statistics.

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Remember, statistics are things that we

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can get from samples, and
we're trying to estimate

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true population parameters.

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Well, what would be the
true population parameters

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we're trying to estimate?

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Well, imagine a world,

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imagine a world here,

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that you are able to find out
the true linear relationship,

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or maybe there is some
true linear relationship

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between shoe size and height.

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You could get it if
theoretically you could

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measure every human being on the planet.

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And depending what you
define as a population,

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it could be all living people

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or all people who will ever live.

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This isn't practical,

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but let's just say that
you actually could.

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And you would have billions
of data points here

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for the true population.

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And then if you were to fit
a regression line to that,

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you could view this as the true
population regression line.

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And so that would be y hat is equal to,

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and to make it clear that here,

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the y-intercept and the slope,

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this would be the true
population parameters.

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Instead of saying a, we say alpha.

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And instead of saying
b, we say beta times x.

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But it's very hard to come up exactly

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with what alpha and beta are,

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and so that's why we
estimate it with a's and b's

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based on a sample.

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Now, what's interesting,
with this in mind,

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is we can start to make
inferences based on our sample.

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So we know that, for example,

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b sub two is unlikely to be exactly beta.

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But how confident can we be

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that there is at least a
positive linear relationship

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or a nonzero linear relationship?

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Or can we create a confidence
interval around this statistic

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in order to have a good sense of where

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the true parameter might actually be?

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And the simple answer is yes.

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And to do so, we'll use
the same exact ideas

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that we did when we made
inferences based on proportions

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or based on means.

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The way that you can make
an inference, for example,

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for your true population
slope of your regression line,

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say, okay, I took a sample,

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I got this slope right over here,

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so I'll just call that b two,

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and then I could create a
confidence interval around that.

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And so that confidence interval

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is going to be based on
some critical value times,

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ideally, the standard deviation

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of the sampling distribution
of your sample statistic.

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In this case, it would be the
sample regression line slope.

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But because we don't know
exactly what this is,

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we can't figure out precisely
what this is going to be

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from a sample, we are going to estimate it

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with what's known as the
standard error of the statistic.

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And we'll go into more depth
in this in future videos.

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And since we're estimating here,

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we're going to use a
critical t-value here,

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which we have studied before.

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And so based on your confidence
level you want to have,

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let's say it's 95%,

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based on the degrees of
freedom, which we'll see

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will come out of how
many data points we have,

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we can figure this out.

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And from our sample,
we can figure this out,

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and we can figure this out.

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And then we would have
constructed a confidence interval.

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We'll also see that you could
do hypothesis testing here.

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You could say, hey, let's
set up a null hypothesis,

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and the null hypothesis is going to be

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that there is no nonzero
linear relationship

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or that the true population
slope of the regression line

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or slope of the population
regression line is equal to zero

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and that the alternative hypothesis

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is that the true
relationship could either be

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greater than zero, it's a
positive linear relationship,

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or that it's just nonzero.

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And then what you could
do is, assuming this,

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you could see what's the
probability of getting a statistic

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that is at least this
extreme or more extreme?

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And if that's below some threshold,

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you might reject the null hypothesis,

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which would suggest the alternative.

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So this and this are things
that we have done before

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where you're creating
a confidence interval

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around a statistic or you're
doing hypothesis testing,

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making assumptions about a true parameter.

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The only difference here
is that the parameter

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that we're trying to
estimate are going to be

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the parameters for a theoretical
population regression line,

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and we're going to do that
using sample statistics

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for a sample regression line.


Using a confidence interval to test slope

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:02.590
- [Instructor] Hashem obtained
a random sample of students

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and noticed a positive linear relationship

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between their ages and
their backpack weights.

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A 95% confidence interval for the slope

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of the regression line was
0.39 plus or minus 0.23.

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Hashem wants to use this interval to test

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the null hypothesis that the true slope

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of the population regression line,

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so this is a population
parameter right here

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for the slope of the
population regression line,

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is equal to zero versus
the alternative hypothesis

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is that the true slope of the
population regression line

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is not equal to zero at
the alpha is equal to 0.05

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level of significance.

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Assume that all conditions
for inference have been met.

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So given the information that we just have

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about what Hashem is doing,
what would be his conclusion?

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Would he reject the null hypothesis,

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which would suggest the alternative?

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Or would have be unable to
reject the null hypothesis?

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Well let's just think
about this a little bit.

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We have a 95% confidence interval,

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let me write this down,

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so our 95% confidence interval,

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we could write it like this

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or you could say that it
goes from 0.39 minus 0.23,

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so that'd be 0.16, it goes from 0.16

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until 0.39 plus 0.23

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is going to be 0.62.

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Now what a 95% confidence
interval tells us

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is that 95% of the time
that we take a sample

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and we construct a 95% confidence interval

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that 95% of the time we do this,

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it should overlap with the
true population parameter

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that we are trying to estimate.

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But in this hypothesis test, remember,

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we're assuming that the
true population parameter

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is equal to zero and that does not overlap

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with this confidence interval.

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So assuming, let me write this down,

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assuming null hypothesis is true,

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we are in the less than or equal

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to 5% chance of situations

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where beta not overlap

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with 95% intervals.

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And the whole notion of
hypothesis testing is

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you assume the null hypothesis,

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you take your sample, and
then if you get statistics,

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if the probability of
getting those statistics

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for something even more
extreme than those statistics

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is less than your significance level,

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then you reject the null hypothesis

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and that's exactly what's happening here.

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This null hypothesis value
is nowhere even close

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to overlapping, it's over 0.16ths

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below the low end of this bound.

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And so because of that, we
would reject the null hypothesis

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which suggests the alternative,

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which suggests the alternative hypothesis

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and one way to interpret
this alternative hypothesis

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that beta is not equal to zero

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is that there is

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a non zero linear relationship

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between ages and backpack weights.

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And we are done.


Supply and demand curves in foreign exchange

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.380
- [Instructor] In a previous video,

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we've given an intuition

00:00:02.700 --> 00:00:05.640
on what foreign exchange
markets are all about.

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In particular, we talked about
the foreign exchange market

00:00:08.640 --> 00:00:12.740
between the U.S. dollar
and the Chinese yuan.

00:00:12.740 --> 00:00:14.470
What we're going to do
in this video is think

00:00:14.470 --> 00:00:17.730
about the same idea, but think
about it in terms of graphs

00:00:17.730 --> 00:00:20.800
and the types of economic
models that we're used to seeing

00:00:20.800 --> 00:00:23.930
in an introductory macroeconomics course.

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So what we're going to
focus on in this video

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is the foreign exchange market.

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Foreign exchange

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market

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for the Chinese

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yuan.

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Now, we're going to think about it

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in terms of supply and demand curves.

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It can be a little bit confusing

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because we're gonna be thinking
of the price of the yuan

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in terms of another currency,
in this case the dollar,

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although you could do it in
terms of other currencies,

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the pound or the euro or whatever else.

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Now, this can be a little bit confusing

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because we're going to be thinking

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about currency on both axes.

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But let's first think
about the horizontal axis

00:01:03.920 --> 00:01:05.620
that when we're thinking
about most markets,

00:01:05.620 --> 00:01:07.490
that is our quantity axis.

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And here once we're going
to think about quantity.

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We're gonna think about the quantity.

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Quantity of Chinese yuan.

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And then our vertical axis,

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we're essentially going to
be thinking about the price

00:01:22.370 --> 00:01:24.000
of the Chinese yuan.

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But how do you think about
the price of a currency?

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Well, we're going to think of it

00:01:27.720 --> 00:01:29.520
in terms of another currency.

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And for the sake of this video,

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that other currency is
going to be the U.S. dollar.

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So this is going to be U.S. dollars

00:01:40.230 --> 00:01:42.940
per Chinese yuan.

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And I encourage you, pause this video.

00:01:46.830 --> 00:01:50.110
Think deeply about why it's
U.S. dollars per Chinese yuan,

00:01:50.110 --> 00:01:52.900
as opposed to Chinese
yuan per U.S. dollars.

00:01:52.900 --> 00:01:56.075
And think about why I put the
quantity of Chinese yuan here

00:01:56.075 --> 00:01:57.760
instead of the quantity of U.S. dollars,

00:01:57.760 --> 00:02:00.300
because this is the
foreign exchange market

00:02:00.300 --> 00:02:02.320
for the Chinese yuan.

00:02:02.320 --> 00:02:03.410
I could have done another chart

00:02:03.410 --> 00:02:06.130
where it's the foreign exchange
market for the U.S. dollar,

00:02:06.130 --> 00:02:09.030
in which case then my
quantity would be U.S. dollar.

00:02:09.030 --> 00:02:10.850
And then I would think of how much

00:02:10.850 --> 00:02:13.000
of some other currency per U.S. dollar.

00:02:13.000 --> 00:02:16.270
So I would say maybe how much
Chinese yuan per U.S. dollar?

00:02:16.270 --> 00:02:17.610
But here it's the other way around.

00:02:17.610 --> 00:02:20.570
I'm in the market for the Chinese yuan.

00:02:20.570 --> 00:02:22.730
So let's think about the
supply and demand curves

00:02:22.730 --> 00:02:24.770
and which way they would work.

00:02:24.770 --> 00:02:29.270
Well, imagine that people are
offering very few U.S. dollars

00:02:29.270 --> 00:02:30.570
per Chinese yuan.

00:02:30.570 --> 00:02:33.070
Well, in that world, a lot of
people might not wanna convert

00:02:33.070 --> 00:02:34.010
their yuan into dollars.

00:02:34.010 --> 00:02:37.080
They might not offer them up for supply

00:02:37.080 --> 00:02:39.710
to be converted into U.S. dollars.

00:02:39.710 --> 00:02:42.230
The quantity of Chinese yuan,

00:02:42.230 --> 00:02:44.750
if the price for the Chinese yuan is low,

00:02:44.750 --> 00:02:46.050
might be pretty low.

00:02:46.050 --> 00:02:48.280
And as the price people are willing to pay

00:02:48.280 --> 00:02:50.440
in terms of dollars goes up,

00:02:50.440 --> 00:02:53.980
well, more and more people
might be willing to transact.

00:02:53.980 --> 00:02:57.300
So our supply curve, and here
we're talking about the supply

00:02:57.300 --> 00:03:01.200
for Chinese yuan, is likely to increase

00:03:01.200 --> 00:03:04.190
as people are willing to
pay more for those yuans.

00:03:04.190 --> 00:03:06.260
And this is like many markets
that we've seen before.

00:03:06.260 --> 00:03:08.400
It's just a little bit less intuitive

00:03:08.400 --> 00:03:11.020
because we're thinking about
markets for one currency

00:03:11.020 --> 00:03:13.080
in terms of another currency.

00:03:13.080 --> 00:03:15.480
Now, what about the demand curve?

00:03:15.480 --> 00:03:16.650
Well, the demand curve is gonna look

00:03:16.650 --> 00:03:18.780
like a lot of demand curves we've seen.

00:03:18.780 --> 00:03:21.700
If the price of a Chinese yuan is high,

00:03:21.700 --> 00:03:23.830
well, very few people
are going to demand it.

00:03:23.830 --> 00:03:25.670
And as the price of the Chinese yuan

00:03:25.670 --> 00:03:28.040
in terms of dollars is lower and lower,

00:03:28.040 --> 00:03:31.850
more and more people might
demand more Chinese yuan,

00:03:31.850 --> 00:03:34.510
go like, "Hey, it's cheaper
now in terms of U.S. dollars."

00:03:34.510 --> 00:03:37.160
So this is what a demand
curve might look like.

00:03:37.160 --> 00:03:38.610
And as you can imagine,

00:03:38.610 --> 00:03:41.060
this point is our equilibrium point,

00:03:41.060 --> 00:03:44.350
and it would tell us our
equilibrium exchange rate.

00:03:44.350 --> 00:03:47.320
We could call that our
equilibrium exchange rate,

00:03:47.320 --> 00:03:52.320
and this would be our
equilibrium quantity.

00:03:52.830 --> 00:03:56.940
So, for example, let's say
that our equilibrium quantity,

00:03:56.940 --> 00:03:59.720
and let's say this is the
quantity that changes hands

00:03:59.720 --> 00:04:02.450
in some time period, so let's say per day.

00:04:02.450 --> 00:04:07.450
Let's say or equilibrium
quantity is equal to 1,000 yuan.

00:04:08.220 --> 00:04:11.113
1,000 yuan.

00:04:12.280 --> 00:04:14.150
And let me just call this Q sub one,

00:04:14.150 --> 00:04:15.230
is 1,000 yuan.

00:04:15.230 --> 00:04:16.200
These numbers are very low.

00:04:16.200 --> 00:04:18.620
Real exchange markets, we
might be talking about billions

00:04:18.620 --> 00:04:19.940
or tens or hundreds of billions

00:04:19.940 --> 00:04:22.810
or even sometimes trillions
of various currency.

00:04:22.810 --> 00:04:25.270
But let's just say for
argument it's 1,000 yuan

00:04:25.270 --> 00:04:29.410
is our current equilibrium
exchange quantity per day.

00:04:29.410 --> 00:04:31.970
And let's say this
exchange rate, e sub one,

00:04:31.970 --> 00:04:34.800
is equal to 10 cents per yuan.

00:04:34.800 --> 00:04:38.780
So 10 cents,

00:04:38.780 --> 00:04:43.610
or 1/10 of a U.S. dollar per Chinese yuan.

00:04:43.610 --> 00:04:45.630
So that's our current exchange rate.

00:04:45.630 --> 00:04:47.310
Now, let's say for some reason,

00:04:47.310 --> 00:04:51.940
all of a sudden Americans
become increasingly interested

00:04:51.940 --> 00:04:53.960
in converting their currency.

00:04:53.960 --> 00:04:55.870
Maybe they want to invest in China.

00:04:55.870 --> 00:04:57.677
Maybe all of a sudden the Chinese say,

00:04:57.677 --> 00:04:59.880
"Hey, Americans, come
buy property in China."

00:04:59.880 --> 00:05:01.700
A lot of people are interested.

00:05:01.700 --> 00:05:03.410
Well, what would happen here?

00:05:03.410 --> 00:05:05.640
Well, then the demand
for yuan would increase

00:05:05.640 --> 00:05:08.930
because you could only buy that
property in China with yuan,

00:05:08.930 --> 00:05:11.090
not with U.S. dollars.

00:05:11.090 --> 00:05:12.320
What would happen here?

00:05:12.320 --> 00:05:14.560
Well, your demand curve
would shift to the right

00:05:14.560 --> 00:05:16.140
like we've seen before.

00:05:16.140 --> 00:05:17.920
If we call this D one,

00:05:17.920 --> 00:05:20.160
then we could get to a new demand curve

00:05:20.160 --> 00:05:24.670
that might look something
like this, D two.

00:05:24.670 --> 00:05:26.000
Now, what would happen

00:05:26.000 --> 00:05:29.900
if our equilibrium exchange
rate doesn't change?

00:05:29.900 --> 00:05:31.600
Well, if this is our exchange rate,

00:05:31.600 --> 00:05:34.070
if this were to stay our exchange rate,

00:05:34.070 --> 00:05:37.670
now all of a sudden a higher
quantity is being demanded

00:05:37.670 --> 00:05:39.430
than is being supplied.

00:05:39.430 --> 00:05:40.780
The Americans in this situation,

00:05:40.780 --> 00:05:42.420
or it actually doesn't
even have to be Americans.

00:05:42.420 --> 00:05:44.660
It could just be whoever's
holding U.S. dollars,

00:05:44.660 --> 00:05:47.810
there's demand for more
than 1,000 yuan per day.

00:05:47.810 --> 00:05:51.123
Maybe this is 1,500 yuan
or whatever it might be.

00:05:52.110 --> 00:05:55.500
What you would naturally see
is that price of the yuan

00:05:55.500 --> 00:05:58.420
in terms of U.S. dollars will go up

00:05:58.420 --> 00:06:00.260
until you get to an equilibrium point.

00:06:00.260 --> 00:06:02.830
And on the first video when
we talk about the intuition

00:06:02.830 --> 00:06:04.180
of foreign exchange markets,

00:06:04.180 --> 00:06:06.310
we talk about why this would be.

00:06:06.310 --> 00:06:09.050
So you would then get
to a new equilibrium,

00:06:09.050 --> 00:06:11.760
right over here, this is e sub two,

00:06:11.760 --> 00:06:14.580
and a new equilibrium quantity.

00:06:14.580 --> 00:06:16.820
Let's call this Q two.

00:06:16.820 --> 00:06:19.360
Our new equilibrium quantity, Q two,

00:06:19.360 --> 00:06:24.060
might be 1,200 yuan per day
versus 1,000 yuan per day.

00:06:24.060 --> 00:06:26.370
And our new equilibrium exchange rate,

00:06:26.370 --> 00:06:31.070
maybe this is now equal
to 15 cents per yuan

00:06:31.070 --> 00:06:33.860
instead of 10 cents per yuan.

00:06:33.860 --> 00:06:37.150
So big picture, you can think
of the foreign exchange market

00:06:37.150 --> 00:06:38.270
in a lot of ways like we've looked

00:06:38.270 --> 00:06:41.163
at other markets in macroeconomics.

00:06:42.598 --> 00:06:43.870
It takes a little bit of an intuitive leap

00:06:43.870 --> 00:06:46.340
to just think about the
market for one currency

00:06:46.340 --> 00:06:47.693
in terms of another.


Ideologies of political parties in the United States

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:01.460
- [Instructor] What we're
going to do in this video

00:00:01.460 --> 00:00:04.280
is talk about the two major ideologies

00:00:04.280 --> 00:00:06.190
you will hear about in the United States,

00:00:06.190 --> 00:00:08.220
and that's the Liberal ideology

00:00:08.220 --> 00:00:10.500
and the Conservative ideology.

00:00:10.500 --> 00:00:13.270
And the Liberal ideology
is often associated

00:00:13.270 --> 00:00:17.380
with the Democratic party,
and the Conservative ideology

00:00:17.380 --> 00:00:20.260
is often associated with the Republicans.

00:00:20.260 --> 00:00:21.670
But be very careful.

00:00:21.670 --> 00:00:23.580
This does not mean that every Democrat

00:00:23.580 --> 00:00:25.700
will have a stereotypically Liberal view

00:00:25.700 --> 00:00:28.470
on every major issue,
or that every Republican

00:00:28.470 --> 00:00:30.510
would have a stereotypically
Conservative view

00:00:30.510 --> 00:00:31.410
on every issue.

00:00:31.410 --> 00:00:33.860
We're all individuals,
and many of us will have

00:00:33.860 --> 00:00:35.680
different opinions on these issues,

00:00:35.680 --> 00:00:37.620
and might lean Conservative on some,

00:00:37.620 --> 00:00:40.163
and might lean Liberal on others.

00:00:41.000 --> 00:00:44.700
But just to get a sense of
what Liberal and Conservative

00:00:44.700 --> 00:00:46.920
tend to mean in the United States,

00:00:46.920 --> 00:00:48.970
on issues that you will hear about,

00:00:48.970 --> 00:00:50.740
let's go through this list of issues.

00:00:50.740 --> 00:00:52.280
And I encourage you to pause this video,

00:00:52.280 --> 00:00:54.450
and have your own go at what do you think

00:00:54.450 --> 00:00:56.720
would be the Liberal or
the Conservative viewpoint

00:00:56.720 --> 00:00:58.870
on each of these issues.

00:00:58.870 --> 00:01:01.300
So let's start with arguably
one of the most contentious

00:01:01.300 --> 00:01:05.090
issues in the United States,
and that's of abortion.

00:01:05.090 --> 00:01:08.820
A Liberal would view this as
a reproductive rights issue.

00:01:08.820 --> 00:01:11.500
They view this as the
right of a woman to choose

00:01:11.500 --> 00:01:13.220
what to do with her body.

00:01:13.220 --> 00:01:16.540
So they tend to be pro
the right of a woman

00:01:16.540 --> 00:01:18.860
to have an abortion, to
choose to have an abortion,

00:01:18.860 --> 00:01:22.453
and will self-label themselves pro-choice.

00:01:23.740 --> 00:01:25.820
Conservatives on the other hand,

00:01:25.820 --> 00:01:29.120
will often view this as a life issue,

00:01:29.120 --> 00:01:33.030
that they view the
developing fetus as a life,

00:01:33.030 --> 00:01:36.080
and like any life, has certain rights.

00:01:36.080 --> 00:01:38.700
And so they would typically
be against abortion,

00:01:38.700 --> 00:01:43.210
and would self-label themselves pro-life.

00:01:43.210 --> 00:01:46.170
Affirmative action is
another contentious issue.

00:01:46.170 --> 00:01:48.690
This is the idea, and we've
talked about it in other videos,

00:01:48.690 --> 00:01:51.540
that proactive measures should be taken

00:01:51.540 --> 00:01:54.810
in order to address wrongs of the past,

00:01:54.810 --> 00:01:57.660
or in order to address inequality today,

00:01:57.660 --> 00:01:59.210
or discrimination today.

00:01:59.210 --> 00:02:02.480
And it often takes form
in, can race be considered

00:02:02.480 --> 00:02:07.020
as a part of admissions
into, say, higher education?

00:02:07.020 --> 00:02:11.280
Liberals tend to be in
favor of affirmative action,

00:02:11.280 --> 00:02:13.763
while Conservatives tend to be against it.

00:02:15.180 --> 00:02:17.070
The Liberal point of view is yes,

00:02:17.070 --> 00:02:18.620
we have a very unequal society,

00:02:18.620 --> 00:02:20.320
there's a lot of discrimination,

00:02:20.320 --> 00:02:22.410
race should be considered a factor

00:02:22.410 --> 00:02:25.000
in order to ease that discrimination.

00:02:25.000 --> 00:02:27.030
A Conservative today might argue,

00:02:27.030 --> 00:02:30.380
wait a second, we wanna be
a racially blind society,

00:02:30.380 --> 00:02:32.833
and so we do not want race to be a factor.

00:02:33.920 --> 00:02:37.310
Guns is another very contentious issue.

00:02:37.310 --> 00:02:40.710
Liberals will often cite
guns as a major factor

00:02:40.710 --> 00:02:43.430
in a lot of the crime and
shootings in the United States,

00:02:43.430 --> 00:02:47.573
and they would tend to be pro gun control.

00:02:48.970 --> 00:02:50.530
Conservatives on the other hand,

00:02:50.530 --> 00:02:52.520
will often cite the Second Amendment,

00:02:52.520 --> 00:02:54.540
the right to bear arms,
and they would say,

00:02:54.540 --> 00:02:57.430
look, when we see mass shootings or crime,

00:02:57.430 --> 00:03:00.890
it's not a gun issue,
there's other issues at play.

00:03:00.890 --> 00:03:05.890
And so they would be more pro gun rights.

00:03:06.750 --> 00:03:09.340
They would generally
be against gun control,

00:03:09.340 --> 00:03:11.770
and want more rights for gun owners.

00:03:11.770 --> 00:03:14.270
On a related note, on crime,

00:03:14.270 --> 00:03:18.100
Liberals tend to view
it as a social issue.

00:03:18.100 --> 00:03:19.670
What are the underlying causes,

00:03:19.670 --> 00:03:23.120
say, poverty or something
else that is causing crime?

00:03:23.120 --> 00:03:26.610
They would also be very concerned
with defendants' rights,

00:03:26.610 --> 00:03:29.670
citing examples where
certain ethnic groups,

00:03:29.670 --> 00:03:32.790
certain races might be
disproportionately accused,

00:03:32.790 --> 00:03:35.640
or disproportionately punished.

00:03:35.640 --> 00:03:37.150
Conservatives on the other hand,

00:03:37.150 --> 00:03:41.023
tend to favor tougher
policing and tougher laws,

00:03:42.390 --> 00:03:44.090
saying that okay, it's interesting

00:03:44.090 --> 00:03:45.210
to look at the social issues,

00:03:45.210 --> 00:03:47.050
and yes we do need certain rights,

00:03:47.050 --> 00:03:47.940
but at the end of the day,

00:03:47.940 --> 00:03:49.960
you don't wanna be too easy on crime.

00:03:49.960 --> 00:03:54.440
So tougher policing, and tougher laws.

00:03:55.820 --> 00:03:57.800
Moving into the economic sphere,

00:03:57.800 --> 00:04:01.200
the Liberal view on taxes
is generally that it's okay,

00:04:01.200 --> 00:04:02.730
and some Liberals will even view it

00:04:02.730 --> 00:04:05.130
as a tool for building social equality.

00:04:05.130 --> 00:04:05.963
They would say hey look,

00:04:05.963 --> 00:04:07.220
if someone's gotta pay for something,

00:04:07.220 --> 00:04:08.550
maybe it should be the rich.

00:04:08.550 --> 00:04:09.970
They can afford to pay more,

00:04:09.970 --> 00:04:12.410
and it'll help level the
playing field a little bit.

00:04:12.410 --> 00:04:16.700
And so okay, especially on, and
I'll put the rich in quotes,

00:04:16.700 --> 00:04:18.910
because different people
would have different standards

00:04:18.910 --> 00:04:20.730
for what it means to be rich.

00:04:20.730 --> 00:04:24.333
And Conservatives generally
are not okay with taxes.

00:04:25.300 --> 00:04:28.500
They would say that it is
a disincentive to work.

00:04:28.500 --> 00:04:31.030
They would say that it's
a friction on the economy.

00:04:31.030 --> 00:04:33.400
They would say that it
would reduce investment

00:04:33.400 --> 00:04:35.380
that could create jobs.

00:04:35.380 --> 00:04:37.240
The Liberal and Conservative
views on spending

00:04:37.240 --> 00:04:38.970
is related to that on taxes.

00:04:38.970 --> 00:04:40.940
A stereotypical view of Liberals

00:04:40.940 --> 00:04:43.760
is that, hey they might like
to tax, especially on the rich,

00:04:43.760 --> 00:04:46.500
and then spend it, especially
on social programs.

00:04:46.500 --> 00:04:51.153
Okay, especially on social programs.

00:04:52.210 --> 00:04:55.170
Maybe welfare system, or free healthcare,

00:04:55.170 --> 00:04:57.420
these would all be things
that a Liberal point of view

00:04:57.420 --> 00:04:59.600
would more likely to support.

00:04:59.600 --> 00:05:02.680
A Conservative viewpoint is not okay

00:05:02.680 --> 00:05:03.710
with government spending,

00:05:03.710 --> 00:05:05.890
that the government should
be as small as possible,

00:05:05.890 --> 00:05:07.890
and spend as little as possible.

00:05:07.890 --> 00:05:10.980
Perhaps the only exception,
or one of the few exceptions,

00:05:10.980 --> 00:05:13.630
would be except on military spending.

00:05:13.630 --> 00:05:18.630
Except on military, where a
Conservative might be in favor

00:05:19.120 --> 00:05:21.150
of more spending there in order to have

00:05:21.150 --> 00:05:23.820
a strong and muscular military.

00:05:23.820 --> 00:05:26.880
Now related to this idea
on the size of government,

00:05:26.880 --> 00:05:28.470
is that of regulation.

00:05:28.470 --> 00:05:32.390
Liberals tend to be in
favor of regulation,

00:05:32.390 --> 00:05:34.260
especially if they're in regards

00:05:34.260 --> 00:05:37.030
to things like preserving the environment.

00:05:37.030 --> 00:05:40.490
Conservatives tend to
be against regulation,

00:05:40.490 --> 00:05:43.540
saying that okay, maybe some
baseline regulation is okay,

00:05:43.540 --> 00:05:44.880
but it needs to be minimal,

00:05:44.880 --> 00:05:46.710
because the more regulations you have,

00:05:46.710 --> 00:05:48.830
once again, these are
frictions on the economy.

00:05:48.830 --> 00:05:50.570
It makes it harder to start a business,

00:05:50.570 --> 00:05:52.070
it makes it harder to grow,

00:05:52.070 --> 00:05:54.280
it makes it more expensive to do things.

00:05:54.280 --> 00:05:56.313
So they tend to be against regulation.

00:05:57.330 --> 00:05:59.220
Last but not least, and
this is just a survey

00:05:59.220 --> 00:06:01.050
of some of the big issues
you will hear about

00:06:01.050 --> 00:06:03.060
in the United States, is the military.

00:06:03.060 --> 00:06:04.490
And I already touched on it.

00:06:04.490 --> 00:06:06.900
A Liberal is likely to want a military,

00:06:06.900 --> 00:06:08.860
but they would probably want something

00:06:08.860 --> 00:06:11.210
that does just the base services

00:06:11.210 --> 00:06:12.630
of what you would expect from a military.

00:06:12.630 --> 00:06:14.480
If people are invading our borders,

00:06:14.480 --> 00:06:17.200
that the military is there to protect us.

00:06:17.200 --> 00:06:22.200
So maybe we could call it basic military.

00:06:22.510 --> 00:06:23.900
While a Conservative would say,

00:06:23.900 --> 00:06:25.240
hey, you can't get complacent.

00:06:25.240 --> 00:06:26.780
Even if we're in a time of peace,

00:06:26.780 --> 00:06:30.010
even if people haven't crossed
our borders in a long time,

00:06:30.010 --> 00:06:31.940
you don't know what's around the corner.

00:06:31.940 --> 00:06:33.480
Look at history, there's all sorts

00:06:33.480 --> 00:06:36.630
of complacent societies
that eventually get overrun.

00:06:36.630 --> 00:06:41.110
And so they would want a
strong and muscular military.

00:06:41.110 --> 00:06:42.370
And they'll make the argument

00:06:42.370 --> 00:06:44.730
that the stronger your military is,

00:06:44.730 --> 00:06:47.360
perhaps maybe the less likely
that you would have to use it,

00:06:47.360 --> 00:06:49.900
because people would not
even want to mess with you.

00:06:49.900 --> 00:06:51.910
They wouldn't even wanna
think about messing with you

00:06:51.910 --> 00:06:55.070
if you have a strong and muscular posture.

00:06:55.070 --> 00:06:57.490
So I'll leave you there
on these general ideas,

00:06:57.490 --> 00:07:01.020
and once again, these
are stereotypical views.

00:07:01.020 --> 00:07:03.360
You will meet many people,
including yourself,

00:07:03.360 --> 00:07:05.220
you might have a mix of these views.

00:07:05.220 --> 00:07:07.430
For example, it would not be too difficult

00:07:07.430 --> 00:07:11.000
to find a Democrat who for
example might be pro-choice,

00:07:11.000 --> 00:07:12.820
but against affirmative action,

00:07:12.820 --> 00:07:16.240
who is pro gun control,
who is maybe someplace

00:07:16.240 --> 00:07:18.470
in the middle here on crime,

00:07:18.470 --> 00:07:23.110
who is economically Conservative,
who's not okay with taxes,

00:07:23.110 --> 00:07:25.050
who wants them to be
as little as possible,

00:07:25.050 --> 00:07:27.540
who is not okay with government spending,

00:07:27.540 --> 00:07:30.280
who's maybe middle of
the road on regulation,

00:07:30.280 --> 00:07:33.350
and maybe they like a
strong muscular military.

00:07:33.350 --> 00:07:35.340
And likewise, you could find Republicans

00:07:35.340 --> 00:07:37.963
who similarly have a mix of viewpoints.


Using a P-value to make conclusions in a test about slope

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.705
- [Narrator] "Alicia took a
random sample of mobile phones"

00:00:02.705 --> 00:00:05.012
"and found a positive linear relationship"

00:00:05.012 --> 00:00:08.733
"between their processor
speeds and their prices."

00:00:08.733 --> 00:00:11.359
"Here is computer output
from a least-squares"

00:00:11.359 --> 00:00:14.309
"regression analysis on her sample."

00:00:14.309 --> 00:00:16.344
So just to be clear what's going on,

00:00:16.344 --> 00:00:18.722
she took a sample of phones,
they're not telling us

00:00:18.722 --> 00:00:22.290
exactly how many, but she
took a number of phones

00:00:22.290 --> 00:00:24.071
and she found a linear relationship

00:00:24.071 --> 00:00:26.198
between processor speed and prices.

00:00:26.198 --> 00:00:30.463
So this is price right over
here, and this is processor

00:00:30.463 --> 00:00:33.107
speed right over here.

00:00:33.107 --> 00:00:36.021
And then she plotted her sample.

00:00:36.021 --> 00:00:38.129
For every phone would be a data point,

00:00:38.129 --> 00:00:40.676
and so you see that, and then
she put those data points

00:00:40.676 --> 00:00:42.996
into her computer, and
it was able to come up

00:00:42.996 --> 00:00:47.218
with a line, a regression
line, for her sample.

00:00:47.218 --> 00:00:49.373
And her regression line for her sample,

00:00:49.373 --> 00:00:51.831
if we say that's going to be y is,

00:00:51.831 --> 00:00:55.919
or y hat is going to be a plus bx,

00:00:55.919 --> 00:01:00.919
for her sample, a is going to be 127.092,

00:01:01.025 --> 00:01:02.323
so that's that over there,

00:01:02.323 --> 00:01:05.401
and for her sample, the
slope of the regression line

00:01:05.401 --> 00:01:07.658
is going to be the coefficient on speed.

00:01:07.658 --> 00:01:08.911
Another way to think about it,

00:01:08.911 --> 00:01:10.556
this x variable right over here is speed,

00:01:10.556 --> 00:01:13.552
so the coefficient on that is the slope.

00:01:13.552 --> 00:01:15.406
But we have to remind ourselves

00:01:15.406 --> 00:01:17.453
that these are estimates of maybe

00:01:17.453 --> 00:01:20.073
some true truth in the universe.

00:01:20.073 --> 00:01:22.226
If she were able to sample every phone

00:01:22.226 --> 00:01:24.298
in the market, then she would get

00:01:24.298 --> 00:01:25.959
the true population parameters,

00:01:25.959 --> 00:01:28.227
but since this is a sample,
it's just an estimate.

00:01:28.227 --> 00:01:30.682
And she, just because she sees this

00:01:30.682 --> 00:01:32.865
positive linear relationship in her sample

00:01:32.865 --> 00:01:34.987
doesn't necessarily mean that this is

00:01:34.987 --> 00:01:37.395
the case for the entire population.

00:01:37.395 --> 00:01:39.984
She might have just
happened to sample things

00:01:39.984 --> 00:01:42.161
that had this positive
linear relationship.

00:01:42.161 --> 00:01:44.829
And so that's why she's
doing this hypothesis test.

00:01:44.829 --> 00:01:47.155
And in a hypothesis
test, you actually assume

00:01:47.155 --> 00:01:49.490
that there isn't a relationship

00:01:49.490 --> 00:01:51.674
between processor speed and price.

00:01:51.674 --> 00:01:54.103
So Beta right over here, this would be

00:01:54.103 --> 00:01:56.253
the true population parameter for

00:01:56.253 --> 00:01:58.177
regression on the population.

00:01:58.177 --> 00:02:00.705
So if this is the
population right over here,

00:02:00.705 --> 00:02:03.782
and if somehow, where it's
price on the vertical axis,

00:02:03.782 --> 00:02:07.684
and processor speed on
the horizontal axis,

00:02:07.684 --> 00:02:11.159
and if you were able to look
at the entire population,

00:02:11.159 --> 00:02:12.353
I don't know how many phones there are,

00:02:12.353 --> 00:02:13.809
but it might be billions of phone,

00:02:13.809 --> 00:02:16.033
and then do a regression line,

00:02:16.033 --> 00:02:18.370
then our null hypothesis
is that the slope of

00:02:18.370 --> 00:02:21.829
the regression line is going to be zero.

00:02:21.829 --> 00:02:24.149
So the regression line might
look something like that,

00:02:24.149 --> 00:02:26.223
where the equation of the regression line

00:02:26.223 --> 00:02:28.873
for the population, y hat would be

00:02:28.873 --> 00:02:33.873
Alpha plus Beta times, times x.

00:02:34.022 --> 00:02:36.546
And so our null hypothesis
is that Beta's equal to zero,

00:02:36.546 --> 00:02:38.099
and the alternative hypothesis,

00:02:38.099 --> 00:02:41.896
which is her suspicion,
is that the true slope

00:02:41.896 --> 00:02:45.164
of the regression line is
actually greater than zero.

00:02:45.164 --> 00:02:48.561
"Assume that all conditions
for inference have been met."

00:02:48.561 --> 00:02:53.561
"At the Alpha equals 0.01
level of significance,"

00:02:53.813 --> 00:02:56.772
"is there sufficient evidence to conclude"

00:02:56.772 --> 00:03:01.145
"a positive linear relationship
between these variables"

00:03:01.145 --> 00:03:02.620
"for all mobile phones?"

00:03:02.620 --> 00:03:03.499
"Why?"

00:03:03.499 --> 00:03:06.554
So pause this video and see
if you can have a go at it.

00:03:06.554 --> 00:03:08.556
Well, in order to do this hypothesis task,

00:03:08.556 --> 00:03:12.325
we have to say, "Well,
assuming the null hypothesis"

00:03:12.325 --> 00:03:15.890
"is true, assuming this is
the actual slope of the"

00:03:15.890 --> 00:03:18.246
"population regression line,"
I guess you could think

00:03:18.246 --> 00:03:21.781
about it, "what is the
probability of us getting"

00:03:21.781 --> 00:03:23.751
"this result right over here?"

00:03:23.751 --> 00:03:26.949
And what we can do is use this information

00:03:26.949 --> 00:03:30.268
and our estimate of the
sampling distribution

00:03:30.268 --> 00:03:33.437
of the sample regression line slope,

00:03:33.437 --> 00:03:36.986
and we can come up with a T-statistic.

00:03:36.986 --> 00:03:39.956
And for this situation where
our alternative hypothesis

00:03:39.956 --> 00:03:44.497
is that our true
population regression slope

00:03:44.497 --> 00:03:47.247
is greater than zero, our P-value

00:03:48.237 --> 00:03:52.902
can be viewed as the probability
of getting a T-statistic

00:03:52.902 --> 00:03:54.654
greater than or equal to this.

00:03:54.654 --> 00:03:56.825
So getting a T-statistic greater than

00:03:56.825 --> 00:04:00.612
or equal to 2.999.

00:04:00.612 --> 00:04:02.515
Now, you could be tempted
to say, "Hey, look"

00:04:02.515 --> 00:04:05.115
"there's this column
that gives us a P-value,"

00:04:05.115 --> 00:04:06.739
"maybe they just figured out for us"

00:04:06.739 --> 00:04:10.074
"that this probability is 0.004."

00:04:10.074 --> 00:04:12.049
And we have to be very, very careful here,

00:04:12.049 --> 00:04:13.902
because here, they're actually giving us,

00:04:13.902 --> 00:04:16.534
I guess you could call
it a two-sided P-value.

00:04:16.534 --> 00:04:19.883
If you think of a T-distribution,

00:04:19.883 --> 00:04:23.231
and they would do it for the
appropriate degrees of freedom,

00:04:23.231 --> 00:04:25.058
this is saying, "What's the probability"

00:04:25.058 --> 00:04:26.557
"of getting a result where the"

00:04:26.557 --> 00:04:30.582
"absolute value is 2.999 or greater?"

00:04:30.582 --> 00:04:34.061
So if this is T equals zero
right here in the middle,

00:04:34.061 --> 00:04:39.061
and this is 2.999, we
care about this region.

00:04:39.558 --> 00:04:41.235
We care about this right tail.

00:04:41.235 --> 00:04:43.920
This P-value right over
here, this is giving us

00:04:43.920 --> 00:04:46.019
not just the right tail,
but it's also saying,

00:04:46.019 --> 00:04:48.636
"Well, what about getting
something less than"

00:04:48.636 --> 00:04:52.040
"negative 2.999, or
including negative 2.99?"

00:04:52.040 --> 00:04:54.500
So it's giving us both of these areas,

00:04:54.500 --> 00:04:57.686
so if you want the
P-value for this scenario,

00:04:57.686 --> 00:04:59.395
we would just look at this.

00:04:59.395 --> 00:05:01.248
And as you can see,
because this distribution

00:05:01.248 --> 00:05:03.120
is symmetric, the
T-distribution is going to be

00:05:03.120 --> 00:05:05.714
symmetric, you take half of this.

00:05:05.714 --> 00:05:10.125
So this is going to be equal to 0.002.

00:05:10.125 --> 00:05:12.257
And what you'd do in any significance test

00:05:12.257 --> 00:05:16.249
is then compare your P-value
to your level of significance.

00:05:16.249 --> 00:05:20.357
And so if you look at 0.002 and compare it

00:05:20.357 --> 00:05:25.274
to 0.01, which of these is greater?

00:05:26.488 --> 00:05:27.756
Well, at first your eyes might say,

00:05:27.756 --> 00:05:28.795
"Hey, two is greater than one,"

00:05:28.795 --> 00:05:32.303
but this is two thousandths
versus one hundredth.

00:05:32.303 --> 00:05:35.263
This is 10 thousandths right over here.

00:05:35.263 --> 00:05:37.806
So in this situation, our P-value is less

00:05:37.806 --> 00:05:41.463
than our level of significance,
and so we're saying,

00:05:41.463 --> 00:05:44.893
"Hey, the probability of getting a result"

00:05:44.893 --> 00:05:48.647
"this extreme or more extreme is so low,"

00:05:48.647 --> 00:05:51.105
"if we assume our null hypothesis,"

00:05:51.105 --> 00:05:54.127
"that in this situation we will reject."

00:05:54.127 --> 00:05:56.329
"We will decide to reject
our null hypothesis,"

00:05:56.329 --> 00:05:58.486
"which would suggest the alternative."

00:05:58.486 --> 00:06:00.222
So, "Is there sufficient
evidence to conclude"

00:06:00.222 --> 00:06:01.731
"a positive linear relationship"

00:06:01.731 --> 00:06:04.011
"between these variables
for all mobile phones?"

00:06:04.011 --> 00:06:05.214
Yes.

00:06:05.214 --> 00:06:06.312
"Why?"

00:06:06.312 --> 00:06:11.312
Because, because P-value

00:06:11.380 --> 00:06:14.879
is less than our significance level,

00:06:14.879 --> 00:06:18.962
and so we reject our null hypothesis,

00:06:18.962 --> 00:06:23.629
which suggests, suggests
our alternative hypothesis.


Calculating t statistic for slope of regression line

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.440
- [Instructor] Jian obtained
a random sample of data

00:00:02.440 --> 00:00:05.180
on how long it took each of 24 students

00:00:05.180 --> 00:00:07.460
to complete a timed reaction game

00:00:07.460 --> 00:00:09.160
and a timed memory game.

00:00:09.160 --> 00:00:12.170
He noticed a positive linear relationship

00:00:12.170 --> 00:00:14.250
between the times on each task.

00:00:14.250 --> 00:00:17.470
Here is a computer output
on the sample data.

00:00:17.470 --> 00:00:20.940
So, we have some statistics calculated

00:00:20.940 --> 00:00:23.800
on the reaction time, on the memory time.

00:00:23.800 --> 00:00:27.500
And then he had his
computer do a regression

00:00:27.500 --> 00:00:29.520
for the data that he collected.

00:00:29.520 --> 00:00:31.790
And then we're told
assume that all conditions

00:00:31.790 --> 00:00:33.480
for inference have been met.

00:00:33.480 --> 00:00:36.430
Calculate the test statistic
that should be used

00:00:36.430 --> 00:00:38.910
for testing a null hypothesis

00:00:38.910 --> 00:00:42.280
that the population
slope is actually zero.

00:00:42.280 --> 00:00:45.930
So pause this video and have a go at it.

00:00:45.930 --> 00:00:48.110
All right, so let's just
make sure we understand

00:00:48.110 --> 00:00:49.770
what is going on.

00:00:49.770 --> 00:00:52.590
So let's first think about the population.

00:00:52.590 --> 00:00:54.870
So I'll do that right over here.

00:00:54.870 --> 00:00:56.420
So in the population,

00:00:56.420 --> 00:00:59.660
there might be some true
linear relationships.

00:00:59.660 --> 00:01:01.700
So in theory,

00:01:01.700 --> 00:01:04.500
on our x-axis, we would
have our reaction time

00:01:04.500 --> 00:01:07.740
and on our y-axis, you
have your memory time.

00:01:07.740 --> 00:01:11.830
If you were able to plot every
single possible data point,

00:01:11.830 --> 00:01:13.370
it might even be an infinite,

00:01:13.370 --> 00:01:14.280
or near infinite, (laughs)

00:01:14.280 --> 00:01:15.570
so it would be very hard to do it.

00:01:15.570 --> 00:01:16.403
But if you were,

00:01:16.403 --> 00:01:17.870
if there was just some
truth in the universe,

00:01:17.870 --> 00:01:19.960
it says yes, there actually is

00:01:19.960 --> 00:01:21.940
a positive linear relationship.

00:01:21.940 --> 00:01:23.620
And it looks like this.

00:01:23.620 --> 00:01:26.630
And you could describe that
regression line as y hat.

00:01:26.630 --> 00:01:27.910
It's a regression line.

00:01:27.910 --> 00:01:31.260
Is equal to some true population paramater

00:01:31.260 --> 00:01:32.870
which would be this y intercept.

00:01:32.870 --> 00:01:35.330
So we could call that alpha

00:01:35.330 --> 00:01:37.890
plus some true population parameter

00:01:37.890 --> 00:01:39.650
that would be the slope
of this regression line

00:01:39.650 --> 00:01:41.330
we could call that beta.

00:01:41.330 --> 00:01:43.260
Times x.

00:01:43.260 --> 00:01:46.300
Now we don't know what this
truth of the universe is

00:01:46.300 --> 00:01:47.500
of the linear relationship

00:01:47.500 --> 00:01:49.630
between reaction time and memory time.

00:01:49.630 --> 00:01:51.040
But we can try to estimate it.

00:01:51.040 --> 00:01:53.240
And that's what Jian is trying to do.

00:01:53.240 --> 00:01:56.010
So he's taking a sample of 24,

00:01:56.010 --> 00:01:57.200
so samples

00:01:58.170 --> 00:02:01.630
samples 24 data.

00:02:01.630 --> 00:02:03.400
Data points.

00:02:03.400 --> 00:02:05.180
And that's much easier to then,

00:02:05.180 --> 00:02:09.540
you can even visualize it
on a scatter plot like this.

00:02:09.540 --> 00:02:11.280
So you'd have one, two,
three, four, five, six, seven,

00:02:11.280 --> 00:02:12.558
eight, nine, 10, 11, 12, 13,

00:02:12.558 --> 00:02:16.930
14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24.

00:02:16.930 --> 00:02:19.080
You input those data points into computer

00:02:19.080 --> 00:02:20.180
and it does a regression line.

00:02:20.180 --> 00:02:22.060
And it's trying to minimize
the squared distance

00:02:22.060 --> 00:02:23.190
to all of these points.

00:02:23.190 --> 00:02:25.890
And so let's say it gets a regression line

00:02:25.890 --> 00:02:28.400
that looks something like this.

00:02:28.400 --> 00:02:30.290
Where this regression line

00:02:30.290 --> 00:02:34.730
can be described as some
estimate of the true y intercept.

00:02:34.730 --> 00:02:36.780
So this would actually be a
statistic right over here.

00:02:36.780 --> 00:02:39.040
That's estimating this parameter.

00:02:39.040 --> 00:02:42.760
Plus some estimate of the true
slope of the regression line.

00:02:42.760 --> 00:02:44.230
So this is just a statistic,

00:02:44.230 --> 00:02:46.300
this b, is just a statistic

00:02:46.300 --> 00:02:50.270
that is trying to estimate
the true parameter, beta.

00:02:50.270 --> 00:02:52.030
Now when we went

00:02:52.030 --> 00:02:54.110
and inputted these data
points into a computer,

00:02:54.110 --> 00:02:56.890
we got values for a and b right over here.

00:02:56.890 --> 00:02:59.410
A is equal to this, the
constant coefficient.

00:02:59.410 --> 00:03:01.370
And then the reaction coefficient,

00:03:01.370 --> 00:03:02.203
this is just telling us,

00:03:02.203 --> 00:03:04.430
hey, for every incremental
change in the reaction,

00:03:04.430 --> 00:03:07.180
how much would we expect
the memory time to change.

00:03:07.180 --> 00:03:08.720
Or for every change in x,

00:03:08.720 --> 00:03:10.400
how much would we expect
for a change in y.

00:03:10.400 --> 00:03:12.390
So this is actually our estimate

00:03:12.390 --> 00:03:15.060
of the slope of the regression line.

00:03:15.060 --> 00:03:16.010
Now you could imagine,

00:03:16.010 --> 00:03:17.510
every time you take a different sample,

00:03:17.510 --> 00:03:20.040
you might get a different
estimate of these things.

00:03:20.040 --> 00:03:22.510
And when we're doing
inferential statistics,

00:03:22.510 --> 00:03:23.900
we set up hypotheses.

00:03:23.900 --> 00:03:26.430
You set up a null and an
alternative hypothesis.

00:03:26.430 --> 00:03:29.230
And then null hypothesis
is always the no news here.

00:03:29.230 --> 00:03:31.270
And no news, when you're
dealing with regressions

00:03:31.270 --> 00:03:33.160
is that even though you might suspect

00:03:33.160 --> 00:03:34.490
there's a positive linear relationship,

00:03:34.490 --> 00:03:37.090
even though you might see
it in the data you got,

00:03:37.090 --> 00:03:38.820
it's for your null hypothesis,

00:03:38.820 --> 00:03:39.653
you wanna assume

00:03:39.653 --> 00:03:41.430
that there is no positive
linear relationship.

00:03:41.430 --> 00:03:43.840
So our null hypothesis here

00:03:43.840 --> 00:03:47.600
would be that the true slope
of the true regression line,

00:03:47.600 --> 00:03:49.850
this, the parameter right over here,

00:03:49.850 --> 00:03:51.690
is equal to zero.

00:03:51.690 --> 00:03:53.490
So beta is equal to zero.

00:03:53.490 --> 00:03:56.220
So our null hypothesis actually might be

00:03:56.220 --> 00:03:57.500
that our true regression line

00:03:57.500 --> 00:03:59.760
might look something like this.

00:03:59.760 --> 00:04:04.060
That what y is, is somewhat
independent of what x is.

00:04:04.060 --> 00:04:06.000
And that if you suspect

00:04:06.000 --> 00:04:08.890
that there is a positive
linear relationship,

00:04:08.890 --> 00:04:10.210
you could say something like,

00:04:10.210 --> 00:04:12.050
well, my alternative hypothesis

00:04:12.050 --> 00:04:13.840
is that my beta is greater than zero.

00:04:13.840 --> 00:04:14.673
Or if you suspect

00:04:14.673 --> 00:04:15.850
that there's just some
linear relationship,

00:04:15.850 --> 00:04:17.460
you don't know if it's
positive or negative,

00:04:17.460 --> 00:04:19.910
then you might say that the
beta is not equal to zero.

00:04:19.910 --> 00:04:22.020
But here it says he noticed or he suspects

00:04:22.020 --> 00:04:23.480
a positive linear relationship.

00:04:23.480 --> 00:04:25.950
So this would be his
alternative hypothesis.

00:04:25.950 --> 00:04:27.940
But what you then want to do

00:04:27.940 --> 00:04:30.030
to test your null hypothesis,

00:04:30.030 --> 00:04:32.390
which we've done multiple, multiple times,

00:04:32.390 --> 00:04:35.430
is find a test statistic

00:04:35.430 --> 00:04:39.020
that is associated with
the statistic for b

00:04:39.020 --> 00:04:41.540
that you actually got.

00:04:41.540 --> 00:04:45.250
Now ideally, you would take your b,

00:04:45.250 --> 00:04:47.270
you would take your b,

00:04:47.270 --> 00:04:48.330
and from that,

00:04:48.330 --> 00:04:50.420
subtract the

00:04:50.420 --> 00:04:52.750
slope assumed in the null hypothesis,

00:04:52.750 --> 00:04:54.340
so the slope of the
regression line you get

00:04:54.340 --> 00:04:55.675
minus the slope

00:04:55.675 --> 00:04:58.170
that's assumed from the null hypothesis.

00:04:58.170 --> 00:05:01.520
And then divide by the standard deviation

00:05:01.520 --> 00:05:03.670
of the sampling distribution

00:05:03.670 --> 00:05:05.610
of the slope of the regression line.

00:05:05.610 --> 00:05:07.050
And if you did this,

00:05:07.050 --> 00:05:08.010
you would get a,

00:05:08.010 --> 00:05:10.850
it would be appropriate to use

00:05:10.850 --> 00:05:13.360
a z statistic over here.

00:05:13.360 --> 00:05:14.700
Now, the problem is,

00:05:14.700 --> 00:05:17.490
is that we don't know exactly what

00:05:17.490 --> 00:05:20.440
the standard deviation of
the sampling distribution is.

00:05:20.440 --> 00:05:21.770
But we can estimate it.

00:05:21.770 --> 00:05:23.570
We can calculate

00:05:23.570 --> 00:05:27.730
the slope that we got for
our sample regression line

00:05:27.730 --> 00:05:31.180
minus the slope we're assuming
in our null hypothesis,

00:05:31.180 --> 00:05:32.430
which is going to be equal to zero,

00:05:32.430 --> 00:05:33.920
so we know what we're assuming.

00:05:33.920 --> 00:05:37.960
And we can calculate the standard error

00:05:37.960 --> 00:05:39.660
of the sampling distribution.

00:05:39.660 --> 00:05:41.980
And in fact, our computer
has already done it for us.

00:05:41.980 --> 00:05:45.050
And this is an estimate of this.

00:05:45.050 --> 00:05:46.550
And we know what number that is.

00:05:46.550 --> 00:05:48.500
So we know what all of these numbers are,

00:05:48.500 --> 00:05:50.340
but if you're using an estimate

00:05:50.340 --> 00:05:52.630
of the standard deviation of
the sampling distribution,

00:05:52.630 --> 00:05:53.680
we've seen this before.

00:05:53.680 --> 00:05:55.810
When we've done inferential
statistics using,

00:05:55.810 --> 00:06:00.810
for means, it is appropriate
to use a t statistic.

00:06:00.810 --> 00:06:02.160
But with that said, pause the video.

00:06:02.160 --> 00:06:04.400
What is this going to be equal to?

00:06:04.400 --> 00:06:06.130
Well, this is going to be equal to

00:06:06.130 --> 00:06:08.460
the slope for our sample regression line,

00:06:08.460 --> 00:06:12.640
we know it's 14.686

00:06:12.640 --> 00:06:15.600
minus our assumed true
population parameter,

00:06:15.600 --> 00:06:17.820
the slope of the true regression line,

00:06:17.820 --> 00:06:19.630
well, we're assuming that it's zero.

00:06:19.630 --> 00:06:21.880
So minus zero.

00:06:21.880 --> 00:06:26.140
And then we divide that
by the standard error

00:06:26.140 --> 00:06:27.140
which is going to be,

00:06:27.140 --> 00:06:29.170
we can use this as a standard error for b.

00:06:29.170 --> 00:06:34.170
And so this is divided by 13.329.

00:06:34.667 --> 00:06:39.080
So it's just going to be
14.686 divided by 13.329.

00:06:39.080 --> 00:06:40.100
And if we assume,

00:06:40.100 --> 00:06:42.320
if we're doing a one sided test here,

00:06:42.320 --> 00:06:44.500
what we would then do
is take this t statistic

00:06:44.500 --> 00:06:46.460
and think about the degrees of freedom.

00:06:46.460 --> 00:06:48.660
And then say, and then
calculate a p value.

00:06:48.660 --> 00:06:51.180
What is the probability
of getting a result

00:06:51.180 --> 00:06:55.010
at least this far above
t is equal to zero,

00:06:55.010 --> 00:06:58.040
or what is the probability
of getting a t statistic

00:06:58.040 --> 00:07:00.030
this high or higher

00:07:00.030 --> 00:07:01.330
and that would be our p value.

00:07:01.330 --> 00:07:03.270
And if that's below some threshold,

00:07:03.270 --> 00:07:04.440
that's hey, that's pretty unlikely.

00:07:04.440 --> 00:07:06.600
Then we would reject the null and

00:07:06.600 --> 00:07:08.230
that which would suggest the alternative.

00:07:08.230 --> 00:07:10.010
But they're not asking
us to do all of that.

00:07:10.010 --> 00:07:12.070
They're just asking us to calculate

00:07:12.070 --> 00:07:14.010
an appropriate test statistic,

00:07:14.010 --> 00:07:15.210
which we have just done.


Brown v. Board of Education of Topeka

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:02.350
- [Kim] Hi, this is Kim from Khan Academy,

00:00:02.350 --> 00:00:03.750
and today we're learning more

00:00:03.750 --> 00:00:07.100
about Brown v. Board
of Education of Topeka.

00:00:07.100 --> 00:00:12.100
Decided in 1954, Brown v.
Board was a landmark case

00:00:12.140 --> 00:00:14.310
that opened the door for desegregation

00:00:14.310 --> 00:00:16.420
and the Modern Civil Rights Movement.

00:00:16.420 --> 00:00:18.470
In Brown, the Supreme Court ruled

00:00:18.470 --> 00:00:21.900
that segregated schools for
white and black children,

00:00:21.900 --> 00:00:24.630
which had been prevalent
throughout the American South

00:00:24.630 --> 00:00:28.040
since the 1896 decision
in Plessy v. Ferguson

00:00:28.040 --> 00:00:32.370
legalized segregation, were
in fact inherently unequal.

00:00:32.370 --> 00:00:35.290
The named plaintiff in
this case, Oliver Brown,

00:00:35.290 --> 00:00:37.440
was the father of Linda, a third-grader

00:00:37.440 --> 00:00:40.440
who had to take a bus to a
segregated elementary school

00:00:40.440 --> 00:00:42.350
that was much farther from her home

00:00:42.350 --> 00:00:44.520
than the nearby school for white children.

00:00:44.520 --> 00:00:46.510
To learn more about the Brown case,

00:00:46.510 --> 00:00:48.510
I sought out the help of two experts.

00:00:48.510 --> 00:00:51.430
Theodore Shaw is the Julius L. Chambers

00:00:51.430 --> 00:00:53.230
Distinguished Professor of Law,

00:00:53.230 --> 00:00:55.350
and Director of the
Center for Civil Rights

00:00:55.350 --> 00:00:58.100
at the University of North
Carolina School of Law.

00:00:58.100 --> 00:00:59.950
Michael McConnell is the Richard

00:00:59.950 --> 00:01:01.860
and Frances Mallery Professor,

00:01:01.860 --> 00:01:04.550
and Director of the
Constitutional Law Center

00:01:04.550 --> 00:01:06.910
at Stanford Law School,
and a Senior Fellow

00:01:06.910 --> 00:01:08.590
at the Hoover Institution.

00:01:08.590 --> 00:01:12.010
So Prof. Shaw, could you just
kinda set the stage for us?

00:01:12.010 --> 00:01:15.110
What was the overall social
and political context

00:01:15.110 --> 00:01:16.190
behind this case?

00:01:16.190 --> 00:01:19.630
- [Prof. Shaw] Well Brown was
decided in the Cold War era.

00:01:19.630 --> 00:01:23.110
I think that's important for
reasons that I'll come back to.

00:01:23.110 --> 00:01:27.040
But it also was decided,
I guess it was 58 years

00:01:27.040 --> 00:01:31.730
after the Supreme Court
ruled in Plessy v. Ferguson

00:01:31.730 --> 00:01:36.330
that separate but equal railway
cars were constitutional.

00:01:36.330 --> 00:01:41.330
Of course that extended to all
walks of life in the South.

00:01:42.400 --> 00:01:46.600
And so what you had was
Jim Crow segregation

00:01:46.600 --> 00:01:51.330
sustained and set in place by law.

00:01:51.330 --> 00:01:55.870
And Brown was the case
that cracked the edifice

00:01:55.870 --> 00:01:58.430
of Jim Crow segregation.

00:01:58.430 --> 00:02:02.530
So, it was an enormously important case.

00:02:02.530 --> 00:02:05.030
It came after a long campaign

00:02:05.030 --> 00:02:09.837
by the NAACP Legal Defense
Fund to overturn Plessy.

00:02:09.837 --> 00:02:12.210
- [Kim] One thing that I
think is really confusing

00:02:12.210 --> 00:02:15.550
about Plessy v. Ferguson
is how it interacts

00:02:15.550 --> 00:02:17.050
with the 14th Amendment.

00:02:17.050 --> 00:02:19.800
So immediately after the Civil War,

00:02:19.800 --> 00:02:24.210
Congress passed, and the states
ratified, the 14th Amendment

00:02:24.210 --> 00:02:29.070
which gives equal protection
under the law to all citizens.

00:02:29.070 --> 00:02:33.280
But then Plessy v. Ferguson
just a couple decades later,

00:02:33.280 --> 00:02:35.750
legalizes segregation.

00:02:35.750 --> 00:02:37.120
Can you talk a little bit more

00:02:37.120 --> 00:02:41.500
about how the equal protection clause

00:02:41.500 --> 00:02:45.052
of the 14th Amendment has
been interpreted in Plessy,

00:02:45.052 --> 00:02:47.530
and then later in Brown v. Board?

00:02:47.530 --> 00:02:50.260
- [Prof. McConnell] The
doctrine of separate but equal

00:02:50.260 --> 00:02:55.260
was itself a departure from
the prior understanding

00:02:55.730 --> 00:02:58.420
of what the 14th Amendment meant.

00:02:58.420 --> 00:03:01.860
In the immediate wake
of the 14th Amendment,

00:03:01.860 --> 00:03:06.300
segregation was understood
to be a violation

00:03:06.300 --> 00:03:07.573
of equal rights.

00:03:08.654 --> 00:03:10.340
The Congress of the United States

00:03:10.340 --> 00:03:13.600
passed the Civil Rights Act of 1875

00:03:13.600 --> 00:03:17.820
which prohibited mandatory segregation

00:03:19.680 --> 00:03:21.840
of among other things, railroads.

00:03:21.840 --> 00:03:25.660
Now that was struck down in 1883,

00:03:25.660 --> 00:03:29.040
I believe it was 1886, but for reasons

00:03:29.040 --> 00:03:33.070
not related to separate
but equal or segregation.

00:03:33.070 --> 00:03:37.680
But it's only in the late 1880s and 1890s

00:03:37.680 --> 00:03:41.270
that states began passing
Jim Crow legislation,

00:03:41.270 --> 00:03:44.820
like the law in Plessy v. Ferguson.

00:03:44.820 --> 00:03:49.210
I think Plessy very
likely would've come out

00:03:49.210 --> 00:03:53.430
a different way, had it
come up 25 years earlier.

00:03:53.430 --> 00:03:55.330
It's really very difficult to understand

00:03:55.330 --> 00:04:00.330
the logic of Plessy, because
Plessy takes the point of view

00:04:02.300 --> 00:04:07.300
that segregation is okay
because it treats everyone

00:04:07.380 --> 00:04:12.380
the same way, and the sense
of inferiority or insult

00:04:12.730 --> 00:04:17.730
that comes from segregation,
the court just ignored.

00:04:19.130 --> 00:04:21.210
You can read the opinion different ways,

00:04:21.210 --> 00:04:25.650
and people do and have read
the opinion different ways.

00:04:25.650 --> 00:04:27.870
A portion of the opinion makes it sound

00:04:27.870 --> 00:04:31.990
as though this is kind of
social science conclusion,

00:04:31.990 --> 00:04:36.990
that segregation breeds
a feeling of inferiority,

00:04:39.100 --> 00:04:41.710
that has a deleterious effect

00:04:41.710 --> 00:04:46.090
on the black students'
performance in school.

00:04:46.090 --> 00:04:48.050
- [Prof. Shaw] There's a famous footnote

00:04:48.050 --> 00:04:49.760
in Brown v. Board of Education,

00:04:49.760 --> 00:04:53.033
Footnote 11 in which
some social scientists,

00:04:54.019 --> 00:04:58.900
and particularly Dr. Kenneth Clark,

00:04:58.900 --> 00:05:02.120
did a series of tests, the doll tests,

00:05:02.120 --> 00:05:04.490
in which they showed black and white dolls

00:05:04.490 --> 00:05:08.770
who were otherwise identical
to little children,

00:05:08.770 --> 00:05:11.820
little black children, and they
asked who was the good doll,

00:05:11.820 --> 00:05:13.930
who was the bad doll,
who was the pretty doll,

00:05:13.930 --> 00:05:15.930
who was the ugly doll.

00:05:15.930 --> 00:05:19.320
And the black children
would invariably choose

00:05:19.320 --> 00:05:21.360
the white dolls as the better dolls,

00:05:21.360 --> 00:05:24.330
and the black dolls as the inferior dolls.

00:05:24.330 --> 00:05:27.740
And then finally the bombshell question,

00:05:27.740 --> 00:05:30.510
which one looks like you?

00:05:30.510 --> 00:05:35.510
And they would stop and
choose the black doll,

00:05:36.370 --> 00:05:39.610
and many times, tears
would come to their eyes.

00:05:39.610 --> 00:05:43.470
And so this old notion of the inferiority

00:05:43.470 --> 00:05:48.470
that segregation imbued
in black school children

00:05:48.830 --> 00:05:51.060
was important, and the Supreme Court

00:05:51.060 --> 00:05:53.830
had some very eloquent
language about that.

00:05:53.830 --> 00:05:56.600
- [Prof. McConnell] Everyone
knew that the only reason

00:05:56.600 --> 00:06:00.673
to have laws requiring black
and white to be separate,

00:06:01.690 --> 00:06:04.630
was because of the feeling

00:06:04.630 --> 00:06:07.100
on the part of the dominant white group,

00:06:07.100 --> 00:06:10.680
that the African race was inferior.

00:06:10.680 --> 00:06:14.100
I think it was a failure
of political will really,

00:06:14.100 --> 00:06:17.740
rather than a matter of
constitutional or legal logic

00:06:17.740 --> 00:06:19.080
that led to that.

00:06:19.080 --> 00:06:22.530
And then it took almost 60 years

00:06:22.530 --> 00:06:25.760
before the Supreme Court
would turn itself around.

00:06:25.760 --> 00:06:28.650
- [Prof. Shaw] The late
A. Leon Higginbotham

00:06:28.650 --> 00:06:31.010
who sat on the Court of Appeals

00:06:31.010 --> 00:06:33.790
for the Third Circuit in Philadelphia,

00:06:33.790 --> 00:06:36.863
he used to say that Plessy was
wrong the day it was decided.

00:06:37.770 --> 00:06:41.150
And I believe he was right about that.

00:06:41.150 --> 00:06:43.710
If you go back and look at the rationale

00:06:43.710 --> 00:06:46.593
stated in Plessy, it didn't hold up.

00:06:47.668 --> 00:06:49.983
It's not an intellectually
honest decision,

00:06:50.870 --> 00:06:55.870
and having said that,
if you look at Brown,

00:06:56.790 --> 00:07:01.170
it really did not overrule
Plessy on its face

00:07:02.810 --> 00:07:07.633
with respect to segregation
and Jim Crow across the board.

00:07:08.820 --> 00:07:12.690
What it did was say that in
the field of public education,

00:07:12.690 --> 00:07:15.460
separate but equal is unconstitutional.

00:07:15.460 --> 00:07:19.250
Separate schools cannot be equal.

00:07:19.250 --> 00:07:22.690
And what followed Brown
was a series of cases

00:07:22.690 --> 00:07:25.990
in all kinds of areas in which the court

00:07:25.990 --> 00:07:28.210
sometimes without much of an opinion,

00:07:28.210 --> 00:07:31.990
cited Brown, but struck down
segregation in public libraries

00:07:31.990 --> 00:07:36.630
and public accommodations,
and transportation, et cetera.

00:07:36.630 --> 00:07:41.630
The case that settled
the long-running boycott

00:07:43.560 --> 00:07:48.560
in Montgomery, Alabama, that
catapulted Martin Luther King

00:07:48.710 --> 00:07:51.603
into the national and world stage,

00:07:53.094 --> 00:07:54.930
it was decided, it was resolved finally.

00:07:54.930 --> 00:07:58.630
The boycott was important, but
it was resolved by this case

00:07:58.630 --> 00:08:02.093
that the Supreme Court
decided and had cited Brown.

00:08:02.093 --> 00:08:04.940
- [Kim] And so, Thurgood Marshall
and the Legal Defense Fund

00:08:04.940 --> 00:08:09.220
had kind of been chipping away
at the edges of segregation,

00:08:09.220 --> 00:08:12.420
and then they turned directly
to school segregation

00:08:12.420 --> 00:08:13.820
in the Brown cases.

00:08:13.820 --> 00:08:16.460
So what kind of arguments did they use

00:08:16.460 --> 00:08:19.180
to challenge the system
of school segregation?

00:08:19.180 --> 00:08:22.790
- [Prof. McConnell] Thurgood
Marshall and his co-council

00:08:22.790 --> 00:08:27.500
in the NAACP established
a very careful strategy

00:08:27.500 --> 00:08:30.330
of one case at a time, sort
of pushing the envelope

00:08:30.330 --> 00:08:34.780
of this idea that
segregation is inconsistent

00:08:34.780 --> 00:08:38.240
with demands of equal protection.

00:08:38.240 --> 00:08:43.240
And the interesting
thing about Brown itself,

00:08:43.370 --> 00:08:48.370
the case challenging
segregation in Topeka, Kansas,

00:08:48.440 --> 00:08:50.980
you know, why Topeka, Kansas?

00:08:50.980 --> 00:08:55.980
Why was this such an excellent
place to raise the challenge?

00:08:57.320 --> 00:09:02.320
And the reason is that
the schools in Topeka

00:09:03.230 --> 00:09:08.080
were very substantially equal in quality.

00:09:08.080 --> 00:09:11.020
In Topeka they actually
had integrated high schools

00:09:11.020 --> 00:09:11.900
and middle schools.

00:09:11.900 --> 00:09:16.100
It was only the elementary
schools that were segregated,

00:09:16.100 --> 00:09:19.160
and the District Court found,

00:09:19.160 --> 00:09:21.070
and the Supreme Court did not disagree

00:09:21.070 --> 00:09:25.380
that in terms of their
material advantages,

00:09:25.380 --> 00:09:28.140
the black school was
essentially just as good

00:09:28.140 --> 00:09:29.600
as the white school.

00:09:29.600 --> 00:09:34.150
And that's why the NAACP
wanted to go after Topeka,

00:09:34.150 --> 00:09:37.070
because then it went to
the principle of the thing.

00:09:37.070 --> 00:09:40.180
You know it's one thing
to attack segregation

00:09:40.180 --> 00:09:43.280
when separate but equal
is really just a fiction,

00:09:43.280 --> 00:09:46.110
but when separate but equal is a reality,

00:09:46.110 --> 00:09:49.280
at least pretty close to it, as in Topeka,

00:09:49.280 --> 00:09:51.760
you have to go after the actual principle

00:09:51.760 --> 00:09:54.780
that segregation is wrong in principle

00:09:54.780 --> 00:09:57.560
and not just wrong because
it tends most of the time

00:09:57.560 --> 00:10:00.900
to lead to material disadvantage.

00:10:00.900 --> 00:10:03.580
- [Prof. Shaw] Brown
was actually five cases.

00:10:03.580 --> 00:10:06.620
The Brown case itself
came from Topeka, Kansas.

00:10:06.620 --> 00:10:08.590
In many ways, the Supreme Court

00:10:08.590 --> 00:10:11.490
might've chosen to lead
with the Brown case

00:10:11.490 --> 00:10:13.530
because it wasn't part of the South,

00:10:13.530 --> 00:10:15.870
and so in some ways it
might've been viewed

00:10:15.870 --> 00:10:18.640
as taking pressure off of the South.

00:10:18.640 --> 00:10:22.740
But there were five cases altogether.

00:10:22.740 --> 00:10:26.080
There was a case out of South Carolina,

00:10:26.080 --> 00:10:29.780
a case out of Virginia,
a case out of Delaware,

00:10:29.780 --> 00:10:34.050
a case out of Washington DC,
and then the Topeka case.

00:10:34.050 --> 00:10:38.820
And the Brown family was
actually not even the only family

00:10:38.820 --> 00:10:41.370
named Brown in the Topeka case.

00:10:41.370 --> 00:10:46.370
And the other Brown case
did not have their father

00:10:48.770 --> 00:10:53.770
available as the lead voice,
and they wanted a man,

00:10:53.980 --> 00:10:57.080
and so Oliver Brown
was the one they chose.

00:10:57.080 --> 00:11:00.670
They stipulated that the
schools were in fact equal

00:11:00.670 --> 00:11:03.540
in all respects, and
of course they weren't.

00:11:03.540 --> 00:11:06.780
It was never possible to make
separate but equal, equal.

00:11:06.780 --> 00:11:10.040
But they stipulated that
these schools were equal,

00:11:10.040 --> 00:11:11.470
the black and the white schools,

00:11:11.470 --> 00:11:13.030
because they wanted the court

00:11:13.030 --> 00:11:18.030
to be able to, or to have
to, confront the issue

00:11:18.160 --> 00:11:22.490
of whether separate but equal
per se was unconstitutional.

00:11:22.490 --> 00:11:24.470
- [Kim] Interesting, so it's this place

00:11:24.470 --> 00:11:27.810
where they're hoping not to have a ruling

00:11:27.810 --> 00:11:31.280
that for example, oh the
schools for black children

00:11:31.280 --> 00:11:33.090
need to be brought up to code,

00:11:33.090 --> 00:11:34.620
to be the same as white children,

00:11:34.620 --> 00:11:36.470
but they're trying to strike down

00:11:36.470 --> 00:11:39.440
the idea of segregation altogether.

00:11:39.440 --> 00:11:42.360
And indeed they rule unanimously

00:11:42.360 --> 00:11:47.300
that separate but equal
facilities are inherently unequal.

00:11:47.300 --> 00:11:49.810
- [Prof. McConnell] The
Supreme Court in Brown

00:11:49.810 --> 00:11:54.330
says well whatever may
have been the status

00:11:54.330 --> 00:11:59.330
of education as a civil right back then,

00:11:59.370 --> 00:12:04.150
back in the 19th century, by
now it surely is a civil right,

00:12:04.150 --> 00:12:08.220
and therefore there can
be no discrimination

00:12:08.220 --> 00:12:10.051
with regard to it.

00:12:10.051 --> 00:12:11.850
And one other thing that changed

00:12:11.850 --> 00:12:14.410
that I think the Supreme
Court doesn't refer to,

00:12:14.410 --> 00:12:17.840
but many historians think
was surely in the back

00:12:17.840 --> 00:12:21.520
of their minds, has to do with
foreign policy of all things,

00:12:21.520 --> 00:12:24.830
that during this time the United States

00:12:24.830 --> 00:12:26.820
in the wake of World War II was playing

00:12:26.820 --> 00:12:31.810
a very major role in opposing colonialism,

00:12:31.810 --> 00:12:36.770
opposing communism, in
trying to spread the gospel

00:12:36.770 --> 00:12:39.610
of liberty around the world.

00:12:39.610 --> 00:12:44.610
And when we had segregation at home,

00:12:44.840 --> 00:12:47.170
this looked like rank hypocrisy.

00:12:47.170 --> 00:12:49.890
- [Prof. Shaw] This was
the Cold War period.

00:12:49.890 --> 00:12:54.660
These soldiers came home from
fighting a form of racism

00:12:54.660 --> 00:12:59.660
in Europe, Nazi Germany,
you know the Holocaust.

00:13:00.290 --> 00:13:02.420
And when they came back,

00:13:02.420 --> 00:13:04.563
they came back to segregated America.

00:13:05.440 --> 00:13:08.550
And there are many
stories that are painful

00:13:08.550 --> 00:13:11.250
about what their experiences were.

00:13:11.250 --> 00:13:12.700
But meanwhile the United States

00:13:12.700 --> 00:13:16.560
is having this Cold War
with the Soviet Union,

00:13:16.560 --> 00:13:21.520
and the Soviet Union was
exploiting this inconsistency

00:13:21.520 --> 00:13:24.430
in what America said it
was, and what it was.

00:13:24.430 --> 00:13:27.400
And though the Supreme Court
never said a word about that

00:13:27.400 --> 00:13:32.400
in the Brown cases, that
was very much context

00:13:32.510 --> 00:13:33.840
for that decision.

00:13:33.840 --> 00:13:36.970
- [Kim] So what did it
take for desegregation

00:13:36.970 --> 00:13:39.130
to really come about?

00:13:39.130 --> 00:13:41.990
Do you think we've achieved desegregation?

00:13:41.990 --> 00:13:43.290
- [Prof. Shaw] Yes and no.

00:13:44.200 --> 00:13:47.873
There was a long battle
to implement Brown,

00:13:48.710 --> 00:13:51.230
and in fact for the most part,

00:13:51.230 --> 00:13:56.170
Brown was not implemented
fully to the extent it ever was

00:13:56.170 --> 00:13:59.890
until the early 1970s when
the Supreme Court decided

00:13:59.890 --> 00:14:03.010
a case called Swann v.
Charlotte-Mecklenburg

00:14:03.010 --> 00:14:05.580
Board of Education in North Carolina,

00:14:05.580 --> 00:14:07.780
and sanctioned the use of busing.

00:14:07.780 --> 00:14:08.760
- [Prof. McConnell] This does not depend

00:14:08.760 --> 00:14:12.350
upon what social scientists might think

00:14:12.350 --> 00:14:14.440
about how people do in school.

00:14:14.440 --> 00:14:16.900
This is a much more fundamental principle

00:14:16.900 --> 00:14:19.300
of equality under the law,

00:14:19.300 --> 00:14:24.300
that our law must treat
black and white the same,

00:14:25.760 --> 00:14:28.640
that for the government to
draw racial distinctions

00:14:28.640 --> 00:14:33.450
is fundamentally unequal,
and unconstitutional,

00:14:33.450 --> 00:14:37.530
and wrong, and unjust, fundamentally so

00:14:37.530 --> 00:14:42.530
and not just as a matter
of social science evidence.

00:14:44.130 --> 00:14:47.360
- [Prof. Shaw] I don't
wanna sound despairing,

00:14:47.360 --> 00:14:50.600
you know, but I think if anybody looks

00:14:50.600 --> 00:14:55.600
at our public schools today,
and many of our communities,

00:14:56.940 --> 00:14:59.940
and our towns, and
cities, and our counties,

00:14:59.940 --> 00:15:02.060
we have to acknowledge honestly

00:15:02.060 --> 00:15:05.250
that there's still a
great deal of segregation,

00:15:05.250 --> 00:15:10.250
and there are many schools
that are identifiably white,

00:15:10.590 --> 00:15:12.320
or identifiably black.

00:15:12.320 --> 00:15:17.320
And now Latinos, Hispanics,

00:15:17.740 --> 00:15:20.380
large degree of segregation there also.

00:15:20.380 --> 00:15:24.220
And even voluntary desegregation efforts

00:15:24.220 --> 00:15:25.910
have been found by the Supreme Court

00:15:25.910 --> 00:15:30.250
a few years ago to be
largely unconstitutional

00:15:30.250 --> 00:15:32.053
and illegal, and that's tragic.

00:15:33.080 --> 00:15:36.300
But that doesn't mean that
Brown wasn't significant.

00:15:36.300 --> 00:15:39.750
I was born in the year of
Brown v. Board of Education.

00:15:39.750 --> 00:15:41.760
The America that I grew up in

00:15:41.760 --> 00:15:46.510
is a very different country
than the one that proceeded it.

00:15:46.510 --> 00:15:49.640
In spite of our flaws and our warts,

00:15:49.640 --> 00:15:51.830
no Brown v. Board of Education,

00:15:51.830 --> 00:15:55.560
no Civil Rights Movement, I would say.

00:15:55.560 --> 00:15:59.920
And ultimately no
desegregation of our society

00:15:59.920 --> 00:16:01.380
to the extent that we've accomplished it,

00:16:01.380 --> 00:16:05.210
and we have to a great
degree, and no Barack Obama.

00:16:05.210 --> 00:16:08.760
So I think Brown stands, as I said,

00:16:08.760 --> 00:16:13.720
as a dividing point in
America, but we can do better.

00:16:13.720 --> 00:16:15.620
- [Kim] So we've learned
that under the leadership

00:16:15.620 --> 00:16:19.340
of Thurgood Marshall, and
the NAACP Legal Defense Fund,

00:16:19.340 --> 00:16:22.070
the decision in Brown
v. Board of Education

00:16:22.070 --> 00:16:25.120
struck a major blow to
the system of segregation.

00:16:25.120 --> 00:16:28.400
After World War II, amidst
the ideological struggle

00:16:28.400 --> 00:16:31.280
of the Cold War, the
Supreme Court overturned

00:16:31.280 --> 00:16:33.870
the precedent set in Plessy v. Ferguson,

00:16:33.870 --> 00:16:35.870
and ruled that segregated education

00:16:35.870 --> 00:16:39.280
constituted a violation of
the equal protection clause

00:16:39.280 --> 00:16:40.780
of the 14th Amendment.

00:16:40.780 --> 00:16:43.210
To learn more about Brown v. Board,

00:16:43.210 --> 00:16:45.260
visit the National Constitution Center's

00:16:45.260 --> 00:16:48.380
interactive Constitution,
and Khan Academy's resources

00:16:48.380 --> 00:16:50.353
on US government and politics.


Confidence interval for the slope of a regression line

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.430
- [Instructor] Musa is
interested in the relationship

00:00:02.430 --> 00:00:06.210
between hours spent studying
and caffeine consumption

00:00:06.210 --> 00:00:07.870
among students at his school.

00:00:07.870 --> 00:00:10.800
He randomly selects 20
students at his school

00:00:10.800 --> 00:00:13.480
and records their caffeine
intake in milligrams

00:00:13.480 --> 00:00:16.610
and the amount of time
studying in a given week.

00:00:16.610 --> 00:00:18.250
Here is a computer output

00:00:18.250 --> 00:00:22.210
from a least-squares regression
analysis on his sample.

00:00:22.210 --> 00:00:25.320
Assume that all conditions
for inference have been met.

00:00:25.320 --> 00:00:28.020
What is the 95% confidence interval

00:00:28.020 --> 00:00:32.030
for the slope of the
least-squares regression line?

00:00:32.030 --> 00:00:34.160
So if you feel inspired, pause the video

00:00:34.160 --> 00:00:35.470
and see if you can have a go at it.

00:00:35.470 --> 00:00:38.010
Otherwise, we'll do this together.

00:00:38.010 --> 00:00:41.460
Okay, so let's first remind
ourselves what's even going on.

00:00:41.460 --> 00:00:44.530
So let's visualize the regression.

00:00:44.530 --> 00:00:47.940
So our horizontal axis, or our x-axis,

00:00:47.940 --> 00:00:50.720
that would be our caffeine

00:00:50.720 --> 00:00:51.553
intake

00:00:51.553 --> 00:00:52.540
in milligrams.

00:00:52.540 --> 00:00:56.060
And then our y-axis, or our vertical axis,

00:00:56.060 --> 00:00:57.610
that would be the,

00:00:57.610 --> 00:00:59.170
I would assume it's in hours.

00:00:59.170 --> 00:01:00.230
So time

00:01:01.160 --> 00:01:02.403
time studying.

00:01:03.870 --> 00:01:05.370
And Musa here,

00:01:05.370 --> 00:01:08.200
he randomly selects 20 students.

00:01:08.200 --> 00:01:09.730
And so for each of those students,

00:01:09.730 --> 00:01:11.340
he sees how much caffeine they consumed

00:01:11.340 --> 00:01:12.530
and how much time they spent studying

00:01:12.530 --> 00:01:13.660
and plots them here.

00:01:13.660 --> 00:01:15.060
And so there'll be 20 data points.

00:01:15.060 --> 00:01:19.031
One, two, three, four, five,
six, seven, eight, nine, 10,

00:01:19.031 --> 00:01:21.840
11, 12, 13, 14, 15,

00:01:21.840 --> 00:01:25.410
16, 17, 18, 19, 20.

00:01:25.410 --> 00:01:27.960
He inputs these data
points into a computer.

00:01:27.960 --> 00:01:31.260
In order to fit a
least-squares regression line.

00:01:31.260 --> 00:01:33.100
And let's say the
least-squares regression line

00:01:33.100 --> 00:01:35.550
looks something like this.

00:01:35.550 --> 00:01:37.120
And a least-squares regression line

00:01:37.120 --> 00:01:39.660
comes from trying to
minimize the square distance

00:01:39.660 --> 00:01:40.680
between the line

00:01:40.680 --> 00:01:42.750
and all of these points.

00:01:42.750 --> 00:01:44.740
And then this is giving us information

00:01:44.740 --> 00:01:46.720
on that least-squares regression line.

00:01:46.720 --> 00:01:48.430
And the most valuable things here,

00:01:48.430 --> 00:01:51.420
if we really wanna help
visualize or understand the line

00:01:51.420 --> 00:01:54.360
is what we get in this column.

00:01:54.360 --> 00:01:56.620
The constant coefficient
tells us essentially

00:01:56.620 --> 00:01:57.990
what is the y-intercept here.

00:01:57.990 --> 00:02:00.110
So 2.544.

00:02:00.110 --> 00:02:02.690
And then the coefficient on the caffeine,

00:02:02.690 --> 00:02:04.420
this is, one way of thinking about,

00:02:04.420 --> 00:02:06.630
well for every incremental
increase in caffeine,

00:02:06.630 --> 00:02:08.560
how much does the time studying increase?

00:02:08.560 --> 00:02:10.920
Or you might recognize this as the slope

00:02:10.920 --> 00:02:13.080
of the least-squares regression line.

00:02:13.080 --> 00:02:15.240
So this is

00:02:15.240 --> 00:02:16.530
the slope

00:02:16.530 --> 00:02:21.530
and this would be equal to 0.164.

00:02:22.260 --> 00:02:24.630
Now this information right over here,

00:02:24.630 --> 00:02:27.930
it tells us how well our
least-squares regression line

00:02:27.930 --> 00:02:29.270
fits the data.

00:02:29.270 --> 00:02:31.280
R-squared, you might
already be familiar with,

00:02:31.280 --> 00:02:35.500
it says how much of the
variance in the y variable

00:02:35.500 --> 00:02:38.180
is explainable by the x variable.

00:02:38.180 --> 00:02:40.450
If it was one or 100%,

00:02:40.450 --> 00:02:42.180
that means all of it could be explained.

00:02:42.180 --> 00:02:43.320
And it's a very good fit.

00:02:43.320 --> 00:02:45.310
If it was zero, that means
none of it can be explained,

00:02:45.310 --> 00:02:47.140
and it'd be a very bad fit.

00:02:47.140 --> 00:02:50.920
Capital S, this is the standard
deviation of the residuals.

00:02:50.920 --> 00:02:54.850
And it's another measure of
how much these data points vary

00:02:54.850 --> 00:02:57.240
from this regression line.

00:02:57.240 --> 00:02:59.130
Now this column right over here

00:02:59.130 --> 00:03:00.770
is going to prove to be useful

00:03:00.770 --> 00:03:03.340
for answering the question at hand.

00:03:03.340 --> 00:03:06.470
This gives us the standard
error of the coefficient.

00:03:06.470 --> 00:03:08.590
And the coefficient that
we really care about,

00:03:08.590 --> 00:03:10.460
the statistic that we really care about

00:03:10.460 --> 00:03:12.840
is the slope of the regression line.

00:03:12.840 --> 00:03:15.160
And this gives us the standard error

00:03:15.160 --> 00:03:17.390
for the slope of the regression line.

00:03:17.390 --> 00:03:19.890
You could view this as the estimate

00:03:19.890 --> 00:03:23.830
of the standard deviation
of the sampling distribution

00:03:23.830 --> 00:03:25.780
of the slope of the regression line.

00:03:25.780 --> 00:03:28.840
Remember, we took a
sample of 20 folks here,

00:03:28.840 --> 00:03:30.320
and we calculated a statistic

00:03:30.320 --> 00:03:32.620
which is the slope of the regression line.

00:03:32.620 --> 00:03:34.580
Every time you do a different sample,

00:03:34.580 --> 00:03:37.350
you will likely get a different slope.

00:03:37.350 --> 00:03:39.620
And this slope is an estimate

00:03:39.620 --> 00:03:42.300
of some true parameter in the population.

00:03:42.300 --> 00:03:44.780
This would sometimes also
be called the standard error

00:03:44.780 --> 00:03:47.980
of the slope of the
least-squares regression line.

00:03:47.980 --> 00:03:50.310
Now these last two columns,
you don't have to worry about

00:03:50.310 --> 00:03:52.540
in the context of this video.

00:03:52.540 --> 00:03:54.880
This is useful if you were saying

00:03:54.880 --> 00:03:57.480
well, assuming that
there is no relationship

00:03:57.480 --> 00:04:00.430
between caffeine intake and time studying,

00:04:00.430 --> 00:04:02.140
what is the associated T statistic

00:04:02.140 --> 00:04:05.030
for the statistics that
I actually calculated

00:04:05.030 --> 00:04:06.610
and what would be the probability

00:04:06.610 --> 00:04:09.980
of getting something that
extreme or more extreme

00:04:09.980 --> 00:04:12.460
assuming that there is no association.

00:04:12.460 --> 00:04:13.700
Assuming that for example,

00:04:13.700 --> 00:04:16.600
the actual slope of the
regression line is zero.

00:04:16.600 --> 00:04:18.120
And this says, well the probability,

00:04:18.120 --> 00:04:20.660
if we would assume that,
is actually quite low.

00:04:20.660 --> 00:04:22.410
It's about a 1% chance

00:04:22.410 --> 00:04:24.600
that you would've gotten these results

00:04:24.600 --> 00:04:27.210
if there truly was not a relationship

00:04:27.210 --> 00:04:29.770
between caffeine intake and time studying.

00:04:29.770 --> 00:04:31.210
But with all of that out of the way,

00:04:31.210 --> 00:04:33.420
let's actually answer the question.

00:04:33.420 --> 00:04:37.330
Well, to construct a confidence
interval around a statistic,

00:04:37.330 --> 00:04:39.130
you would take the value of the statistic

00:04:39.130 --> 00:04:41.120
that you calculated from your sample.

00:04:41.120 --> 00:04:42.740
So 0.164

00:04:44.510 --> 00:04:48.360
and then it would be plus
or minus a critical t value

00:04:48.360 --> 00:04:50.290
and then this would be driven by the fact

00:04:50.290 --> 00:04:53.130
that you care about a
95% confidence interval

00:04:53.130 --> 00:04:54.770
and by the degrees of freedom,

00:04:54.770 --> 00:04:56.760
and I'll talk about that in a second.

00:04:56.760 --> 00:04:58.230
And then you would multiply that

00:04:58.230 --> 00:05:00.900
times the standard error of the statistic.

00:05:00.900 --> 00:05:03.130
And in this case, the
statistic that we care about

00:05:03.130 --> 00:05:04.790
is the slope.

00:05:04.790 --> 00:05:07.190
And so this is 0.057.

00:05:07.190 --> 00:05:09.680
Times 0.057.

00:05:09.680 --> 00:05:11.550
And the reason why we're
using a critical t value

00:05:11.550 --> 00:05:13.910
instead of a critical z value

00:05:13.910 --> 00:05:16.510
is because our standard
error of the statistic

00:05:16.510 --> 00:05:17.510
is an estimate.

00:05:17.510 --> 00:05:19.680
We don't actually know
the standard deviation

00:05:19.680 --> 00:05:21.870
of the sampling distribution.

00:05:21.870 --> 00:05:23.160
So the last thing we
have to do is figure out

00:05:23.160 --> 00:05:24.940
what is this critical t value.

00:05:24.940 --> 00:05:26.990
You can figure it out
using either a calculator

00:05:26.990 --> 00:05:28.350
or using a table.

00:05:28.350 --> 00:05:30.030
I'll do it using a table.

00:05:30.030 --> 00:05:32.490
And to do that we need to know
what the degrees of freedom.

00:05:32.490 --> 00:05:36.060
Well, when you're doing this
with a regression slope,

00:05:36.060 --> 00:05:37.890
like we're doing right now,

00:05:37.890 --> 00:05:39.250
your degrees of freedom are going to be

00:05:39.250 --> 00:05:42.990
the number of data points
you have minus two.

00:05:42.990 --> 00:05:45.860
So our degrees of freedom
are gonna be 20 minus two.

00:05:45.860 --> 00:05:47.900
Which is equal to 18.

00:05:47.900 --> 00:05:49.970
I'm not gonna go into a
bunch of depth right now.

00:05:49.970 --> 00:05:53.690
It actually is beyond the
scope of this video for sure,

00:05:53.690 --> 00:05:56.260
as to why you subtract two here.

00:05:56.260 --> 00:05:58.810
But just so that we can
look it up on a table,

00:05:58.810 --> 00:06:00.770
this is our degrees of freedom.

00:06:00.770 --> 00:06:04.760
So we care about a 95% confidence level.

00:06:04.760 --> 00:06:08.850
That's equivalent to having
a 2 1/2% tail on either side.

00:06:08.850 --> 00:06:12.890
And our degrees of freedom is 18.

00:06:12.890 --> 00:06:17.470
So our critical t value is 2.101.

00:06:17.470 --> 00:06:20.870
And so, our 95% confidence interval

00:06:20.870 --> 00:06:25.740
is going to be 0.164 plus or
minus our critical t value

00:06:25.740 --> 00:06:30.300
2.101 times the standard
error of the statistic.

00:06:30.300 --> 00:06:31.500
Times,

00:06:31.500 --> 00:06:33.553
I'll just put it in parentheses, 0.057.

00:06:35.280 --> 00:06:36.890
And you could type this into a calculator

00:06:36.890 --> 00:06:39.050
if you wanted to figure
out the exact values here.

00:06:39.050 --> 00:06:42.500
But the way to interpret
a 95% confidence interval

00:06:42.500 --> 00:06:45.260
is that 95% of the time,

00:06:45.260 --> 00:06:48.240
that you calculated 95%
confidence interval,

00:06:48.240 --> 00:06:51.890
it is going to overlap with the true value

00:06:51.890 --> 00:06:54.553
of the parameter that we are estimating.