Worked example: sequence recursive formula

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.480
- [Instructor] A sequence is
defined recursively as follows.

00:00:03.480 --> 00:00:06.410
So A sub N is equal to A sub N minus one

00:00:06.410 --> 00:00:07.850
times A sub N minus two

00:00:07.850 --> 00:00:09.220
or another way of thinking about it.

00:00:09.220 --> 00:00:12.410
the Nth term is equal to
the N minus oneth term

00:00:12.410 --> 00:00:14.357
times the N minus two-th term

00:00:15.660 --> 00:00:17.310
with the zeroth term

00:00:17.310 --> 00:00:19.180
where A sub zero is equal to two

00:00:19.180 --> 00:00:21.420
and A sub one is equal to three.

00:00:21.420 --> 00:00:22.430
Find A sub four.

00:00:22.430 --> 00:00:24.000
So let's write this down.

00:00:24.000 --> 00:00:27.400
They're telling us A
sub zero is equal to two

00:00:27.400 --> 00:00:31.140
and they also tell us that
A sub one is equal to three.

00:00:31.140 --> 00:00:33.010
So they've kind of given
us our starting conditions

00:00:33.010 --> 00:00:34.430
or our base conditions.

00:00:34.430 --> 00:00:38.820
Now we can think about what A sub two is.

00:00:38.820 --> 00:00:40.630
They tell us that A sub two is going to be

00:00:40.630 --> 00:00:42.390
A sub two minus one,

00:00:42.390 --> 00:00:44.180
so that's A sub one.

00:00:44.180 --> 00:00:49.180
It's A sub one times A sub two minus two.

00:00:49.390 --> 00:00:51.890
That's A sub zero.

00:00:51.890 --> 00:00:55.290
A times A sub zero.

00:00:55.290 --> 00:00:56.590
They already told us what A sub one

00:00:56.590 --> 00:00:58.310
and A sub zero is.

00:00:58.310 --> 00:01:00.080
This thing is three,

00:01:00.080 --> 00:01:02.220
this thing is two.

00:01:02.220 --> 00:01:03.846
It's three times two

00:01:03.846 --> 00:01:07.150
which is equal to six.

00:01:07.150 --> 00:01:12.150
Now let's move on to A sub three.

00:01:12.980 --> 00:01:15.120
A sub three is going to be the product

00:01:15.120 --> 00:01:17.020
of the previous two terms.

00:01:17.020 --> 00:01:18.580
It's going to be A sub two,

00:01:18.580 --> 00:01:20.120
three minus one is two,

00:01:20.120 --> 00:01:21.410
three minus two is one.

00:01:21.410 --> 00:01:23.330
So it's A sub two.

00:01:23.330 --> 00:01:26.490
A sub two times A sub one.

00:01:26.490 --> 00:01:28.470
Times A sub one.

00:01:28.470 --> 00:01:33.410
So it's equal to six times three

00:01:33.410 --> 00:01:35.250
six times three.

00:01:35.250 --> 00:01:37.140
which is equal to 18.

00:01:37.140 --> 00:01:39.280
Which is equal to 18.

00:01:39.280 --> 00:01:41.580
And then finally A sub four,

00:01:41.580 --> 00:01:43.210
which I will do in a color that I'll use,

00:01:43.210 --> 00:01:44.043
I'll do it in yellow.

00:01:44.043 --> 00:01:47.810
A sub four is going to
be equal to A sub three,

00:01:47.810 --> 00:01:49.978
A sub three times A sub two.

00:01:49.978 --> 00:01:52.720
So four minus one is three,

00:01:52.720 --> 00:01:55.030
four minus two is two.

00:01:55.030 --> 00:01:57.700
So times A sub two,

00:01:57.700 --> 00:01:59.810
times A,

00:01:59.810 --> 00:02:01.960
and then a blue color.

00:02:01.960 --> 00:02:05.420
Times A sub two,

00:02:05.420 --> 00:02:08.820
which is equal to 18,

00:02:08.820 --> 00:02:11.860
18 times six.

00:02:11.860 --> 00:02:15.840
18 times six which is equal to,

00:02:15.840 --> 00:02:16.673
let's see,

00:02:16.673 --> 00:02:20.600
six times eight is 48 plus 60,

00:02:20.600 --> 00:02:23.260
or six times 10 is 100,

00:02:23.260 --> 00:02:24.980
108.

00:02:24.980 --> 00:02:25.813
And we're done,

00:02:25.813 --> 00:02:27.593
A sub four is equal to 108.


Worked example: sequence explicit formula

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.050
- [Instructor] If a sub
n is equal to n squared

00:00:03.050 --> 00:00:05.320
minus 10 over n plus one,

00:00:05.320 --> 00:00:09.110
determine a sub four plus a sub nine.

00:00:09.110 --> 00:00:11.060
Well, let's just think about
each of these independently.

00:00:11.060 --> 00:00:14.920
A sub four, a, let me write it this way.

00:00:14.920 --> 00:00:15.910
A,

00:00:15.910 --> 00:00:18.770
the fourth term, so a sub four,

00:00:18.770 --> 00:00:21.770
so our n, our lowercase
n, is going to be four,

00:00:21.770 --> 00:00:24.310
is going to be equal to
everywhere we see an n

00:00:24.310 --> 00:00:27.070
in this explicit definition
for this sequence.

00:00:27.070 --> 00:00:30.120
Everywhere we see an n, we
would replace it with a four.

00:00:30.120 --> 00:00:34.410
So it's going to be equal to four squared

00:00:34.410 --> 00:00:36.130
minus 10

00:00:36.130 --> 00:00:37.280
over

00:00:37.280 --> 00:00:38.470
four,

00:00:38.470 --> 00:00:39.870
over four

00:00:39.870 --> 00:00:41.310
plus one,

00:00:41.310 --> 00:00:45.070
over four plus one, which is equal to,

00:00:45.070 --> 00:00:48.770
well, let's see, that's 16 minus 10

00:00:48.770 --> 00:00:50.870
over five,

00:00:50.870 --> 00:00:54.340
which is equal to six over five.

00:00:54.340 --> 00:00:55.650
So that is a sub four.

00:00:55.650 --> 00:00:57.540
That is the fourth term.

00:00:57.540 --> 00:01:00.260
Now let's think about a sub nine.

00:01:00.260 --> 00:01:01.580
So a

00:01:01.580 --> 00:01:03.900
sub nine,

00:01:03.900 --> 00:01:07.380
so, once again, everywhere
that we see an n,

00:01:07.380 --> 00:01:08.860
we would replace it with a nine.

00:01:08.860 --> 00:01:11.720
We're looking at when
lowercase n is equal to nine,

00:01:11.720 --> 00:01:13.750
or we're looking at the ninth term.

00:01:13.750 --> 00:01:16.030
So it's going to be nine squared.

00:01:16.030 --> 00:01:19.130
Let me do that blue color just
so we see what we're doing.

00:01:19.130 --> 00:01:20.733
Nine squared,

00:01:21.700 --> 00:01:24.470
we'll do it in a green color, minus 10

00:01:24.470 --> 00:01:26.410
over nine plus one,

00:01:26.410 --> 00:01:27.810
over nine

00:01:27.810 --> 00:01:28.710
plus one

00:01:30.090 --> 00:01:34.040
is equal to, well, in the
numerator, we have 81 minus 10,

00:01:34.040 --> 00:01:36.060
81 minus 10

00:01:36.060 --> 00:01:38.730
over 10, over nine plus one.

00:01:38.730 --> 00:01:42.710
And so this is going to
be equal to 71 over 10.

00:01:42.710 --> 00:01:44.830
Now they want us to sum these two things,

00:01:44.830 --> 00:01:46.420
so that's going to be equal to,

00:01:46.420 --> 00:01:48.380
it's going to be equal to 6/5.

00:01:48.380 --> 00:01:50.870
A sub four is 6/5,

00:01:50.870 --> 00:01:53.513
plus a sub nine, which is 71 over 10.

00:01:54.610 --> 00:01:56.020
71 over 10.

00:01:56.020 --> 00:02:00.347
Well, we can rewrite 6/5
as being equal to 12/10,

00:02:00.347 --> 00:02:05.347
12/10 and then 71/10, so plus 71

00:02:05.550 --> 00:02:08.470
over 10, which is equal to,

00:02:08.470 --> 00:02:10.890
well, if I have 12/10 and
then I have another 71/10,

00:02:10.890 --> 00:02:13.445
now I'm going to have 83/10,

00:02:13.445 --> 00:02:15.940
83/10.

00:02:15.940 --> 00:02:16.773
And we're done.


Chi-square test for association (independence)

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:01.423
- [Instructor] We're already familiar

00:00:01.423 --> 00:00:03.060
with the chi-squared statistic.

00:00:03.060 --> 00:00:04.640
If you're not, I encourage you to review

00:00:04.640 --> 00:00:05.880
the videos on that.

00:00:05.880 --> 00:00:08.040
And we've already done
some hypothesis testing

00:00:08.040 --> 00:00:09.980
with the chi-squared statistic,

00:00:09.980 --> 00:00:12.166
and we've even done
some hypothesis testing

00:00:12.166 --> 00:00:14.580
based on two-way tables.

00:00:14.580 --> 00:00:15.960
And now we're going to extend that

00:00:15.960 --> 00:00:17.915
by thinking about a chi-squared test

00:00:17.915 --> 00:00:22.915
for association between two variables.

00:00:23.160 --> 00:00:27.030
So let's say that we suspect
that someone's foot length

00:00:27.030 --> 00:00:30.100
is related to their hand length.

00:00:30.100 --> 00:00:33.220
That these things are not independent.

00:00:33.220 --> 00:00:35.840
Well, what we can do is
set up a hypothesis test.

00:00:35.840 --> 00:00:37.793
And remember, the null hypothesis

00:00:37.793 --> 00:00:41.860
in a hypothesis test, is
to always assume no news.

00:00:41.860 --> 00:00:43.560
So what we could say is here is

00:00:43.560 --> 00:00:45.520
that there is no association.

00:00:45.520 --> 00:00:49.543
No association between,

00:00:50.420 --> 00:00:55.420
between foot and hand length.

00:00:57.930 --> 00:01:01.640
Another way to think about it
is that they are independent.

00:01:01.640 --> 00:01:04.330
And oftentimes what
we're doing is called a

00:01:04.330 --> 00:01:06.930
chi-squared test for independence.

00:01:06.930 --> 00:01:10.280
And then our alternative
hypothesis would be

00:01:10.280 --> 00:01:13.610
our suspicion there is an association.

00:01:13.610 --> 00:01:17.840
There is an association.

00:01:17.840 --> 00:01:21.920
So, foot and hand length
are not independent.

00:01:21.920 --> 00:01:26.920
So what we can then do
is go to a population,

00:01:27.070 --> 00:01:28.990
and we can randomly sample it.

00:01:28.990 --> 00:01:32.930
And so let's say we
randomly sample 100 folks.

00:01:32.930 --> 00:01:34.900
And for all of those 100 folks,

00:01:34.900 --> 00:01:37.000
we figure out whether
their right hand is longer,

00:01:37.000 --> 00:01:39.380
their left hand is longer,
or both hands are the same.

00:01:39.380 --> 00:01:41.820
And we also do that for the feet,

00:01:41.820 --> 00:01:44.790
and we tabulate all of the data.

00:01:44.790 --> 00:01:48.230
And this is the data that we actually get.

00:01:48.230 --> 00:01:50.090
Now it's worth thinking
about this for a second

00:01:50.090 --> 00:01:52.810
on how what we just did is different

00:01:52.810 --> 00:01:55.910
from a chi-squared test for homogeneity.

00:01:55.910 --> 00:01:58.290
And a chi-squared test for homogeneity,

00:01:58.290 --> 00:02:01.370
we sample from two different populations

00:02:01.370 --> 00:02:03.248
where we look at two different groups,

00:02:03.248 --> 00:02:05.400
and we see whether the distribution

00:02:05.400 --> 00:02:06.780
of a certain variable amongst those

00:02:06.780 --> 00:02:08.660
two different groups is the same.

00:02:08.660 --> 00:02:11.610
Here we are just sampling from one group,

00:02:11.610 --> 00:02:14.360
but we're thinking about
two different variables

00:02:14.360 --> 00:02:16.090
for that one group.

00:02:16.090 --> 00:02:18.080
We're thinking about feet length,

00:02:18.080 --> 00:02:20.030
and we're thinking about hand length.

00:02:20.030 --> 00:02:23.850
And so you can see here,
that 11 folks had both their

00:02:23.850 --> 00:02:26.630
right hand longer and
their right foot longer.

00:02:26.630 --> 00:02:28.430
Three folks had their right hand longer,

00:02:28.430 --> 00:02:30.500
but their left foot was longer.

00:02:30.500 --> 00:02:32.900
And then eight folks had
their right hand longer,

00:02:32.900 --> 00:02:34.780
but both feet were the same.

00:02:34.780 --> 00:02:38.660
Likewise, we had nine
people where their left foot

00:02:38.660 --> 00:02:41.020
and hand was longer,
but you had two people

00:02:41.020 --> 00:02:42.350
where the left hand was longer,

00:02:42.350 --> 00:02:43.760
but the right foot was longer.

00:02:43.760 --> 00:02:45.750
And we can go through all of these.

00:02:45.750 --> 00:02:47.510
But to do our chi-squared test,

00:02:47.510 --> 00:02:49.470
we would've said, what
would be the expected value

00:02:49.470 --> 00:02:52.520
of each of these data points if we assumed

00:02:52.520 --> 00:02:54.560
that the null hypothesis was true?

00:02:54.560 --> 00:02:58.440
That there was no association
between foot and hand length.

00:02:58.440 --> 00:03:01.100
So to help us do that,
I'm going to make a total

00:03:01.960 --> 00:03:06.960
of our columns here, and also
a total of our rows here.

00:03:08.380 --> 00:03:13.000
Let me draw a line here,
so we know what's going on.

00:03:13.000 --> 00:03:15.310
And so, what are the
total number of people

00:03:15.310 --> 00:03:17.490
who had a longer right hand?

00:03:17.490 --> 00:03:20.010
Well, it's going to be
11 plus three plus eight,

00:03:20.010 --> 00:03:21.800
which is 22.

00:03:21.800 --> 00:03:24.420
The total number of people
who had a longer left hand

00:03:24.420 --> 00:03:28.920
is two plus nine plus 14, which is 25.

00:03:28.920 --> 00:03:32.550
And then the total number
of people whose hands had

00:03:32.550 --> 00:03:36.023
the same length, 12 plus 13 plus 28,

00:03:36.973 --> 00:03:40.600
25 plus 28, that is 53.

00:03:40.600 --> 00:03:42.950
And then if I were to total this column,

00:03:42.950 --> 00:03:47.950
22 plus 25 is 47, plus 53,
we get 100, right over here.

00:03:48.530 --> 00:03:50.440
And then if we total the number of people

00:03:50.440 --> 00:03:54.090
who had a longer right
foot, 11 plus two plus 12,

00:03:54.090 --> 00:03:57.300
is 13 plus 12, that is 25.

00:03:57.300 --> 00:03:59.970
Longer left foot, three plus nine plus 13,

00:03:59.970 --> 00:04:01.820
that's also 25.

00:04:01.820 --> 00:04:03.720
And then we can either add these up,

00:04:03.720 --> 00:04:05.770
and we would get 50, or we could say,

00:04:05.770 --> 00:04:08.690
hey 25 plus 25 plus what is 100?

00:04:08.690 --> 00:04:11.150
Well, that is going to be equal to 50.

00:04:11.150 --> 00:04:13.450
Now to figure out these expected values,

00:04:13.450 --> 00:04:15.810
remember, we're going to
figure out the expected values

00:04:15.810 --> 00:04:18.840
assuming that the null hypothesis is true.

00:04:18.840 --> 00:04:21.660
Assuming that these
distributions are independent.

00:04:21.660 --> 00:04:26.660
That feet length and hand length
are independent variables.

00:04:26.810 --> 00:04:29.770
Well, if they are independent,
which we are assuming,

00:04:29.770 --> 00:04:34.770
then our best estimate is that
22% have a longer right hand,

00:04:34.840 --> 00:04:38.680
and our best estimate is that
25% have a longer right foot.

00:04:38.680 --> 00:04:40.620
And so out of 100, you would expect

00:04:40.620 --> 00:04:45.620
0.22 times 0.25 times 100 to have a

00:04:47.240 --> 00:04:49.640
longer right hand and foot.

00:04:49.640 --> 00:04:51.180
I'm just multiplying the probabilities,

00:04:51.180 --> 00:04:54.850
which you would do if these
were independent variables.

00:04:54.850 --> 00:04:59.020
And so 0.22 times 0.25, let's see,

00:05:00.378 --> 00:05:03.970
one fourth of 22 is 5 1/2,

00:05:03.970 --> 00:05:08.970
so this is going to be equal to 5.5.

00:05:09.050 --> 00:05:11.110
Now what number would you expect to have a

00:05:11.110 --> 00:05:15.050
longer right hand, but a longer left foot?

00:05:15.050 --> 00:05:20.050
So that would be 0.22
times 0.25 times 100.

00:05:21.240 --> 00:05:22.910
Well, we already calculated
what that would be.

00:05:22.910 --> 00:05:24.060
That would be 5.5.

00:05:25.130 --> 00:05:26.670
And then to figure out the expected number

00:05:26.670 --> 00:05:29.010
that would have a longer right hand,

00:05:29.010 --> 00:05:31.500
but both feet would be the same length,

00:05:31.500 --> 00:05:35.180
we could multiply 22 out
of 100 times 50 out of 100

00:05:35.180 --> 00:05:38.470
times 100, which is
going to be half of 22,

00:05:38.470 --> 00:05:40.560
which is equal to 11.

00:05:40.560 --> 00:05:43.030
And we can keep going.

00:05:43.030 --> 00:05:45.740
This value right over here would be

00:05:45.740 --> 00:05:50.740
0.25 times 0.25 times
100, 25 times 25 is 625,

00:05:52.230 --> 00:05:55.160
so that would be 6.25.

00:05:55.160 --> 00:05:59.950
This value right over here
would be 0.25 times 0.25

00:05:59.950 --> 00:06:02.573
times 100, which is again, 6.25.

00:06:03.890 --> 00:06:05.570
And then this value right over here,

00:06:05.570 --> 00:06:07.250
a couple of ways we can get it.

00:06:07.250 --> 00:06:12.250
We can multiply 0.25 times 50 times 100,

00:06:12.740 --> 00:06:15.940
which would get us to 12.5,

00:06:15.940 --> 00:06:17.920
or we could have said
this plus this plus this

00:06:17.920 --> 00:06:21.881
has to equal 25, so this would be 12.5.

00:06:21.881 --> 00:06:24.170
And on this expected
value, we can figure out

00:06:24.170 --> 00:06:29.170
because 5.5 plus 6.25 plus
this is going to equal 25.

00:06:29.360 --> 00:06:34.360
So let's see, 5.5 plus 6.25 is 11.75.

00:06:34.524 --> 00:06:39.524
11.75 plus 13.25 is equal to 25.

00:06:39.960 --> 00:06:41.460
Same thing over here.

00:06:41.460 --> 00:06:44.030
This would be 13.25, 'cause this is 11.75

00:06:45.210 --> 00:06:48.610
plus 13.25 is equal to 25.

00:06:48.610 --> 00:06:53.200
If we add these two together, we get 26.5.

00:06:54.140 --> 00:06:58.220
26.5 plus what is equal to 53?

00:06:58.220 --> 00:07:02.000
Well, it'd be equal to another 26.5.

00:07:02.000 --> 00:07:05.130
Now once you figure out all
of your expected values,

00:07:05.130 --> 00:07:07.890
that's a good time to
test your conditions.

00:07:07.890 --> 00:07:10.020
The first condition is that
you took a random sample.

00:07:10.020 --> 00:07:11.970
So let's assume we had done that.

00:07:11.970 --> 00:07:14.930
The second condition is
that your expected value

00:07:14.930 --> 00:07:18.510
for any of the data points has
to be at least equal to five.

00:07:18.510 --> 00:07:20.840
And we can see that all
of our expected values are

00:07:20.840 --> 00:07:22.310
at least equal to five.

00:07:22.310 --> 00:07:23.700
The actual data points we got

00:07:23.700 --> 00:07:25.080
do not have to be equal to five.

00:07:25.080 --> 00:07:26.960
So it's okay that we got a two here,

00:07:26.960 --> 00:07:30.440
because the expected value
here is five or larger.

00:07:30.440 --> 00:07:33.800
And then the last condition
is the independence condition.

00:07:33.800 --> 00:07:36.750
That either we are
sampling with replacement

00:07:36.750 --> 00:07:38.680
or that we have to feel comfortable

00:07:38.680 --> 00:07:41.920
that our sample size is no more
than 10% of the population.

00:07:41.920 --> 00:07:44.040
So let's assume that
that happened as well.

00:07:44.040 --> 00:07:46.100
So assuming we met all
of those conditions,

00:07:46.100 --> 00:07:50.170
we are ready to calculate
our chi-squared statistic.

00:07:50.170 --> 00:07:51.530
And so what we're going to do,

00:07:51.530 --> 00:07:53.430
is for every data point,
we're going to find the

00:07:53.430 --> 00:07:55.370
difference between the data point,

00:07:55.370 --> 00:07:59.320
11 minus the expected, minus 5.5,

00:07:59.320 --> 00:08:03.900
squared over the expected,
so I did that one.

00:08:03.900 --> 00:08:05.230
Now I'll do this one.

00:08:05.230 --> 00:08:10.230
So plus three minus 5.5 squared over 5.5

00:08:12.720 --> 00:08:14.590
plus, now I'll do this one,

00:08:14.590 --> 00:08:19.440
eight minus 11 squared over 11,

00:08:19.440 --> 00:08:24.440
then I'll do this one, two
minus 6.25 squared over 6.25.

00:08:27.030 --> 00:08:27.910
And I'll keep doing it.

00:08:27.910 --> 00:08:30.890
I'm going to do it for all
nine of these data points.

00:08:30.890 --> 00:08:32.830
And I actually calculated
this ahead of time

00:08:32.830 --> 00:08:34.460
to save some time.

00:08:34.460 --> 00:08:37.070
And so if you do this for
all nine of the data points,

00:08:37.070 --> 00:08:42.070
you're going to get a
chi-squared statistic of 11.942.

00:08:42.090 --> 00:08:44.140
Now before we calculate the P-value,

00:08:44.140 --> 00:08:44.973
we're going to have to think about

00:08:44.973 --> 00:08:47.300
what are our degrees of freedom?

00:08:47.300 --> 00:08:50.120
Now we have a three-by-three table here,

00:08:50.120 --> 00:08:51.700
so one way to think about it,

00:08:51.700 --> 00:08:54.290
it's the number of rows minus one,

00:08:54.290 --> 00:08:56.710
times the number of columns minus one,

00:08:56.710 --> 00:08:59.050
and this is two times two,
which is equal to four.

00:08:59.050 --> 00:09:00.700
Another way to think
about it is if you know

00:09:00.700 --> 00:09:03.350
four of these data points
and you know the totals,

00:09:03.350 --> 00:09:06.670
then you can figure out
the other five data points.

00:09:06.670 --> 00:09:09.420
And so now we are ready
to calculate a P-value.

00:09:09.420 --> 00:09:11.050
And you can do that using a calculator,

00:09:11.050 --> 00:09:13.730
and you can do that using
a chi-squared table,

00:09:13.730 --> 00:09:16.220
but let's say we did
it using a calculator,

00:09:16.220 --> 00:09:20.873
and we get a P-value of 0.018

00:09:22.050 --> 00:09:24.450
And just to remind ourselves what this is,

00:09:24.450 --> 00:09:26.810
this is the probability of getting a

00:09:26.810 --> 00:09:31.810
chi-squared statistic at
least this large or larger.

00:09:32.020 --> 00:09:33.810
And so next, we do what we always do

00:09:33.810 --> 00:09:35.170
with hypothesis testing.

00:09:35.170 --> 00:09:37.540
We compare this to our significance level.

00:09:37.540 --> 00:09:39.500
And we actually should have
set our significance level

00:09:39.500 --> 00:09:40.333
from the beginning.

00:09:40.333 --> 00:09:42.430
So let's just assume that when we set up

00:09:42.430 --> 00:09:44.110
our hypotheses here, we also said

00:09:44.110 --> 00:09:47.290
that we want a significance level of 0.05.

00:09:47.290 --> 00:09:50.270
You really should do this before
you calculate all of this.

00:09:50.270 --> 00:09:53.550
But then you compare your P-value

00:09:53.550 --> 00:09:55.680
to your significance level, and we see

00:09:55.680 --> 00:09:58.830
that this P-value is a good bit less

00:09:58.830 --> 00:10:00.320
than our significance level.

00:10:00.320 --> 00:10:02.060
And so one way to think about it is,

00:10:02.060 --> 00:10:04.370
we got all these expected values assuming

00:10:04.370 --> 00:10:06.660
that the null hypothesis was true.

00:10:06.660 --> 00:10:08.450
But the probability of getting a result

00:10:08.450 --> 00:10:12.850
this extreme or more
extreme is less than 2%,

00:10:12.850 --> 00:10:15.020
which is lower than
our significance level.

00:10:15.020 --> 00:10:20.020
And so this will lead us to
reject our null hypothesis

00:10:20.520 --> 00:10:24.640
and it suggests to us that
there is an association

00:10:24.640 --> 00:10:27.153
between hand length and foot length.


Introduction to the chi-square test for homogeneity

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.780
- We've already been introduced
to the Chi-squared statistic

00:00:03.780 --> 00:00:05.120
in other videos.

00:00:05.120 --> 00:00:09.230
Now we're going to use it
for a test for homogeneity.

00:00:09.230 --> 00:00:11.510
And homogeneity or homogeneity,

00:00:11.510 --> 00:00:14.730
in everyday language this
means how similar things are

00:00:14.730 --> 00:00:16.500
and that's what we're
essentially going to test here.

00:00:16.500 --> 00:00:18.020
We're gonna look at two different groups

00:00:18.020 --> 00:00:20.710
and see whether the
distributions of those groups

00:00:20.710 --> 00:00:24.030
for a certain variable are similar or not.

00:00:24.030 --> 00:00:26.370
And so the question I'm
going to think about,

00:00:26.370 --> 00:00:29.210
or we're going to think about
together in this video is,

00:00:29.210 --> 00:00:30.060
let's say

00:00:30.060 --> 00:00:31.410
we were thinking about left-handed

00:00:31.410 --> 00:00:33.920
versus right-handed people
and we're wondering,

00:00:33.920 --> 00:00:37.090
do they have the same
preferences for subject domains?

00:00:37.090 --> 00:00:38.560
Are they equally inclined

00:00:38.560 --> 00:00:41.060
to science, technology,
engineering, math, humanities,

00:00:41.060 --> 00:00:42.230
or neither.

00:00:42.230 --> 00:00:45.860
And so we can set up our null
and alternative hypotheses.

00:00:45.860 --> 00:00:47.720
Our null hypothesis is that

00:00:47.720 --> 00:00:49.960
there is no difference in
the distribution between

00:00:49.960 --> 00:00:51.670
left-handed and right-handed people

00:00:51.670 --> 00:00:54.340
in terms of their preference
for subject domains.

00:00:54.340 --> 00:00:57.170
So, no difference

00:00:58.800 --> 00:01:00.740
in subject

00:01:01.670 --> 00:01:03.250
subject preference

00:01:05.070 --> 00:01:06.610
for left and right.

00:01:06.610 --> 00:01:08.350
For left

00:01:09.410 --> 00:01:13.350
and right-handed folks.

00:01:13.350 --> 00:01:15.180
And then the alternative hypothesis.

00:01:15.180 --> 00:01:17.660
Well, no, there is a difference.

00:01:17.660 --> 00:01:22.283
So there is a difference.

00:01:23.200 --> 00:01:25.440
So how would we go about testing this?

00:01:25.440 --> 00:01:27.810
Well, we've done hypothesis
testing many times

00:01:27.810 --> 00:01:29.360
in many videos already.

00:01:29.360 --> 00:01:32.680
But here, we're going to sample
from two different groups.

00:01:32.680 --> 00:01:35.810
So let's say that this
is the population of

00:01:35.810 --> 00:01:40.560
right-handed folks and
this is the population of

00:01:40.560 --> 00:01:42.410
left-handed folks.

00:01:42.410 --> 00:01:44.820
Let's say from that sample
of right-handed folks,

00:01:44.820 --> 00:01:47.260
I take a sample of 60,

00:01:47.260 --> 00:01:49.780
and then I do the same thing
for the left-handed folks.

00:01:49.780 --> 00:01:52.570
And these don't even have
to be the sample sizes,

00:01:52.570 --> 00:01:57.010
so the left-handed folks,
let's say I sample, 40 folks.

00:01:57.010 --> 00:02:01.100
And here is the data
that I actually collect.

00:02:01.100 --> 00:02:03.980
So for those 60 right-handed folks,

00:02:03.980 --> 00:02:06.880
30 of them prefer the STEM subjects,

00:02:06.880 --> 00:02:08.690
science, technology, engineering, math.

00:02:08.690 --> 00:02:10.400
15 preferred humanities.

00:02:10.400 --> 00:02:13.400
And 15 were indifferent,
they liked them equally.

00:02:13.400 --> 00:02:16.180
And then, for the 40 left-handed folks,

00:02:16.180 --> 00:02:20.880
I got 10 preferring STEM,
25 preferring humanities,

00:02:20.880 --> 00:02:22.830
and 5 viewed them equally.

00:02:22.830 --> 00:02:25.330
And then you see the total
number of right-handed folks,

00:02:25.330 --> 00:02:26.720
total number of left-handed folks,

00:02:26.720 --> 00:02:29.080
and then you have the total
number from both groups

00:02:29.080 --> 00:02:31.500
that preferred STEM, total
number from both groups

00:02:31.500 --> 00:02:34.100
that preferred humanities,
total from both groups

00:02:34.100 --> 00:02:36.090
that had no preference.

00:02:36.090 --> 00:02:39.180
So let's just start thinking
about what the expected data

00:02:39.180 --> 00:02:42.360
would be if we're assuming that
the null hypothesis is true,

00:02:42.360 --> 00:02:44.270
that there's no difference in preference

00:02:44.270 --> 00:02:46.190
between right and left-handed folks.

00:02:46.190 --> 00:02:48.950
This is the right-handed column.

00:02:48.950 --> 00:02:52.170
This is the left-handed column.

00:02:52.170 --> 00:02:54.350
Well assuming that the
null hypothesis is true,

00:02:54.350 --> 00:02:55.680
that there's no difference between

00:02:55.680 --> 00:02:58.180
right and left-handed people
in terms of their preference,

00:02:58.180 --> 00:03:01.350
our best estimate of what the
distribution of preference

00:03:01.350 --> 00:03:03.930
would be in the population generally

00:03:03.930 --> 00:03:06.950
would come from this total column.

00:03:06.950 --> 00:03:08.860
Since we're assuming no difference,

00:03:08.860 --> 00:03:10.430
we would assume that in either group,

00:03:10.430 --> 00:03:14.400
40 out of every 100 would
prefer STEM or 40 percent.

00:03:14.400 --> 00:03:16.510
40 percent would prefer humanities

00:03:16.510 --> 00:03:20.020
and 20 percent would have no preference.

00:03:20.020 --> 00:03:22.700
And so our expected
would be that 40 percent

00:03:22.700 --> 00:03:26.010
of the 60 right-handed
folks would prefer STEM.

00:03:26.010 --> 00:03:28.177
So what's 40 percent of 60?

00:03:28.177 --> 00:03:32.060
0.4 times 60 is 24.

00:03:32.060 --> 00:03:35.050
And similarly, we would expect 40 percent

00:03:35.050 --> 00:03:36.350
preferring humanities.

00:03:36.350 --> 00:03:39.003
40 percent times 60 is 24 again.

00:03:40.646 --> 00:03:42.380
And then we would expect 20 percent

00:03:42.380 --> 00:03:45.010
of the right-handed group
to have no preference.

00:03:45.010 --> 00:03:48.600
So 20 percent of 60 is 12.

00:03:48.600 --> 00:03:51.950
And these, once again, they add up to 60.

00:03:51.950 --> 00:03:53.450
And then for the left-handed folks,

00:03:53.450 --> 00:03:55.410
we would go through the same process.

00:03:55.410 --> 00:03:59.350
We would expect that 40
percent of them prefer STEM,

00:03:59.350 --> 00:04:02.920
40 percent of 40, that is 16.

00:04:02.920 --> 00:04:07.640
On the humanities, again,
40 percent of 40 is 16.

00:04:07.640 --> 00:04:11.080
And equal, 20 percent of 40 is 8.

00:04:11.080 --> 00:04:13.210
And then, all of these add up to 40.

00:04:13.210 --> 00:04:15.220
Once you calculate these expected values,

00:04:15.220 --> 00:04:17.640
it's a good time to make sure
you're meeting your conditions

00:04:17.640 --> 00:04:20.740
for conducting a Chi-squared test.

00:04:20.740 --> 00:04:22.350
The first is the random condition.

00:04:22.350 --> 00:04:25.120
And so, these need to
be truly random samples,

00:04:25.120 --> 00:04:27.030
so hopefully, we met that condition.

00:04:27.030 --> 00:04:29.560
The second is that the expected value

00:04:29.560 --> 00:04:31.130
for any of these data points

00:04:31.130 --> 00:04:33.350
have to be at least equal to five.

00:04:33.350 --> 00:04:35.930
And so we have met that condition.

00:04:35.930 --> 00:04:38.170
These are all at least equal to five.

00:04:38.170 --> 00:04:41.100
And then the last condition
is the independence condition

00:04:41.100 --> 00:04:43.610
that we are either
sampling with replacement

00:04:43.610 --> 00:04:45.190
or if we're not sampling with replacement,

00:04:45.190 --> 00:04:47.110
we have to feel good that our samples

00:04:47.110 --> 00:04:49.300
are no more than 10
percent of the population.

00:04:49.300 --> 00:04:52.570
So let's assume that
that is the case as well.

00:04:52.570 --> 00:04:56.560
And now, we're ready to calculate
our Chi-squared statistic.

00:04:56.560 --> 00:05:00.470
We would get our Chi-squared statistic

00:05:00.470 --> 00:05:02.130
is going to be equal to

00:05:02.130 --> 00:05:04.650
the difference between what we got

00:05:04.650 --> 00:05:07.200
and the expected, squared.

00:05:07.200 --> 00:05:12.200
So 30 minus 24, squared, divided by

00:05:12.350 --> 00:05:14.730
the expected, divided by 24.

00:05:14.730 --> 00:05:18.320
And we'll do it for all
six of these data points.

00:05:18.320 --> 00:05:20.220
So then, I will go to the next one.

00:05:20.220 --> 00:05:21.803
So then, this is going to be,

00:05:22.780 --> 00:05:26.700
so plus and if I look
at this and this here,

00:05:26.700 --> 00:05:31.360
I'm going to have 10 minus 16, squared

00:05:32.330 --> 00:05:35.530
over expected 16.

00:05:35.530 --> 00:05:39.380
And then, I'm going to have,
I'll look at that data point

00:05:39.380 --> 00:05:42.110
and that expected and I will get

00:05:42.110 --> 00:05:47.110
15 minus 24 squared
over expected, over 24.

00:05:48.380 --> 00:05:49.800
I'm running out of colors.

00:05:49.800 --> 00:05:52.580
And then we would look at
that, those two numbers

00:05:52.580 --> 00:05:53.413
and we would say,

00:05:53.413 --> 00:05:58.400
plus 25 minus 16 squared,
divided by expected,

00:05:59.670 --> 00:06:04.670
and then, we would get, we
would look at these two,

00:06:04.700 --> 00:06:09.630
plus 15 minus 12 squared,

00:06:09.630 --> 00:06:11.830
over expected, over 12.

00:06:11.830 --> 00:06:14.040
And then, last but not least,

00:06:14.040 --> 00:06:16.130
lemme find a color I haven't used.

00:06:16.130 --> 00:06:19.100
We would look at that and
that and we would say,

00:06:19.100 --> 00:06:22.780
plus five minus eight squared

00:06:22.780 --> 00:06:25.070
over expected, over eight.

00:06:25.070 --> 00:06:27.970
Now, once you get that value
for the Chi-squared statistic,

00:06:27.970 --> 00:06:31.030
the next question is, what
are the degrees of freedom?

00:06:31.030 --> 00:06:33.530
Now a simple rule of thumb
is to just look at your data

00:06:33.530 --> 00:06:37.220
and think about the number of
rows and the number of columns

00:06:37.220 --> 00:06:39.510
and we have three rows and two columns.

00:06:39.510 --> 00:06:41.120
And so your degrees of
freedom are going to be

00:06:41.120 --> 00:06:44.280
the number of rows minus
one, three minus one,

00:06:44.280 --> 00:06:48.050
times the number of columns
minus one, two minus one.

00:06:48.050 --> 00:06:50.540
And so this is going to
be equal to two times one

00:06:50.540 --> 00:06:52.640
which is equal to two.

00:06:52.640 --> 00:06:55.120
Now the reason why that
makes intuitive sense

00:06:55.120 --> 00:06:56.370
is think about it,

00:06:56.370 --> 00:06:58.520
if you knew two of these data points,

00:06:58.520 --> 00:07:00.830
and if you knew all of the totals,

00:07:00.830 --> 00:07:03.310
then you could figure out
the other data points.

00:07:03.310 --> 00:07:04.500
If you knew these two data points,

00:07:04.500 --> 00:07:05.333
you could figure out that.

00:07:05.333 --> 00:07:06.720
If you knew this data point
and you knew the total,

00:07:06.720 --> 00:07:07.880
you could figure out that.

00:07:07.880 --> 00:07:09.110
If you knew this data point

00:07:09.110 --> 00:07:09.943
and you knew the total

00:07:09.943 --> 00:07:11.512
you could figure out that

00:07:11.512 --> 00:07:12.690
And if you figured out that and that,

00:07:12.690 --> 00:07:14.620
then you could figure
out this right over here.

00:07:14.620 --> 00:07:16.750
And so that's why this
rule of thumb works.

00:07:16.750 --> 00:07:17.980
The number of rows minus one

00:07:17.980 --> 00:07:19.890
times the number of columns minus one

00:07:19.890 --> 00:07:22.430
gives you your degrees of freedom.

00:07:22.430 --> 00:07:24.850
Now, given this Chi-squared statistic

00:07:24.850 --> 00:07:26.680
that I haven't calculated but you could

00:07:26.680 --> 00:07:28.510
type this into a calculator
and figure it out,

00:07:28.510 --> 00:07:29.880
and this degrees of freedom,

00:07:29.880 --> 00:07:32.530
we could then figure out the P value.

00:07:32.530 --> 00:07:34.510
We could figure out the probability

00:07:34.510 --> 00:07:37.420
of getting a Chi-squared
statistic this extreme

00:07:37.420 --> 00:07:38.880
or more extreme.

00:07:38.880 --> 00:07:42.340
And if this is less than
our significance level

00:07:42.340 --> 00:07:44.290
which we should have set ahead of time,

00:07:44.290 --> 00:07:46.630
then we would reject the null hypothesis

00:07:46.630 --> 00:07:48.410
and it would suggest the alternative.

00:07:48.410 --> 00:07:50.980
If this is not less than
our significance level,

00:07:50.980 --> 00:07:55.203
then it does not allow us to
reject the null hypothesis.


Chi-square goodness-of-fit example

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:01.870
- [Instructor] In the
game rock-paper-scissors,

00:00:01.870 --> 00:00:05.750
Kenny expects to win, tie,
and lose with equal frequency.

00:00:05.750 --> 00:00:07.740
Kenny plays rock-paper-scissors often,

00:00:07.740 --> 00:00:09.540
but he suspect his own games

00:00:09.540 --> 00:00:11.470
were not following that pattern.

00:00:11.470 --> 00:00:13.910
So he took a random sample of 24 games

00:00:13.910 --> 00:00:15.640
and recorded their outcomes.

00:00:15.640 --> 00:00:17.030
Here are his results.

00:00:17.030 --> 00:00:19.810
So out of the 24 games, he won four,

00:00:19.810 --> 00:00:21.930
lost 13, and tied seven times.

00:00:21.930 --> 00:00:23.790
He wants to use these results to carry out

00:00:23.790 --> 00:00:26.190
a chi-squared goodness-of-fit test

00:00:26.190 --> 00:00:29.600
to determine if the
distribution of his outcomes

00:00:29.600 --> 00:00:32.520
disagrees with an even distribution.

00:00:32.520 --> 00:00:34.930
What are the values of the test statistic,

00:00:34.930 --> 00:00:36.680
the chi-squared test statistic,

00:00:36.680 --> 00:00:39.160
and P-value for Kenny's test?

00:00:39.160 --> 00:00:41.910
So pause this video and see
if you can figure that out.

00:00:42.760 --> 00:00:45.290
Okay, so he's essentially
just doing a hypothesis test

00:00:45.290 --> 00:00:47.830
using the chi-squared statistic.

00:00:47.830 --> 00:00:48.940
Because it's a hypothesis

00:00:48.940 --> 00:00:51.240
that's thinking about multiple categories.

00:00:51.240 --> 00:00:54.610
So what would his null hypothesis be?

00:00:54.610 --> 00:00:57.780
Well, his null hypothesis
would be that he has

00:00:57.780 --> 00:01:00.190
that all of the outcomes
are equal probability.

00:01:00.190 --> 00:01:01.240
Outcomes

00:01:02.360 --> 00:01:03.193
equal

00:01:04.380 --> 00:01:06.580
equal probability.

00:01:06.580 --> 00:01:09.070
And then his alternative hypothesis

00:01:09.070 --> 00:01:11.120
would be that his outcomes

00:01:11.120 --> 00:01:12.790
have not equal

00:01:12.790 --> 00:01:14.190
not

00:01:14.190 --> 00:01:15.630
equal

00:01:15.630 --> 00:01:16.630
probability.

00:01:16.630 --> 00:01:19.460
Remember we assume that the
null hypothesis is true.

00:01:19.460 --> 00:01:22.650
And then assuming if the
null hypothesis is true,

00:01:22.650 --> 00:01:25.960
the probability of getting a
result at least this extreme

00:01:25.960 --> 00:01:27.380
is low enough,

00:01:27.380 --> 00:01:29.720
then we would reject our null hypothesis.

00:01:29.720 --> 00:01:30.650
Another way to think about it is

00:01:30.650 --> 00:01:32.850
if our P-value is below threshold,

00:01:32.850 --> 00:01:35.470
we would reject our null hypothesis.

00:01:35.470 --> 00:01:38.380
And so what he did is

00:01:38.380 --> 00:01:40.070
he took a sample of 24 games,

00:01:40.070 --> 00:01:41.750
so n is equal to 24.

00:01:41.750 --> 00:01:44.020
And then this was the data that he got.

00:01:44.020 --> 00:01:46.810
Now before we even calculate
our chi-squared statistic,

00:01:46.810 --> 00:01:48.010
and figure out what's the probability

00:01:48.010 --> 00:01:51.520
of getting a chi-squared
statistic that large or greater,

00:01:51.520 --> 00:01:54.050
let's make sure we meet the
conditions for inference

00:01:54.050 --> 00:01:58.070
for a chi-squared goodness-of-fit test.

00:01:58.070 --> 00:01:59.360
So you've seen some of them,

00:01:59.360 --> 00:02:01.120
but some of them are a
little bit different.

00:02:01.120 --> 00:02:03.220
One is the random condition.

00:02:03.220 --> 00:02:04.330
I'll write 'em up here.

00:02:04.330 --> 00:02:05.563
The random condition.

00:02:06.720 --> 00:02:07.553
And that would be that

00:02:07.553 --> 00:02:09.540
there's truly a random sample of games.

00:02:09.540 --> 00:02:12.160
And it tell us right here,
he took a random sample

00:02:12.160 --> 00:02:14.460
of his 24 games.

00:02:14.460 --> 00:02:16.260
So we meet that condition.

00:02:16.260 --> 00:02:17.250
The second condition,

00:02:17.250 --> 00:02:20.770
when we're talking about
chi-squared hypothesis testing,

00:02:20.770 --> 00:02:22.690
is the large counts.

00:02:22.690 --> 00:02:23.650
Large

00:02:23.650 --> 00:02:24.620
counts

00:02:24.620 --> 00:02:25.453
condition.

00:02:25.453 --> 00:02:27.870
And this is an important
one to appreciate.

00:02:27.870 --> 00:02:31.020
This is that the expected number

00:02:31.020 --> 00:02:33.270
of each category of outcomes

00:02:33.270 --> 00:02:36.330
is at least equal to five.

00:02:36.330 --> 00:02:37.410
Now you might say, hey, wait, wait,

00:02:37.410 --> 00:02:38.500
I only got four wins.

00:02:38.500 --> 00:02:41.790
Or Kenny only got four wins
out of his sample of 24.

00:02:41.790 --> 00:02:45.060
But that does not violate
the large counts condition.

00:02:45.060 --> 00:02:46.940
Remember, what is the expected number

00:02:46.940 --> 00:02:48.870
of wins, losses, and ties?

00:02:48.870 --> 00:02:51.240
Well if you were assuming
the null hypothesis,

00:02:51.240 --> 00:02:53.410
where the outcomes have equal probability,

00:02:53.410 --> 00:02:55.300
so the expected

00:02:55.300 --> 00:02:56.500
the expected

00:02:56.500 --> 00:02:57.840
I could write right over here.

00:02:57.840 --> 00:02:59.780
It would be that it's 1/3, 1/3, 1/3.

00:02:59.780 --> 00:03:03.220
And so 1/3 of 24 is 8, 8, and 8.

00:03:03.220 --> 00:03:04.910
That's what Kenny would expect.

00:03:04.910 --> 00:03:08.090
And since because all of these
are at least equal to five,

00:03:08.090 --> 00:03:10.400
we meet the large counts condition.

00:03:10.400 --> 00:03:12.130
And then the last condition

00:03:12.130 --> 00:03:13.980
is the independence condition.

00:03:13.980 --> 00:03:17.010
If we aren't sampling with replacement,

00:03:17.010 --> 00:03:18.410
then we just have to feel good

00:03:18.410 --> 00:03:21.680
that our sample size is no more
than 10% of the population.

00:03:21.680 --> 00:03:25.970
And he can definitely play more
than 240 games in his life,

00:03:25.970 --> 00:03:29.780
so we would assume that we
meet that condition as well.

00:03:29.780 --> 00:03:30.910
And so with that out of the way,

00:03:30.910 --> 00:03:33.890
we can actually calculate
our chi-squared statistic

00:03:33.890 --> 00:03:36.040
and try to make some
inference based on it.

00:03:36.040 --> 00:03:37.000
And so,

00:03:37.000 --> 00:03:39.950
let's see, our chi-squared statistic

00:03:39.950 --> 00:03:41.770
is going to be equal to,

00:03:41.770 --> 00:03:42.920
so for each category,

00:03:42.920 --> 00:03:43.920
it's going to be the difference

00:03:43.920 --> 00:03:47.940
between the expected and what
he got in that sample squared

00:03:47.940 --> 00:03:49.680
divided by the expected.

00:03:49.680 --> 00:03:51.810
So the first category is wins.

00:03:51.810 --> 00:03:54.340
So that's going to be four minus eight,

00:03:54.340 --> 00:03:55.810
four minus eight,

00:03:55.810 --> 00:03:59.600
squared over an expected
number of wins of eight,

00:03:59.600 --> 00:04:04.200
plus losses, so that's 13 minus eight.

00:04:04.200 --> 00:04:06.950
13 is how many he got, how many he lost,

00:04:06.950 --> 00:04:09.390
minus eight expected, squared,

00:04:09.390 --> 00:04:11.620
over the number expected

00:04:11.620 --> 00:04:14.430
plus he got seven ties,

00:04:14.430 --> 00:04:16.840
he would have expected eight, squared,

00:04:16.840 --> 00:04:18.610
all of that over eight.

00:04:18.610 --> 00:04:19.700
And so let's see what is this.

00:04:19.700 --> 00:04:22.460
Four minus eight is negative four,

00:04:22.460 --> 00:04:25.200
you square that, you get 16.

00:04:25.200 --> 00:04:27.610
13 minus eight is five,

00:04:27.610 --> 00:04:31.720
you square that, you get 25.

00:04:31.720 --> 00:04:33.880
Seven minus eight is negative one.

00:04:33.880 --> 00:04:35.750
Square that you get one.

00:04:35.750 --> 00:04:40.750
And 16 divided by eight
is going to be two.

00:04:40.920 --> 00:04:44.580
25 divided by eight is
going to be, let's see,

00:04:44.580 --> 00:04:46.470
that's three and 1/8th,

00:04:46.470 --> 00:04:48.207
so that's 3.125,

00:04:49.690 --> 00:04:52.330
and then 1/8th is 0.125.

00:04:52.330 --> 00:04:54.240
0.125.

00:04:54.240 --> 00:04:55.890
You add these together,

00:04:55.890 --> 00:04:56.770
so let's see,

00:04:56.770 --> 00:04:58.584
it's gonna be two plus 3.125,

00:04:58.584 --> 00:04:59.417
5.125

00:05:00.260 --> 00:05:02.260
plus another 0.125

00:05:02.260 --> 00:05:04.820
so that's going to be 5.25.

00:05:04.820 --> 00:05:09.090
So our chi-squared statistic is 5.25.

00:05:09.090 --> 00:05:11.280
And now to figure out our P-value,

00:05:11.280 --> 00:05:12.930
our P-value

00:05:12.930 --> 00:05:15.650
is going to be equal to the probability

00:05:15.650 --> 00:05:18.470
of getting a chi-squared statistic

00:05:18.470 --> 00:05:21.510
greater than or equal to 5.25.

00:05:21.510 --> 00:05:25.147
And you could use a
chi-squared table for that.

00:05:25.147 --> 00:05:27.600
And we always have to think
about our degrees of freedom.

00:05:27.600 --> 00:05:30.080
We have one, two, three categories.

00:05:30.080 --> 00:05:32.460
So our degrees of freedom is
going to be one less than that,

00:05:32.460 --> 00:05:34.960
or three minus one, which is two.

00:05:34.960 --> 00:05:36.300
So our degrees of freedom

00:05:36.300 --> 00:05:37.180
is going to be equal to two.

00:05:37.180 --> 00:05:38.840
And that makes sense, because you know

00:05:38.840 --> 00:05:39.940
for a certain number of games,

00:05:39.940 --> 00:05:41.240
if you know the number of wins,

00:05:41.240 --> 00:05:42.600
and you know there's a
certain number of losses,

00:05:42.600 --> 00:05:44.100
you can figure out the number of ties.

00:05:44.100 --> 00:05:46.130
Or if you know any two
of these categories,

00:05:46.130 --> 00:05:47.410
you can always figure out the third.

00:05:47.410 --> 00:05:49.530
So that's why you have
two degrees of freedom.

00:05:49.530 --> 00:05:52.960
And so let's get out
our chi-squared table.

00:05:52.960 --> 00:05:55.060
So we have two degrees of freedom,

00:05:55.060 --> 00:05:56.820
so we are in this row.

00:05:56.820 --> 00:05:59.130
And where is 5.25?

00:05:59.130 --> 00:06:03.160
So, 5.25 is right over there.

00:06:03.160 --> 00:06:08.160
And so our probability is going
to be between 0.10 and 0.05.

00:06:10.550 --> 00:06:12.310
So our P-value

00:06:12.310 --> 00:06:17.183
is going to be greater than
0.05 and less than 0.10.

00:06:19.140 --> 00:06:21.470
And so for example, if ahead of time,

00:06:21.470 --> 00:06:22.810
and he should have done
this ahead of time,

00:06:22.810 --> 00:06:25.780
he set a significance level of 5%

00:06:25.780 --> 00:06:29.190
and our P-value here is greater
than 5% which we just saw,

00:06:29.190 --> 00:06:31.200
he would fail to reject in this situation

00:06:31.200 --> 00:06:32.130
the null hypothesis.

00:06:32.130 --> 00:06:34.550
But they're not asking us that here.

00:06:34.550 --> 00:06:36.790
All they're asking us is
what is our chi-squared value

00:06:36.790 --> 00:06:39.360
and what range is our P-value in.

00:06:39.360 --> 00:06:42.650
Well, let's see, 5.25
are both of these values,

00:06:42.650 --> 00:06:46.140
and we saw we got a
P-value between 5% and 10%.

00:06:46.140 --> 00:06:48.493
So it is choice A right over there.


Chi-square statistic for hypothesis testing

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:01.430
- [Instructor] Let's
say there's some type of

00:00:01.430 --> 00:00:03.440
standardized exam where every question

00:00:03.440 --> 00:00:05.240
on the test has four choices,

00:00:05.240 --> 00:00:07.490
choice A, choice B,

00:00:07.490 --> 00:00:09.970
choice C, and choice D.

00:00:09.970 --> 00:00:13.520
And the test makers assure
folks that, over many years,

00:00:13.520 --> 00:00:15.990
there's an equal probability
that the correct answer

00:00:15.990 --> 00:00:19.070
for any one of the items is A, B, C, or D.

00:00:19.070 --> 00:00:22.280
It essentially is a 25%
chance of any of them.

00:00:22.280 --> 00:00:24.190
Now, let's say you have a hunch that,

00:00:24.190 --> 00:00:27.760
well, maybe it is skewed
towards one letter or another.

00:00:27.760 --> 00:00:29.650
How could you test this?

00:00:29.650 --> 00:00:31.060
Well, you could start with a null

00:00:31.060 --> 00:00:32.480
and alternative hypothesis,

00:00:32.480 --> 00:00:35.300
and then we can actually
do a hypothesis test.

00:00:35.300 --> 00:00:40.300
So let's say that our null
hypothesis is equal distribution,

00:00:40.340 --> 00:00:43.160
equal distribution

00:00:43.160 --> 00:00:45.080
of correct choices,

00:00:45.080 --> 00:00:47.750
correct choices.

00:00:47.750 --> 00:00:49.550
Or another way of thinking about it is

00:00:49.550 --> 00:00:52.150
A would be correct 25% of the time,

00:00:52.150 --> 00:00:54.930
B would be correct 25% of the time,

00:00:54.930 --> 00:00:57.300
C would be correct 25% of the time,

00:00:57.300 --> 00:01:00.460
and D would be correct 25% of the time.

00:01:00.460 --> 00:01:03.750
Now, what would be our
alternative hypothesis?

00:01:03.750 --> 00:01:05.870
Well, alternative hypothesis

00:01:05.870 --> 00:01:08.170
would be not equal distribution,

00:01:08.170 --> 00:01:09.360
not

00:01:09.360 --> 00:01:11.453
equal distribution.

00:01:12.350 --> 00:01:14.950
Now, how are we going
to actually test this?

00:01:14.950 --> 00:01:17.280
Well, we've seen this show before,

00:01:17.280 --> 00:01:19.200
at least the beginnings of the show.

00:01:19.200 --> 00:01:23.250
You have the population of all
of your potential items here,

00:01:23.250 --> 00:01:24.730
and you could take a sample.

00:01:24.730 --> 00:01:27.990
And so let's say we take
a sample of 100 items.

00:01:27.990 --> 00:01:31.250
So n is equal to 100.

00:01:31.250 --> 00:01:33.210
And let's write down the data that we get

00:01:33.210 --> 00:01:35.110
when we look at that sample.

00:01:35.110 --> 00:01:38.280
So this is the correct choice,

00:01:38.280 --> 00:01:40.083
correct choice.

00:01:41.020 --> 00:01:45.200
And then this would be the expected number

00:01:45.200 --> 00:01:47.050
that you would expect.

00:01:47.050 --> 00:01:49.710
And then this is the actual number.

00:01:49.710 --> 00:01:51.050
And if this doesn't make sense yet,

00:01:51.050 --> 00:01:51.883
we'll see it in a second.

00:01:51.883 --> 00:01:56.883
So there's four different
choices, A, B, C, D

00:01:57.377 --> 00:01:59.030
and a sample of 100.

00:01:59.030 --> 00:02:00.600
Remember, in any hypothesis test,

00:02:00.600 --> 00:02:03.580
we start assuming that the
null hypothesis is true.

00:02:03.580 --> 00:02:05.950
So the expected number
where A is a correct choice

00:02:05.950 --> 00:02:08.090
would be 25% of this 100.

00:02:08.090 --> 00:02:11.180
So you would expect 25 times
the A to be the correct choice,

00:02:11.180 --> 00:02:13.490
25 times B to be the correct choice,

00:02:13.490 --> 00:02:15.580
25 times C to be the correct choice,

00:02:15.580 --> 00:02:18.770
and 25 times D to be the correct choice.

00:02:18.770 --> 00:02:20.390
But let's say our actual results,

00:02:20.390 --> 00:02:21.990
when we look at these 100 items,

00:02:21.990 --> 00:02:24.480
we get that A is the
correct choice 20 times,

00:02:24.480 --> 00:02:26.580
B is the correct choice 20 times,

00:02:26.580 --> 00:02:28.710
C is the correct choice 25 times,

00:02:28.710 --> 00:02:32.220
and D is the correct choice 35 times.

00:02:32.220 --> 00:02:33.230
So if you just look at this,

00:02:33.230 --> 00:02:36.260
just look, hey, maybe there's
a higher frequency of D,

00:02:36.260 --> 00:02:38.110
but maybe you'd say, well,
this is just a sample.

00:02:38.110 --> 00:02:39.630
And just through random chance,

00:02:39.630 --> 00:02:42.260
it might have just
gotten more Ds than not.

00:02:42.260 --> 00:02:44.590
There's some probability
of getting this result,

00:02:44.590 --> 00:02:48.000
even assuming that the
null hypothesis is true.

00:02:48.000 --> 00:02:50.070
And that's the goal of
these hypothesis tests,

00:02:50.070 --> 00:02:52.020
is say what's the probability
of getting a result

00:02:52.020 --> 00:02:54.200
at least this extreme?

00:02:54.200 --> 00:02:57.660
And if that probability
is below some threshold,

00:02:57.660 --> 00:03:00.220
then we tend to reject the null hypothesis

00:03:00.220 --> 00:03:02.090
and accept an alternative.

00:03:02.090 --> 00:03:04.450
And those thresholds you have seen before.

00:03:04.450 --> 00:03:06.590
We've seen these significance levels.

00:03:06.590 --> 00:03:10.203
Let's say we set a
significance level of 5%, 0.05.

00:03:11.920 --> 00:03:14.010
So if the probability
of getting this result

00:03:14.010 --> 00:03:17.420
or something even more
different than what's expected

00:03:17.420 --> 00:03:19.370
is less than the significance level,

00:03:19.370 --> 00:03:21.740
then we'd reject the null hypothesis.

00:03:21.740 --> 00:03:24.700
But this all leads to one
really interesting question.

00:03:24.700 --> 00:03:26.650
How do we calculate a probability

00:03:26.650 --> 00:03:29.760
of getting a result this
extreme or more extreme?

00:03:29.760 --> 00:03:31.720
How do we even measure that?

00:03:31.720 --> 00:03:35.320
And this is where we're going
to introduce a new statistic

00:03:35.320 --> 00:03:37.970
and also, for many of
you, a new Greek letter,

00:03:37.970 --> 00:03:41.340
and that is the capital Greek letter chi,

00:03:41.340 --> 00:03:42.960
which might look like an X to you.

00:03:42.960 --> 00:03:44.510
But it's a little bit curvier,

00:03:44.510 --> 00:03:46.200
and you could look up more on that.

00:03:46.200 --> 00:03:48.300
You kind of curve that part of the X.

00:03:48.300 --> 00:03:50.220
But it's a chi, not an X.

00:03:50.220 --> 00:03:53.670
And the statistic is called chi-squared,

00:03:53.670 --> 00:03:55.390
and it's a way of taking the difference

00:03:55.390 --> 00:03:57.100
between the actual and the expected

00:03:57.100 --> 00:03:59.090
and translating that into a number.

00:03:59.090 --> 00:04:01.090
And the chi-squared distribution is,

00:04:01.090 --> 00:04:04.510
well, I really should say
distributions are well studied.

00:04:04.510 --> 00:04:07.420
And we can use that to figure
out what is the probability

00:04:07.420 --> 00:04:10.720
of getting a result this
extreme or more extreme?

00:04:10.720 --> 00:04:12.400
And if that's lower than
our significance level,

00:04:12.400 --> 00:04:14.010
we reject the null hypothesis,

00:04:14.010 --> 00:04:15.870
and it suggests the alternative.

00:04:15.870 --> 00:04:19.860
But how do we calculate the
chi-squared statistic here?

00:04:19.860 --> 00:04:22.050
Well, it's reasonably intuitive.

00:04:22.050 --> 00:04:24.980
What we do is, for each
of these categories,

00:04:24.980 --> 00:04:27.430
in this case, it's for
each of these choices,

00:04:27.430 --> 00:04:28.560
we look at the difference

00:04:28.560 --> 00:04:30.610
between the actual and the expected.

00:04:30.610 --> 00:04:31.730
So for choice A,

00:04:31.730 --> 00:04:35.720
we'd say 20 is the actual
minus the expected.

00:04:35.720 --> 00:04:38.060
And then we're going to square that.

00:04:38.060 --> 00:04:42.060
And then we're going to
divide by what was expected.

00:04:42.060 --> 00:04:44.600
And then we're going to
do that for choice B.

00:04:44.600 --> 00:04:48.000
So we're going to say the
actual was 20, expected is 25,

00:04:48.000 --> 00:04:50.320
so 20 minus 25

00:04:50.320 --> 00:04:52.540
squared, over the expected,

00:04:52.540 --> 00:04:54.180
over 25.

00:04:54.180 --> 00:04:56.030
Plus then we do that for choice C.

00:04:56.030 --> 00:04:58.400
25 minus 25,

00:04:58.400 --> 00:05:00.330
we know where that one will end up,

00:05:00.330 --> 00:05:03.690
squared, over the expected, over 25.

00:05:03.690 --> 00:05:06.020
And then finally, for choice D,

00:05:06.020 --> 00:05:09.240
which is going to get us 35

00:05:09.240 --> 00:05:10.650
minus 25

00:05:10.650 --> 00:05:11.690
squared,

00:05:11.690 --> 00:05:14.253
all of that over 25.

00:05:15.430 --> 00:05:18.750
And we are now, let's
see, if we calculate this,

00:05:18.750 --> 00:05:20.740
this is going to be negative five squared.

00:05:20.740 --> 00:05:22.860
So that's going to be 25.

00:05:22.860 --> 00:05:24.800
This is going to be 25.

00:05:24.800 --> 00:05:26.600
This is going to be zero.

00:05:26.600 --> 00:05:30.230
35 minus 25 is 10, squared, that is 100.

00:05:30.230 --> 00:05:31.870
So this is one

00:05:31.870 --> 00:05:33.510
plus one,

00:05:33.510 --> 00:05:34.950
plus zero,

00:05:34.950 --> 00:05:36.490
plus four.

00:05:36.490 --> 00:05:39.110
So our chi-squared
statistic, in this example,

00:05:39.110 --> 00:05:40.370
came out nice and clean,

00:05:40.370 --> 00:05:43.950
this won't always be the case, at six.

00:05:43.950 --> 00:05:45.580
So what do we make of this?

00:05:45.580 --> 00:05:47.350
Well, what we can do is then look

00:05:47.350 --> 00:05:49.850
at a chi-squared distribution

00:05:49.850 --> 00:05:51.670
for the appropriate degrees of freedom,

00:05:51.670 --> 00:05:53.240
and we'll talk about that in a second,

00:05:53.240 --> 00:05:55.480
and say what is the probability

00:05:55.480 --> 00:05:59.930
of getting a chi-squared
statistic six or larger?

00:05:59.930 --> 00:06:01.440
And to understand what a chi-squared

00:06:01.440 --> 00:06:03.200
distribution even looks like,

00:06:03.200 --> 00:06:05.210
these are multiple
chi-squared distributions

00:06:05.210 --> 00:06:08.900
for different values for
the degrees of freedom.

00:06:08.900 --> 00:06:10.770
And to calculate the degrees of freedom,

00:06:10.770 --> 00:06:12.490
you look at the number of categories.

00:06:12.490 --> 00:06:14.100
In this case, we have four categories,

00:06:14.100 --> 00:06:15.760
and you subtract one.

00:06:15.760 --> 00:06:16.980
Now, that makes a lot of sense

00:06:16.980 --> 00:06:20.740
because, if you knew how many
As, Bs, and Cs there are,

00:06:20.740 --> 00:06:23.530
if you knew the proportions,
even the assumed proportions,

00:06:23.530 --> 00:06:25.670
you can always calculate the fourth one.

00:06:25.670 --> 00:06:29.270
That's why it is four minus
one degrees of freedom.

00:06:29.270 --> 00:06:31.360
So in this case, our degrees of freedom

00:06:31.360 --> 00:06:32.870
are going to be equal to three.

00:06:32.870 --> 00:06:35.690
Over here, sometimes you'll
see it described as k,

00:06:35.690 --> 00:06:37.550
so k is equal to three.

00:06:37.550 --> 00:06:41.080
So if we look at, that's
that little light blue,

00:06:41.080 --> 00:06:43.890
so we're looking at this
chi-squared distribution

00:06:43.890 --> 00:06:46.920
where the degree of freedom is three.

00:06:46.920 --> 00:06:49.010
And we want to figure out
what is the probability

00:06:49.010 --> 00:06:52.930
of getting a chi-squared
statistic that is six or greater?

00:06:52.930 --> 00:06:57.070
So we would be looking at
this area right over here.

00:06:57.070 --> 00:06:59.300
And you could figure it
out using a calculator,

00:06:59.300 --> 00:07:00.930
or, if you're taking some type of a test,

00:07:00.930 --> 00:07:03.650
like an AP Statistics exam, for example,

00:07:03.650 --> 00:07:06.280
you could use their tables they give you.

00:07:06.280 --> 00:07:09.970
And so a table like this
could be quite useful.

00:07:09.970 --> 00:07:11.600
Remember, we're dealing with the situation

00:07:11.600 --> 00:07:13.470
where we have three degrees of freedom.

00:07:13.470 --> 00:07:16.290
We had four categories, so
four minus one is three.

00:07:16.290 --> 00:07:18.940
And we got a chi-squared value.

00:07:18.940 --> 00:07:22.000
Our chi-squared statistic was six.

00:07:22.000 --> 00:07:23.830
So this right over here tells us

00:07:23.830 --> 00:07:25.910
the probability of getting a 6.25

00:07:27.070 --> 00:07:31.660
or greater for our
chi-squared value is 10%.

00:07:31.660 --> 00:07:33.560
If we go back to this chart,

00:07:33.560 --> 00:07:38.250
we just learned that this
probability from 6.25 and up,

00:07:38.250 --> 00:07:40.180
when we have three degrees of freedom,

00:07:40.180 --> 00:07:43.130
that this right over here is 10%.

00:07:43.130 --> 00:07:44.840
Well, that's 10%.

00:07:44.840 --> 00:07:46.710
Then the probability,

00:07:46.710 --> 00:07:49.960
the probability of getting
a chi-squared value

00:07:49.960 --> 00:07:51.680
greater than or equal to six

00:07:51.680 --> 00:07:54.280
is going to be greater than 10%,

00:07:54.280 --> 00:07:55.550
greater than 10%.

00:07:55.550 --> 00:07:58.350
And we could also view
this as our P-value.

00:07:58.350 --> 00:08:01.540
And so for our probability,
assuming the null hypothesis

00:08:01.540 --> 00:08:04.090
is greater than 10%, well,
it's definitely going

00:08:04.090 --> 00:08:05.920
to be greater than our significance level.

00:08:05.920 --> 00:08:09.380
And because of that,
we will fail to reject,

00:08:09.380 --> 00:08:11.710
fail to reject.

00:08:11.710 --> 00:08:15.110
And so this is an example of,
even though in your sample,

00:08:15.110 --> 00:08:17.380
you just happened to get more Ds,

00:08:17.380 --> 00:08:20.240
the probability of getting
a result at least as extreme

00:08:20.240 --> 00:08:24.923
as what you saw is going to
be a little bit over 10%.


Current due to closing a switch: worked example

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:01.827
- [Instructor] We are
asked, "How does the current

00:00:01.827 --> 00:00:04.417
"going through R1," so, this resistor,

00:00:04.417 --> 00:00:06.260
"when the switch is open,"

00:00:06.260 --> 00:00:09.327
this switch, "compare to
the current through R1

00:00:09.327 --> 00:00:11.130
"when the switch is closed?"

00:00:11.130 --> 00:00:14.590
Pause this video and see
if you can figure that out.

00:00:14.590 --> 00:00:18.240
Alright, so let's just think
about the two scenarios.

00:00:18.240 --> 00:00:20.560
So, we can view the current as this,

00:00:20.560 --> 00:00:22.750
right over here, this
current that we care about.

00:00:22.750 --> 00:00:24.440
We could either measure it there

00:00:24.440 --> 00:00:27.460
or you could measure it right over there,

00:00:27.460 --> 00:00:29.780
and let's first think about the scenario

00:00:29.780 --> 00:00:32.410
where the switch is open.

00:00:32.410 --> 00:00:36.690
So, our current when our switch is open

00:00:36.690 --> 00:00:38.890
is going to be equal to the voltage

00:00:38.890 --> 00:00:43.280
across the resistors, and
that's going to be our 12 volts,

00:00:43.280 --> 00:00:44.500
12 volts,

00:00:44.500 --> 00:00:47.740
divided by the equivalent
resistance of these resistors.

00:00:47.740 --> 00:00:49.120
When the switch is open, essentially,

00:00:49.120 --> 00:00:52.220
we just have R1 and R2 in series,

00:00:52.220 --> 00:00:55.940
and so this is just gonna be R1+R2.

00:00:55.940 --> 00:00:57.520
If you have two resistors in a series,

00:00:57.520 --> 00:00:58.800
their equivalent resistance is just

00:00:58.800 --> 00:01:01.000
the sum of the resistances.

00:01:01.000 --> 00:01:02.290
Fair enough.

00:01:02.290 --> 00:01:03.980
Now, let's think about the situation

00:01:03.980 --> 00:01:06.320
where the switch is closed.

00:01:06.320 --> 00:01:07.393
Closed.

00:01:08.270 --> 00:01:11.880
So here, our current at
this point of our circuit,

00:01:11.880 --> 00:01:14.850
or the current going through R1,

00:01:14.850 --> 00:01:17.100
so I sub closed,

00:01:17.100 --> 00:01:20.200
is, once again, it's going
to be equal to 12 volts,

00:01:20.200 --> 00:01:22.820
the voltage across the resistors,

00:01:22.820 --> 00:01:25.600
but what are we gonna divide by now?

00:01:25.600 --> 00:01:28.510
When we close the switch, what happens?

00:01:28.510 --> 00:01:30.880
Well, these lines where
we see no resistors

00:01:30.880 --> 00:01:34.340
in circuit diagrams, that's
assumed to be resistance-less,

00:01:34.340 --> 00:01:38.520
so all of the current will
actually flow that way.

00:01:38.520 --> 00:01:40.800
So, by closing this switch,

00:01:40.800 --> 00:01:44.570
you're essentially removing
R2 from the circuit.

00:01:44.570 --> 00:01:46.380
The current will just go through R1,

00:01:46.380 --> 00:01:50.000
and then follow the path of
least resistance, literally.

00:01:50.000 --> 00:01:52.830
And so, in this situation,
our current is going

00:01:52.830 --> 00:01:55.310
to be 12 volt divided
by, you essentially just

00:01:55.310 --> 00:01:58.710
have one resistance, divided by R1.

00:01:58.710 --> 00:02:00.590
So, when you closed the circuit,

00:02:00.590 --> 00:02:03.000
you've essentially taken a resistor out,

00:02:03.000 --> 00:02:05.070
and so if you took a resistor out,

00:02:05.070 --> 00:02:07.480
you're going to increase the current,

00:02:07.480 --> 00:02:09.340
so you could just write it as,

00:02:09.340 --> 00:02:12.590
the current when the switch is open

00:02:12.590 --> 00:02:15.650
is going to be less than,

00:02:15.650 --> 00:02:18.080
is going to be less than the current

00:02:18.080 --> 00:02:21.290
when the switch is closed.

00:02:21.290 --> 00:02:23.400
Once again, why is that?

00:02:23.400 --> 00:02:25.420
Well, just look at the denominators here.

00:02:25.420 --> 00:02:26.640
When the switch is open,

00:02:26.640 --> 00:02:28.570
you're dividing by a larger number

00:02:28.570 --> 00:02:30.130
than when the switch is closed.

00:02:30.130 --> 00:02:31.270
Or, another way of thinking about it,

00:02:31.270 --> 00:02:35.520
when the switch is open, the
R2 resistance is factored in.

00:02:35.520 --> 00:02:38.430
When the switch is closed,
the R2 resistance essentially

00:02:38.430 --> 00:02:41.770
becomes a non-factor and
you have less resistance,

00:02:41.770 --> 00:02:44.663
which would mean you
would have higher current.


Current through resistor in parallel: Worked example

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.710
- [Instructor] So we have
an interesting circuit here.

00:00:01.710 --> 00:00:04.210
The goal of this video is to
figure out what is the current

00:00:04.210 --> 00:00:07.850
that flows through this six-ohm resistor?

00:00:07.850 --> 00:00:10.930
Pause this video, and see
if you can work through it.

00:00:10.930 --> 00:00:13.150
So the way that I am going to tackle it

00:00:13.150 --> 00:00:15.900
is first simplify the circuit.

00:00:15.900 --> 00:00:18.570
Take these two resistors in parallel,

00:00:18.570 --> 00:00:22.390
and think about what the
equivalent resistance would be.

00:00:22.390 --> 00:00:24.240
And we have seen that before.

00:00:24.240 --> 00:00:27.250
One over the equivalent resistance

00:00:27.250 --> 00:00:31.450
is going to be equal to one over 6.0 ohms

00:00:31.450 --> 00:00:35.240
plus one over 12.0 ohms.

00:00:35.240 --> 00:00:39.800
1/6 is the same thing as two over 12.

00:00:39.800 --> 00:00:42.660
So 2/12 plus 1/12 is 3/12.

00:00:42.660 --> 00:00:44.430
And 3/12, you could view that

00:00:44.430 --> 00:00:47.290
as the same thing as one over 4.0 ohms.

00:00:47.290 --> 00:00:49.320
And so one over the equivalent resistance

00:00:49.320 --> 00:00:51.330
is equal to one over four ohms.

00:00:51.330 --> 00:00:53.110
Well, that means that
the equivalent resistance

00:00:53.110 --> 00:00:54.680
is four ohms.

00:00:54.680 --> 00:00:57.590
And so we can simplify our circuit now,

00:00:57.590 --> 00:00:59.860
where we replace these
two resistors in parallel

00:00:59.860 --> 00:01:02.100
with one resistor of the
equivalent resistance,

00:01:02.100 --> 00:01:05.830
and that is going to be
equal to a four-ohm resistor.

00:01:05.830 --> 00:01:08.640
Now the next thing we could
do is we could figure out

00:01:08.640 --> 00:01:11.260
what the current is through
this part of the circuit,

00:01:11.260 --> 00:01:12.800
which would be the same
thing as the current

00:01:12.800 --> 00:01:13.960
right over there.

00:01:13.960 --> 00:01:17.940
We could call that i sub one.

00:01:17.940 --> 00:01:20.890
And we can just use Ohm's law for that.

00:01:20.890 --> 00:01:25.500
We would have i sub one would
be equal to our voltage drop,

00:01:25.500 --> 00:01:27.230
which is 24 volts,

00:01:27.230 --> 00:01:28.870
24 volts

00:01:28.870 --> 00:01:30.170
divided by

00:01:30.170 --> 00:01:35.000
the equivalent resistance of
these two resistors in series.

00:01:35.000 --> 00:01:37.340
And when you have resistors
in series, you just add 'em up

00:01:37.340 --> 00:01:39.060
to figure out the equivalent resistance.

00:01:39.060 --> 00:01:42.720
So this would be divided by 2.0 ohms

00:01:42.720 --> 00:01:44.370
plus four ohms,

00:01:44.370 --> 00:01:47.190
plus 4.0 ohms.

00:01:47.190 --> 00:01:49.240
24 divided by two plus four,

00:01:49.240 --> 00:01:51.600
or 24 divided by six, is four.

00:01:51.600 --> 00:01:54.150
And since we're dealing
with two significant digits,

00:01:54.150 --> 00:01:55.760
it'll be 4.0.

00:01:55.760 --> 00:01:56.900
And we're talking about current,

00:01:56.900 --> 00:02:00.650
so this is 4.0 amps, or 4.0 amperes.

00:02:00.650 --> 00:02:02.570
Now how do we use that information

00:02:02.570 --> 00:02:05.310
to calculate this current right over here?

00:02:05.310 --> 00:02:08.630
We can call that i sub two.

00:02:08.630 --> 00:02:10.600
Now one way to think about it is

00:02:10.600 --> 00:02:13.140
what is going to be your voltage drop

00:02:13.140 --> 00:02:16.060
from this point to this point?

00:02:16.060 --> 00:02:18.890
If you know the voltage drop
from that point to that point

00:02:18.890 --> 00:02:21.050
and if you know that the
voltage drop from this point

00:02:21.050 --> 00:02:23.010
all the way down here is,

00:02:23.010 --> 00:02:25.740
then we can figure out what
the voltage drop from here

00:02:25.740 --> 00:02:27.210
to here is going to be.

00:02:27.210 --> 00:02:28.490
So let's do that.

00:02:28.490 --> 00:02:31.960
So the voltage drop across
this first resistor,

00:02:31.960 --> 00:02:34.220
remember, your change in voltage

00:02:34.220 --> 00:02:38.260
is just equal to your current
times your resistance.

00:02:38.260 --> 00:02:39.670
And so this is going to be,

00:02:39.670 --> 00:02:43.110
your current is going to be four amperes

00:02:44.500 --> 00:02:46.970
times your resistance is two ohms,

00:02:46.970 --> 00:02:51.413
times two ohms, which is going
to be equal to 8.0 volts.

00:02:53.130 --> 00:02:55.360
And so if the voltage
difference between that point

00:02:55.360 --> 00:02:58.210
and that point is 24 volts,

00:02:58.210 --> 00:03:00.060
which we know from this voltage source,

00:03:00.060 --> 00:03:02.780
but if we drop eight volts
as we go to this point,

00:03:02.780 --> 00:03:05.400
well, then the difference
between this point and this point

00:03:05.400 --> 00:03:08.370
or this point and this
point right over here,

00:03:08.370 --> 00:03:11.700
this has got to be a 16-volt drop.

00:03:11.700 --> 00:03:16.540
If our delta V across the
six-ohm resistor is equal to

00:03:16.540 --> 00:03:18.740
16 volts,

00:03:18.740 --> 00:03:21.880
well, then we can use Ohm's
law again to figure out i two.

00:03:21.880 --> 00:03:25.730
I sub two is going to be
equal to our drop in voltage,

00:03:25.730 --> 00:03:29.300
so 16 volts, divided by this resistance,

00:03:29.300 --> 00:03:31.120
six ohms.

00:03:31.120 --> 00:03:33.210
And so what is this going to be equal to?

00:03:33.210 --> 00:03:38.210
16 divided by six is 2 4/6 or 2 2/3,

00:03:38.550 --> 00:03:42.190
or 2.666666, and if we round
to two significant digits,

00:03:42.190 --> 00:03:46.110
you're going to have 2.7 amps.

00:03:46.110 --> 00:03:48.210
And so we just figured out
what we wanted to figure out.

00:03:48.210 --> 00:03:52.980
This right over here is
2.7 amps, or 2.7 amperes.

00:03:52.980 --> 00:03:54.980
But we can keep analyzing it.

00:03:54.980 --> 00:03:56.560
For fun, I encourage you to figure out

00:03:56.560 --> 00:04:00.230
what that current is now,
the current i sub three.

00:04:00.230 --> 00:04:02.490
And use the exact same technique.

00:04:02.490 --> 00:04:05.170
And one thing that you should
feel very comfortable of

00:04:05.170 --> 00:04:06.530
is that this current that is flowing

00:04:06.530 --> 00:04:09.800
through the first resistor,
that four-ampere current,

00:04:09.800 --> 00:04:12.950
that current gets split
between i two and i three.

00:04:12.950 --> 00:04:15.950
And so i two and i three should add up

00:04:15.950 --> 00:04:18.460
to the original four amps.

00:04:18.460 --> 00:04:20.110
So just thinking about it that way,

00:04:20.110 --> 00:04:22.950
if you do the same type
of analysis we just did,

00:04:22.950 --> 00:04:27.620
you should get 1.3 amps
for i three because 2.7

00:04:27.620 --> 00:04:31.323
plus 1.3 is going to
be equal to four amps.


Types of forces and free body diagrams

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:02.400
- [Instructor] In this
video, we're gonna discuss

00:00:02.400 --> 00:00:03.620
different types of forces,

00:00:03.620 --> 00:00:04.950
but we're gonna do it in the context

00:00:04.950 --> 00:00:07.380
of free body diagrams.

00:00:07.380 --> 00:00:09.640
So let's say I have a table here,

00:00:09.640 --> 00:00:14.640
and I have a block that is
sitting stationary on that table.

00:00:14.840 --> 00:00:16.530
What are all of the forces

00:00:16.530 --> 00:00:19.640
that are going to act on this blocK?

00:00:19.640 --> 00:00:21.320
Well, to do that, to think about that,

00:00:21.320 --> 00:00:22.880
I can draw a free body diagram

00:00:22.880 --> 00:00:24.760
where I am only going to draw the block.

00:00:24.760 --> 00:00:26.630
Remember, in free body diagrams,

00:00:26.630 --> 00:00:28.900
you only care about the
forces acting on one

00:00:28.900 --> 00:00:31.060
of the the objects in your system.

00:00:31.060 --> 00:00:33.330
So, if we're looking at only the block,

00:00:33.330 --> 00:00:34.163
what's going on?

00:00:34.163 --> 00:00:35.860
We're going to assume that
the block is on earth,

00:00:35.860 --> 00:00:37.940
we're assuming that it's stationary.

00:00:37.940 --> 00:00:41.010
Well, if it's on earth,
the block has some weight.

00:00:41.010 --> 00:00:44.300
You have the force of
gravity acting on the block.

00:00:44.300 --> 00:00:47.390
And so let me draw that
in my free body diagram.

00:00:47.390 --> 00:00:49.290
So you're gonna have a downward force,

00:00:49.290 --> 00:00:52.090
and it's magnitude is gonna be F sub g.

00:00:52.090 --> 00:00:55.040
We could also call that or w.

00:00:55.040 --> 00:00:57.890
And even though this block
had contact with a table

00:00:57.890 --> 00:01:00.090
which maybe has contact with the earth,

00:01:00.090 --> 00:01:03.560
weight, or the force of
gravity is a long-range force.

00:01:03.560 --> 00:01:05.570
Even if this block was in orbit,

00:01:05.570 --> 00:01:07.950
even if it wasn't in
orbit, it would still have

00:01:07.950 --> 00:01:10.000
gravitational interactions with the earth.

00:01:10.000 --> 00:01:12.650
The earth would still be pulling on it.

00:01:12.650 --> 00:01:15.050
But going back to this free body diagram,

00:01:15.050 --> 00:01:18.050
if this was the only
force acting on the block,

00:01:18.050 --> 00:01:20.410
the block would accelerate downwards.

00:01:20.410 --> 00:01:22.190
But we're assuming that it's stationary.

00:01:22.190 --> 00:01:24.640
So there must be another
force that is netting out

00:01:24.640 --> 00:01:26.313
against the force of gravity.

00:01:26.313 --> 00:01:28.010
Now, what would that be?

00:01:28.010 --> 00:01:29.350
Well, that would be the force

00:01:29.350 --> 00:01:32.030
of the table pushing on the block.

00:01:32.030 --> 00:01:34.900
And this force of pushing in a direction

00:01:34.900 --> 00:01:37.720
that is perpendicular to
the surface of an object,

00:01:37.720 --> 00:01:40.050
that's known as normal force.

00:01:40.050 --> 00:01:44.450
And its magnitude you could
denote as capital F sub N.

00:01:44.450 --> 00:01:46.810
Let's do another example, but this time,

00:01:46.810 --> 00:01:50.070
instead of having the block on a table,

00:01:50.070 --> 00:01:52.720
let's say it is hanging from a string

00:01:52.720 --> 00:01:55.660
which is attached to the ceiling.

00:01:55.660 --> 00:01:58.320
But once again, everything is stationary.

00:01:58.320 --> 00:02:00.491
Draw a free body diagram for that.

00:02:00.491 --> 00:02:05.346
Well, once again, I am only
concerned with the block.

00:02:05.346 --> 00:02:07.630
It's still on earth, we're assuming.

00:02:07.630 --> 00:02:10.050
So you're going to have
the force of gravity

00:02:10.050 --> 00:02:12.810
acting downwards on the block.

00:02:12.810 --> 00:02:15.410
But what's keeping it from
accelerating downwards?

00:02:15.410 --> 00:02:17.060
Well, you might say,
well, you got the string

00:02:17.060 --> 00:02:19.850
that's holding it up,
that is pulling on it.

00:02:19.850 --> 00:02:22.820
And that pulling force
is known as tension.

00:02:22.820 --> 00:02:25.110
So what you would have
here is an upward force

00:02:25.110 --> 00:02:27.500
that nets out against
the force of gravity.

00:02:27.500 --> 00:02:30.950
And sometimes its magnitude
is denoted by capital T

00:02:30.950 --> 00:02:35.950
or it might be a F sub T.

00:02:35.980 --> 00:02:38.690
Now, let's make things a
little bit interesting.

00:02:38.690 --> 00:02:41.430
Let's try to kind of combine these things,

00:02:41.430 --> 00:02:43.950
and we'll actually introduce a new force.

00:02:43.950 --> 00:02:48.830
So let's say that we, this is
the ground right over here.

00:02:48.830 --> 00:02:51.973
I have a block on the ground.

00:02:52.980 --> 00:02:56.070
And I have a situation where I am pulling

00:02:56.070 --> 00:02:58.080
on this block using a rope

00:02:58.930 --> 00:03:02.810
with a force of magnitude,
let's just call this

00:03:02.810 --> 00:03:04.630
the force of tension.

00:03:04.630 --> 00:03:06.410
I am pulling on that block.

00:03:06.410 --> 00:03:08.810
But the block is not moving.

00:03:08.810 --> 00:03:11.963
What would be the free body
diagram for this block?

00:03:13.010 --> 00:03:14.740
Well, I'll do the same thing again.

00:03:14.740 --> 00:03:17.490
I will draw the block.

00:03:17.490 --> 00:03:19.410
Now, in the vertical direction,

00:03:19.410 --> 00:03:22.530
you have the same thing that
you saw in that first scenario.

00:03:22.530 --> 00:03:24.840
You're going to have the force of gravity

00:03:24.840 --> 00:03:28.920
or the weight of the block
pulling downward on the block.

00:03:28.920 --> 00:03:31.940
And that's going to be
counteracted by the normal force

00:03:31.940 --> 00:03:33.430
of the ground on the block.

00:03:33.430 --> 00:03:35.820
The ground is holding up the block

00:03:35.820 --> 00:03:36.950
is one way to think about it,

00:03:36.950 --> 00:03:39.200
keeping it from accelerating downwards.

00:03:39.200 --> 00:03:42.133
So the normal force is acting upwards.

00:03:43.214 --> 00:03:45.900
But what about the horizontal direction?

00:03:45.900 --> 00:03:47.800
I already said that I'm
pulling to the right

00:03:47.800 --> 00:03:50.410
with a force of magnitude F sub T.

00:03:50.410 --> 00:03:53.143
So let me do that on my free body diagram.

00:03:53.980 --> 00:03:56.870
So this would be F sub T.

00:03:56.870 --> 00:03:58.530
But I said it's stationary.

00:03:58.530 --> 00:04:01.820
So there must be something
that is counteracting that,

00:04:01.820 --> 00:04:04.760
that is netting against that,
going in that direction.

00:04:04.760 --> 00:04:07.430
What force would that be?

00:04:07.430 --> 00:04:09.160
Well, that would be the force of friction.

00:04:09.160 --> 00:04:11.070
We've all experienced
trying to pull on something,

00:04:11.070 --> 00:04:12.900
trying to drag something across the ground

00:04:12.900 --> 00:04:15.260
and it doesn't move, and
that's because there's friction

00:04:15.260 --> 00:04:17.220
between the object and the ground.

00:04:17.220 --> 00:04:18.800
And friction, fundamentally, it could be

00:04:18.800 --> 00:04:21.760
because the surfaces of
the two objects are rough

00:04:21.760 --> 00:04:24.050
and you kind of have to
grind them pass each other.

00:04:24.050 --> 00:04:26.760
Or sometimes it can even be
due to molecular interactions

00:04:26.760 --> 00:04:27.940
where they're kind of sticky,

00:04:27.940 --> 00:04:30.090
where the objects are
attracted to each other

00:04:30.090 --> 00:04:32.320
and you gotta pull passed that.

00:04:32.320 --> 00:04:35.490
And so in this situation, you
have the force of friction

00:04:35.490 --> 00:04:39.270
counteracting this pulling
force, the force of tension,

00:04:39.270 --> 00:04:40.570
the force of friction.

00:04:40.570 --> 00:04:42.670
And the force of friction
is really interesting,

00:04:42.670 --> 00:04:45.930
because it always goes against
the direction of sliding,

00:04:45.930 --> 00:04:48.265
it always goes against motion.

00:04:48.265 --> 00:04:50.840
Now, with all of these
examples out of the way,

00:04:50.840 --> 00:04:53.980
let's try to do a more complex scenario.

00:04:53.980 --> 00:04:56.363
Let's say that I have a shelf,

00:04:57.270 --> 00:05:00.990
and it has a weight of

00:05:00.990 --> 00:05:02.570
10 newtons.

00:05:02.570 --> 00:05:06.920
Sitting on that shelf I have an object

00:05:06.920 --> 00:05:10.820
that has a weight of five newtons.

00:05:10.820 --> 00:05:13.420
And let's say I have two wires

00:05:13.420 --> 00:05:14.820
and everything is symmetric,

00:05:15.677 --> 00:05:17.030
but this weight is right on the middle,

00:05:17.030 --> 00:05:20.000
and these wires are at both
of the ends of the shelf,

00:05:20.000 --> 00:05:22.800
and this is wire one and this is wire two

00:05:22.800 --> 00:05:24.990
and they are attached to the ceiling.

00:05:24.990 --> 00:05:26.900
And for the sake of simplicity,

00:05:26.900 --> 00:05:28.790
we're gonna assume that
the wires have no weight.

00:05:28.790 --> 00:05:29.850
In actuality they would,

00:05:29.850 --> 00:05:31.330
but for the sake of this argument,

00:05:31.330 --> 00:05:33.360
let's assume that they are weightless.

00:05:33.360 --> 00:05:35.150
What would be a free body diagram

00:05:35.150 --> 00:05:38.323
for this five newton block
that sits on the shelf?

00:05:39.490 --> 00:05:41.160
Well, that one is actually
pretty straightforward,

00:05:41.160 --> 00:05:44.130
and it's analogous to this
first scenario that we saw.

00:05:44.130 --> 00:05:46.860
You have your block, you
have the force of gravity

00:05:46.860 --> 00:05:50.500
pulling down with a force
of magnitude, five newtons,

00:05:50.500 --> 00:05:54.270
and that's gonna be
counteracted by a normal force

00:05:54.270 --> 00:05:56.570
of the same magnitude but going upwards.

00:05:56.570 --> 00:05:58.630
So make sure I have enough space.

00:05:58.630 --> 00:06:02.210
So that's gonna be counteracted
with the normal force

00:06:02.210 --> 00:06:05.330
which is going to be equal
to five newtons upwards.

00:06:05.330 --> 00:06:07.290
And to be clear, this five newtons,

00:06:07.290 --> 00:06:10.210
this is equal to the weight, the magnitude

00:06:10.210 --> 00:06:12.070
of the weight of the object.

00:06:12.070 --> 00:06:13.200
So that was pretty straightforward,

00:06:13.200 --> 00:06:15.660
the free body diagram for just the block.

00:06:15.660 --> 00:06:16.870
And it's really important to see that,

00:06:16.870 --> 00:06:18.620
because notice, in the free body diagram,

00:06:18.620 --> 00:06:20.160
all you see is the block.

00:06:20.160 --> 00:06:22.977
But now let's draw the free
body diagram for the shelf.

00:06:22.977 --> 00:06:26.543
So if I have a shelf right over here.

00:06:26.543 --> 00:06:28.257
Pause this video and try to do that.

00:06:28.257 --> 00:06:31.410
Well, we know its weight, it's 10 newtons.

00:06:31.410 --> 00:06:33.290
So we can do that first.

00:06:33.290 --> 00:06:35.660
So,

00:06:35.660 --> 00:06:37.130
it has a weight of 10 newtons,

00:06:37.130 --> 00:06:39.370
so the force, the magnitude of the force

00:06:39.370 --> 00:06:42.383
of gravity downwards is 10 newtons.

00:06:42.383 --> 00:06:44.343
Is that the only downward force?

00:06:44.343 --> 00:06:48.534
Well no, you have this
object that's sitting on it,

00:06:48.534 --> 00:06:50.890
and gravity is pulling down on that object

00:06:50.890 --> 00:06:52.420
with a force of five newtons,

00:06:52.420 --> 00:06:56.830
and that causes that object
to push on our shelf.

00:06:56.830 --> 00:06:59.740
So that pushing force is
actually a normal force.

00:06:59.740 --> 00:07:02.680
It's due to the gravity on
that five newton object,

00:07:02.680 --> 00:07:04.820
but the end result of
the five newton object

00:07:04.820 --> 00:07:08.540
is pushing down on our shelf.

00:07:08.540 --> 00:07:12.537
So what you have is another
force that is pushing down.

00:07:12.537 --> 00:07:15.070
And it is going to be a five newton force.

00:07:15.070 --> 00:07:17.760
And really, we should view
that as a normal force.

00:07:17.760 --> 00:07:20.100
It's a contact force, it's a pushing force

00:07:20.100 --> 00:07:22.730
of the five newton object
on the 10 newton shelf.

00:07:22.730 --> 00:07:25.480
So this is going to have a
magnitude of five newtons.

00:07:25.480 --> 00:07:27.740
Assuming that it's completely stationary,

00:07:27.740 --> 00:07:29.600
there must be some
counteracting forces here.

00:07:29.600 --> 00:07:31.310
Where is that gonna come from?

00:07:31.310 --> 00:07:32.143
Well, that's gonna come

00:07:32.143 --> 00:07:35.620
from the pulling forces of these wires.

00:07:35.620 --> 00:07:39.040
So you're going to have
the tension from rope one,

00:07:39.040 --> 00:07:41.430
we could call that T sub one,

00:07:41.430 --> 00:07:43.010
and you're gonna have the tension

00:07:43.010 --> 00:07:46.140
from wire or rope two, T sub two.

00:07:46.140 --> 00:07:48.040
And because this thing is stationary,

00:07:48.040 --> 00:07:50.690
T sub one plus T sub two should be equal

00:07:50.690 --> 00:07:53.430
to 10 newtons plus five newtons.

00:07:53.430 --> 00:07:54.380
So I'll leave you there.

00:07:54.380 --> 00:07:56.870
We've done a nice survey of various forces

00:07:56.870 --> 00:07:58.650
you might see in a first
year physics class.

00:07:58.650 --> 00:07:59.880
And we've been able to think about them

00:07:59.880 --> 00:08:02.193
in the context of free body diagrams.


Third parties in the United States

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:02.800
- Let's talk about third
parties in the United States.

00:00:02.800 --> 00:00:04.890
And I put the word
third in quotation marks

00:00:04.890 --> 00:00:06.980
because there's more than one third party.

00:00:06.980 --> 00:00:08.600
So you could even think
of it as a third, fourth,

00:00:08.600 --> 00:00:10.440
fifth, sixth and seventh parties.

00:00:10.440 --> 00:00:12.850
But the reason why
people say third parties

00:00:12.850 --> 00:00:14.250
is because in the United States,

00:00:14.250 --> 00:00:15.660
you have two dominant parties.

00:00:15.660 --> 00:00:18.340
You have the Republicans
and you have the Democrats.

00:00:18.340 --> 00:00:22.080
And so any other party is
considered to be a third party

00:00:22.080 --> 00:00:25.700
after those first two
big, dominant parties.

00:00:25.700 --> 00:00:28.780
So here are some of the
logos of some of the,

00:00:28.780 --> 00:00:31.990
I would say, major third
parties in the United States.

00:00:31.990 --> 00:00:33.920
And I'm gonna put major in quotes

00:00:33.920 --> 00:00:36.730
because they don't have much of a say

00:00:36.730 --> 00:00:38.320
in our government today.

00:00:38.320 --> 00:00:40.890
You have the Libertarian Party,

00:00:40.890 --> 00:00:44.780
that is very focused on
people's individual liberties.

00:00:44.780 --> 00:00:46.760
They generally think about the government

00:00:46.760 --> 00:00:48.440
doing as little as possible.

00:00:48.440 --> 00:00:51.620
That whenever the government
strives to get bigger,

00:00:51.620 --> 00:00:53.300
it infringes on people's liberties,

00:00:53.300 --> 00:00:57.890
either in the economic sphere
or in the social sphere.

00:00:57.890 --> 00:01:00.780
The Green Party is also
very pro-civil liberties,

00:01:00.780 --> 00:01:01.920
but you can also imagine,

00:01:01.920 --> 00:01:03.500
because it's called the Green Party,

00:01:03.500 --> 00:01:06.820
it is very concerned with the environment.

00:01:06.820 --> 00:01:08.580
Now, this third, third party,

00:01:08.580 --> 00:01:10.560
and there's many more
than just these three.

00:01:10.560 --> 00:01:12.370
This is the Reform Party.

00:01:12.370 --> 00:01:14.640
And the Reform Party is really interesting

00:01:14.640 --> 00:01:19.410
because it was started
by Ross Perot in 1995.

00:01:19.410 --> 00:01:22.290
And this was after, in 1992,

00:01:22.290 --> 00:01:25.850
Ross Perot ran as an Independent
candidate for President.

00:01:25.850 --> 00:01:27.690
And he did surprisingly well.

00:01:27.690 --> 00:01:31.850
He got nearly 20 percent
of the popular vote.

00:01:31.850 --> 00:01:32.910
But an interesting question is

00:01:32.910 --> 00:01:36.440
even though he got 20 percent
of the popular vote in 1992,

00:01:36.440 --> 00:01:38.130
and even though in the Reform Party,

00:01:38.130 --> 00:01:42.520
it had some reasonable support
even in the 1996 election,

00:01:42.520 --> 00:01:44.830
how come we don't see Congress people

00:01:44.830 --> 00:01:46.870
who represent the Reform Party?

00:01:46.870 --> 00:01:49.180
And there's two real answers here.

00:01:49.180 --> 00:01:52.130
One is, the winner-take-all system.

00:01:52.130 --> 00:01:57.120
Winner - take - all.

00:01:57.120 --> 00:01:58.530
So if we're dealing with a situation

00:01:58.530 --> 00:02:01.530
where even if a third party
gets 20 percent of the vote,

00:02:01.530 --> 00:02:03.710
they're not going to get
any representation for it.

00:02:03.710 --> 00:02:05.010
And you can contrast that

00:02:05.010 --> 00:02:07.150
with a proportional representation system

00:02:07.150 --> 00:02:09.200
like you have in some Parliaments.

00:02:09.200 --> 00:02:12.370
So in some countries,
their Parliament is elected

00:02:12.370 --> 00:02:14.400
by proportional representation.

00:02:14.400 --> 00:02:17.670
So let's say that 20 percent vote for

00:02:17.670 --> 00:02:20.290
Party A.

00:02:20.290 --> 00:02:24.350
Let's say 70 percent vote for

00:02:24.350 --> 00:02:26.090
Party B.

00:02:26.090 --> 00:02:30.970
And then the remainder, 10
percent, vote for Party C.

00:02:30.970 --> 00:02:34.070
In a Parliamentary proportional
representation system,

00:02:34.070 --> 00:02:37.790
Party C would get roughly
10 percent of the seats

00:02:37.790 --> 00:02:39.170
in the Parliament.

00:02:39.170 --> 00:02:42.070
But that's not the way it
works in our government.

00:02:42.070 --> 00:02:44.430
In our government, in
almost any jurisdiction,

00:02:44.430 --> 00:02:46.690
if you had a voter breakdown like this,

00:02:46.690 --> 00:02:48.140
well, Party B would win.

00:02:48.140 --> 00:02:49.910
Or maybe sometimes Party A would win,

00:02:49.910 --> 00:02:52.380
and even if Party C does get some votes,

00:02:52.380 --> 00:02:54.950
it's never going to cross the threshold

00:02:54.950 --> 00:02:57.290
to actually get representation.

00:02:57.290 --> 00:02:59.880
Now, with that said, this
doesn't fully explain

00:02:59.880 --> 00:03:02.080
why we don't we see more
third party representation

00:03:02.080 --> 00:03:03.800
in say the United States Congress.

00:03:03.800 --> 00:03:06.060
Because there are countries that have more

00:03:06.060 --> 00:03:07.660
third party representation,

00:03:07.660 --> 00:03:10.590
even though they don't have
proportional representation.

00:03:10.590 --> 00:03:13.410
Now another reason that's often cited for

00:03:13.410 --> 00:03:16.450
why we don't see third
party representation

00:03:16.450 --> 00:03:19.770
is that the major parties,
the Republicans and Democrats

00:03:19.770 --> 00:03:23.850
oftentimes incorporate
the third party's messages

00:03:23.850 --> 00:03:24.793
into their own.

00:03:26.040 --> 00:03:27.670
To get a sense of this, I'm gonna show you

00:03:27.670 --> 00:03:32.200
a little bit of an excerpt
from a Reform Party ad in 1996.

00:03:32.200 --> 00:03:34.240
And at that time, they were saying things

00:03:34.240 --> 00:03:36.030
that neither the Democrats
or the Republicans

00:03:36.030 --> 00:03:37.520
were saying very strongly.

00:03:37.520 --> 00:03:39.530
And I want you to think
about, when you hear it,

00:03:39.530 --> 00:03:42.070
whether some of those
messages have, later on,

00:03:42.070 --> 00:03:45.440
become parts of either the
Republican or the Democratic

00:03:45.440 --> 00:03:47.210
candidate's messages.

00:03:47.210 --> 00:03:50.160
- Washington is selling our
future to the special interests.

00:03:50.160 --> 00:03:52.890
- Don't waste your vote on
someone who will sell you out.

00:03:52.890 --> 00:03:55.530
- Ross Perot is the only
candidate who will work

00:03:55.530 --> 00:03:58.310
- For our interests, instead
of the special interests.

00:03:58.310 --> 00:03:59.860
- Just vote for Ross.

00:03:59.860 --> 00:04:02.430
- Because you own this country.

00:04:02.430 --> 00:04:05.220
- So as you saw in that
ad from the Reform Party,

00:04:05.220 --> 00:04:06.540
there's a lot of talk about

00:04:06.540 --> 00:04:08.400
the influence of special interests

00:04:08.400 --> 00:04:10.790
and how people need to
take their government back.

00:04:10.790 --> 00:04:14.730
And if you fast-forward to the
2016 Presidential election,

00:04:14.730 --> 00:04:16.040
you had two major forces,

00:04:16.040 --> 00:04:17.670
actually one on the Republican side

00:04:17.670 --> 00:04:18.880
and one on the Democratic side.

00:04:18.880 --> 00:04:21.480
On the Republican side,
you have Donald Trump,

00:04:21.480 --> 00:04:25.120
who's echoing the need to
focus on special interests.

00:04:25.120 --> 00:04:28.470
That Washington is a swamp,
that it needs to be drained.

00:04:28.470 --> 00:04:30.250
It wasn't obvious from that ad,

00:04:30.250 --> 00:04:34.150
but Ross Perot in 1992 was
ringing the alarm bells

00:04:34.150 --> 00:04:35.980
about NAFTA and free trade and saying

00:04:35.980 --> 00:04:37.880
why it would be bad for the United States.

00:04:37.880 --> 00:04:41.090
And in 2016, you heard
many of these same things

00:04:41.090 --> 00:04:43.380
from the Republican
candidate, Donald Trump.

00:04:43.380 --> 00:04:46.090
Same thing with Bernie Sanders
on the Democratic side.

00:04:46.090 --> 00:04:49.360
He echoed the need to take
a second look at free trade.

00:04:49.360 --> 00:04:52.363
And that special interests
had taken over Washington.