0:00:00.540,0:00:02.540
Where we left off in the[br]last video I kind

0:00:02.540,0:00:03.400
of gave you a question.

0:00:03.400,0:00:06.240
Find an interval so that we're[br]reasonably confident-- we'll

0:00:06.240,0:00:08.220
talk a little bit more about why[br]I have to give this kind

0:00:08.220,0:00:12.570
of vague wording right here--[br]reasonably confident that

0:00:12.570,0:00:19.030
there's a 95% chance that the[br]true population mean, which is

0:00:19.030,0:00:22.360
p, which is the same thing as[br]the mean of the sampling

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distribution of the[br]sampling mean.

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So there's a 95% chance that[br]the true mean-- and

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let me put this here.

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This is also the same thing as[br]the mean of the sampling

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distribution of the sampling[br]mean is in that interval.

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And to do that let me just[br]throw out a few ideas.

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What is the probability that if[br]I take a sample and I were

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to take a mean of that sample,[br]so the probability that a

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random sample mean is within two[br]standard deviations of the

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sampling mean, of[br]our sample mean?

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So what is this probability[br]right over here?

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Let's just look at our[br]actual distribution.

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So this is our distribution,[br]this right here is our

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sampling mean.

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Maybe I should do it in[br]blue because that's

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the color up here.

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This is our sampling mean.

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And so what is the probability[br]that a random sampling mean is

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going to be two standard[br]deviations?

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Well a random sampling is a[br]sample from this distribution.

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It is a sample from the sampling[br]distribution of the

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sample mean.

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So it's literally what is the[br]probability of finding a

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sample within two standard[br]deviations of the mean?

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That's one standard deviation,[br]that's another standard

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deviation right over there.

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In general, if you haven't[br]committed this to memory

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already, it's not a bad thing[br]to commit to memory, is that

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if you have a normal[br]distribution the probability

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of taking a sample within two[br]standard deviations is 95--

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and if you want to get[br]a little bit more

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accurate it's 95.4%.

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But you could say it's roughly--[br]or maybe I could

0:02:03.860,0:02:06.210
write it like this--[br]it's roughly 95%.

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And really that's all that[br]matters because we have this

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little funny language here[br]called reasonably confident,

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and we have to estimate the[br]standard deviation anyway.

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In fact, we could say if we[br]want, I could say that it's

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going to be exactly[br]equal to 95.4%.

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But in general, two standard[br]deviations, 95%, that's what

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people equate with each other.

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Now this statement is the[br]exact same thing as the

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probability that the sample[br]mean, that the sampling mean--

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not the sample mean, the[br]probability of the mean of the

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sampling distribution is within[br]two standard deviations

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of the sampling distribution of[br]x is also going to be the

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same number, is also going[br]to be equal to 95.4%.

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These are the exact[br]same statements.

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If x is within two standard[br]deviations of this, then this,

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then the mean, is within two[br]standard deviations of x.

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These are just two ways of[br]phrasing the same thing.

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Now we know that the mean of the[br]sampling distribution, the

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same thing as a mean of the[br]population distribution, which

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is the same thing as the[br]parameter p-- the proportion

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of people or the proportion of[br]the population that is a 1.

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So this right here is the same[br]thing as the population mean.

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So this statement right here[br]we can switch this with p.

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So the probability that p is[br]within two standard deviations

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of the sampling distribution[br]of x is 95.4%.

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Now we don't know what this[br]number right here is.

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But we have estimated it.

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Remember, our best estimate of[br]this is the true standard, or

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it is the true standard[br]deviation of the population

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divided by 10.

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We can estimate the true[br]standard deviation of the

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population with our sampling[br]standard deviation, which was

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0.5, 0.5 divided by 10.

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Our best estimate of the[br]standard deviation of the

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sampling distribution of the[br]sample mean is 0.05.

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So now we can say-- and I'll[br]switch colors-- the

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probability that the parameter[br]p, the proportion of the

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population saying 1, is within[br]two times-- remember, our best

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estimate of this right here is[br]0.05 of a sample mean that we

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take is equal to 95.4%.

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And so we could say the[br]probability that p is within 2

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times 0.05 is going to be equal[br]to-- 2.0 is going to be

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0.10 of our mean is equal to[br]95-- and actually let me be a

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little careful here.

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I can't say the equal now,[br]because over here if we knew

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this, if we knew this parameter[br]of the sampling

0:05:01.110,0:05:03.120
distribution of the sample[br]mean, we could

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say that it is 95.4%.

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We don't know it.

0:05:06.280,0:05:09.050
We are just trying to find our[br]best estimator for it.

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So actually what I'm going to[br]do here is actually just say

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is roughly-- and just to show[br]that we don't even have that

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level of accuracy, I'm going[br]to say roughly 95%.

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We're reasonably confident that[br]it's about 95% because

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we're using this estimator that[br]came out of our sample,

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and if the sample is really[br]skewed this is going to be a

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really weird number.

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So this is why we just have to[br]be a little bit more exact

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about what we're doing.

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But this is the tool[br]for at least saying

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how good is our result.

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So this is going to[br]be about 95%.

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Or we could say that the[br]probability that p is within

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0.10 of our sample mean[br]that we actually got.

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So what was the sample mean[br]that we actually got?

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It was 0.43.

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So if we're within 0.1 of 0.43,[br]that means we are within

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0.43 plus or minus 0.1 is[br]also, roughly, we're

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reasonably confident[br]it's about 95%.

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And I want to be very clear.

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Everything that I started all[br]the way from up here in brown

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to yellow and all this magenta,[br]I'm just restating

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the same thing inside of this.

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It became a little bit more[br]loosey-goosey once I went from

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the exact standard deviation of[br]the sampling distribution

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to an estimator for it.

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And that's why this is just[br]becoming-- I kind of put the

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squiggly equal signs there[br]to say we're reasonably

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confident-- and I even got rid[br]of some of the precision.

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But we just found[br]our interval.

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An interval that we can be[br]reasonably confident that

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there's a 95% probability that[br]p is within that, is going to

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be 0.43 plus or minus 0.1.

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Or an interval of-- we have[br]a confidence interval.

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We have a 95% confidence[br]interval of, and we could say,

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0.43 minus 0.1 is 0.33.

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If we write that as a percent[br]we could say 33% to-- and if

0:07:08.710,0:07:16.630
we add the 0.1, 0.43 plus[br]0.1 we get 53%-- to 53%.

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So we are 95% confident.

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So we're not saying kind of[br]precisely that the probability

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of the actual proportion is 95%,[br]but we're 95% confident

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that the true proportion[br]is between 33% and 55%.

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That p is in this[br]range over here.

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Or another way, and you'll see[br]this in a lot of surveys that

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have been done, people will say[br]we did a survey and we got

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43% will vote for number one,[br]and number one in this case is

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candidate B.

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And then the other side, since[br]everyone else voted for

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candidate A, 57% will[br]vote for A.

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And then they're going to[br]put on margin of error.

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And you'll see this in any[br]survey that you see on TV.

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They'll put a margin of error.

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And the margin of error is just[br]another way of describing

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this confidence interval.

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And they'll say that the margin[br]of error in this case

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is 10%, which means that there's[br]a 95% confidence

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interval, if you go plus or[br]minus 10% from that value

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right over there.

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And I really want to emphasize,[br]you can't say with

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certainty that there is a 95%[br]chance that the true result

0:08:48.620,0:08:52.180
will be within 10% of this,[br]because we had to estimate the

0:08:52.180,0:08:55.030
standard deviation of[br]the sampling mean.

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But this is the best measure[br]we can with the information

0:08:58.140,0:09:00.500
you have. If you're going to[br]do a survey of 100 people,

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this is the best kind of[br]confidence that we can get.

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And this number is actually[br]fairly big.

0:09:05.600,0:09:08.560
So if you were to look at this[br]you would say, roughly there's

0:09:08.560,0:09:12.360
a 95% chance that the true[br]value of this number is

0:09:12.360,0:09:15.100
between 33% and 53%.

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So there's actually still a[br]chance that candidate B can

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win, even though only[br]43% of your 100 are

0:09:21.170,0:09:21.920
going to vote for him.

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If you wanted to make it a[br]little bit more precise you

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would want to take[br]more samples.

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You can imagine.

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Instead of taking 100 samples,[br]instead of n being 100, if you

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made n equal 1,000, then you[br]would take this number over

0:09:35.310,0:09:37.800
here, you would take this number[br]here and divide by the

0:09:37.800,0:09:40.570
square root of 1,000 instead[br]of the square root of 100.

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So you'd be dividing[br]by 33 or whatever.

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And so then the size of the[br]standard deviation of your

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sampling distribution[br]will go down.

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And so the distance of two[br]standard deviations will be a

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smaller number, and so[br]then you will have a

0:09:55.480,0:09:57.210
smaller margin of error.

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And maybe you want to get the[br]margin of error small enough

0:10:00.290,0:10:02.920
so that you can figure out[br]decisively who's going to win

0:10:02.920,0:10:04.340
the election.