[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.54,0:00:02.54,Default,,0000,0000,0000,,Where we left off in the\Nlast video I kind
Dialogue: 0,0:00:02.54,0:00:03.40,Default,,0000,0000,0000,,of gave you a question.
Dialogue: 0,0:00:03.40,0:00:06.24,Default,,0000,0000,0000,,Find an interval so that we're\Nreasonably confident-- we'll
Dialogue: 0,0:00:06.24,0:00:08.22,Default,,0000,0000,0000,,talk a little bit more about why\NI have to give this kind
Dialogue: 0,0:00:08.22,0:00:12.57,Default,,0000,0000,0000,,of vague wording right here--\Nreasonably confident that
Dialogue: 0,0:00:12.57,0:00:19.03,Default,,0000,0000,0000,,there's a 95% chance that the\Ntrue population mean, which is
Dialogue: 0,0:00:19.03,0:00:22.36,Default,,0000,0000,0000,,p, which is the same thing as\Nthe mean of the sampling
Dialogue: 0,0:00:22.36,0:00:23.85,Default,,0000,0000,0000,,distribution of the\Nsampling mean.
Dialogue: 0,0:00:23.85,0:00:26.57,Default,,0000,0000,0000,,So there's a 95% chance that\Nthe true mean-- and
Dialogue: 0,0:00:26.57,0:00:27.36,Default,,0000,0000,0000,,let me put this here.
Dialogue: 0,0:00:27.36,0:00:29.91,Default,,0000,0000,0000,,This is also the same thing as\Nthe mean of the sampling
Dialogue: 0,0:00:29.91,0:00:32.79,Default,,0000,0000,0000,,distribution of the sampling\Nmean is in that interval.
Dialogue: 0,0:00:32.79,0:00:36.74,Default,,0000,0000,0000,,And to do that let me just\Nthrow out a few ideas.
Dialogue: 0,0:00:36.74,0:00:46.30,Default,,0000,0000,0000,,What is the probability that if\NI take a sample and I were
Dialogue: 0,0:00:46.30,0:00:48.87,Default,,0000,0000,0000,,to take a mean of that sample,\Nso the probability that a
Dialogue: 0,0:00:48.87,0:00:59.26,Default,,0000,0000,0000,,random sample mean is within two\Nstandard deviations of the
Dialogue: 0,0:00:59.26,0:01:04.50,Default,,0000,0000,0000,,sampling mean, of\Nour sample mean?
Dialogue: 0,0:01:04.50,0:01:07.61,Default,,0000,0000,0000,,So what is this probability\Nright over here?
Dialogue: 0,0:01:07.61,0:01:10.10,Default,,0000,0000,0000,,Let's just look at our\Nactual distribution.
Dialogue: 0,0:01:10.10,0:01:12.95,Default,,0000,0000,0000,,So this is our distribution,\Nthis right here is our
Dialogue: 0,0:01:12.95,0:01:14.02,Default,,0000,0000,0000,,sampling mean.
Dialogue: 0,0:01:14.02,0:01:15.58,Default,,0000,0000,0000,,Maybe I should do it in\Nblue because that's
Dialogue: 0,0:01:15.58,0:01:18.73,Default,,0000,0000,0000,,the color up here.
Dialogue: 0,0:01:18.73,0:01:20.18,Default,,0000,0000,0000,,This is our sampling mean.
Dialogue: 0,0:01:20.18,0:01:23.33,Default,,0000,0000,0000,,And so what is the probability\Nthat a random sampling mean is
Dialogue: 0,0:01:23.33,0:01:24.97,Default,,0000,0000,0000,,going to be two standard\Ndeviations?
Dialogue: 0,0:01:24.97,0:01:28.44,Default,,0000,0000,0000,,Well a random sampling is a\Nsample from this distribution.
Dialogue: 0,0:01:28.44,0:01:30.98,Default,,0000,0000,0000,,It is a sample from the sampling\Ndistribution of the
Dialogue: 0,0:01:30.98,0:01:32.04,Default,,0000,0000,0000,,sample mean.
Dialogue: 0,0:01:32.04,0:01:35.40,Default,,0000,0000,0000,,So it's literally what is the\Nprobability of finding a
Dialogue: 0,0:01:35.40,0:01:37.36,Default,,0000,0000,0000,,sample within two standard\Ndeviations of the mean?
Dialogue: 0,0:01:37.36,0:01:40.15,Default,,0000,0000,0000,,That's one standard deviation,\Nthat's another standard
Dialogue: 0,0:01:40.15,0:01:42.98,Default,,0000,0000,0000,,deviation right over there.
Dialogue: 0,0:01:42.98,0:01:45.14,Default,,0000,0000,0000,,In general, if you haven't\Ncommitted this to memory
Dialogue: 0,0:01:45.14,0:01:48.34,Default,,0000,0000,0000,,already, it's not a bad thing\Nto commit to memory, is that
Dialogue: 0,0:01:48.34,0:01:50.71,Default,,0000,0000,0000,,if you have a normal\Ndistribution the probability
Dialogue: 0,0:01:50.71,0:01:55.17,Default,,0000,0000,0000,,of taking a sample within two\Nstandard deviations is 95--
Dialogue: 0,0:01:55.17,0:01:57.13,Default,,0000,0000,0000,,and if you want to get\Na little bit more
Dialogue: 0,0:01:57.13,0:01:59.60,Default,,0000,0000,0000,,accurate it's 95.4%.
Dialogue: 0,0:01:59.60,0:02:03.86,Default,,0000,0000,0000,,But you could say it's roughly--\Nor maybe I could
Dialogue: 0,0:02:03.86,0:02:06.21,Default,,0000,0000,0000,,write it like this--\Nit's roughly 95%.
Dialogue: 0,0:02:06.21,0:02:08.40,Default,,0000,0000,0000,,And really that's all that\Nmatters because we have this
Dialogue: 0,0:02:08.40,0:02:10.91,Default,,0000,0000,0000,,little funny language here\Ncalled reasonably confident,
Dialogue: 0,0:02:10.91,0:02:13.80,Default,,0000,0000,0000,,and we have to estimate the\Nstandard deviation anyway.
Dialogue: 0,0:02:13.80,0:02:16.13,Default,,0000,0000,0000,,In fact, we could say if we\Nwant, I could say that it's
Dialogue: 0,0:02:16.13,0:02:20.82,Default,,0000,0000,0000,,going to be exactly\Nequal to 95.4%.
Dialogue: 0,0:02:20.82,0:02:24.24,Default,,0000,0000,0000,,But in general, two standard\Ndeviations, 95%, that's what
Dialogue: 0,0:02:24.24,0:02:25.74,Default,,0000,0000,0000,,people equate with each other.
Dialogue: 0,0:02:25.74,0:02:28.68,Default,,0000,0000,0000,,Now this statement is the\Nexact same thing as the
Dialogue: 0,0:02:28.68,0:02:36.17,Default,,0000,0000,0000,,probability that the sample\Nmean, that the sampling mean--
Dialogue: 0,0:02:36.17,0:02:38.44,Default,,0000,0000,0000,,not the sample mean, the\Nprobability of the mean of the
Dialogue: 0,0:02:38.44,0:02:46.70,Default,,0000,0000,0000,,sampling distribution is within\Ntwo standard deviations
Dialogue: 0,0:02:46.70,0:02:51.35,Default,,0000,0000,0000,,of the sampling distribution of\Nx is also going to be the
Dialogue: 0,0:02:51.35,0:02:54.75,Default,,0000,0000,0000,,same number, is also going\Nto be equal to 95.4%.
Dialogue: 0,0:02:54.75,0:02:56.22,Default,,0000,0000,0000,,These are the exact\Nsame statements.
Dialogue: 0,0:02:56.22,0:03:00.17,Default,,0000,0000,0000,,If x is within two standard\Ndeviations of this, then this,
Dialogue: 0,0:03:00.17,0:03:02.58,Default,,0000,0000,0000,,then the mean, is within two\Nstandard deviations of x.
Dialogue: 0,0:03:02.58,0:03:05.40,Default,,0000,0000,0000,,These are just two ways of\Nphrasing the same thing.
Dialogue: 0,0:03:05.40,0:03:09.09,Default,,0000,0000,0000,,Now we know that the mean of the\Nsampling distribution, the
Dialogue: 0,0:03:09.09,0:03:11.76,Default,,0000,0000,0000,,same thing as a mean of the\Npopulation distribution, which
Dialogue: 0,0:03:11.76,0:03:14.94,Default,,0000,0000,0000,,is the same thing as the\Nparameter p-- the proportion
Dialogue: 0,0:03:14.94,0:03:19.67,Default,,0000,0000,0000,,of people or the proportion of\Nthe population that is a 1.
Dialogue: 0,0:03:19.67,0:03:22.78,Default,,0000,0000,0000,,So this right here is the same\Nthing as the population mean.
Dialogue: 0,0:03:22.78,0:03:26.90,Default,,0000,0000,0000,,So this statement right here\Nwe can switch this with p.
Dialogue: 0,0:03:26.90,0:03:32.29,Default,,0000,0000,0000,,So the probability that p is\Nwithin two standard deviations
Dialogue: 0,0:03:32.29,0:03:37.26,Default,,0000,0000,0000,,of the sampling distribution\Nof x is 95.4%.
Dialogue: 0,0:03:37.26,0:03:41.83,Default,,0000,0000,0000,,Now we don't know what this\Nnumber right here is.
Dialogue: 0,0:03:41.83,0:03:43.89,Default,,0000,0000,0000,,But we have estimated it.
Dialogue: 0,0:03:43.89,0:03:48.87,Default,,0000,0000,0000,,Remember, our best estimate of\Nthis is the true standard, or
Dialogue: 0,0:03:48.87,0:03:51.06,Default,,0000,0000,0000,,it is the true standard\Ndeviation of the population
Dialogue: 0,0:03:51.06,0:03:52.11,Default,,0000,0000,0000,,divided by 10.
Dialogue: 0,0:03:52.11,0:03:54.35,Default,,0000,0000,0000,,We can estimate the true\Nstandard deviation of the
Dialogue: 0,0:03:54.35,0:03:57.30,Default,,0000,0000,0000,,population with our sampling\Nstandard deviation, which was
Dialogue: 0,0:03:57.30,0:04:00.00,Default,,0000,0000,0000,,0.5, 0.5 divided by 10.
Dialogue: 0,0:04:00.00,0:04:03.98,Default,,0000,0000,0000,,Our best estimate of the\Nstandard deviation of the
Dialogue: 0,0:04:03.98,0:04:08.19,Default,,0000,0000,0000,,sampling distribution of the\Nsample mean is 0.05.
Dialogue: 0,0:04:08.19,0:04:11.47,Default,,0000,0000,0000,,So now we can say-- and I'll\Nswitch colors-- the
Dialogue: 0,0:04:11.47,0:04:14.67,Default,,0000,0000,0000,,probability that the parameter\Np, the proportion of the
Dialogue: 0,0:04:14.67,0:04:21.72,Default,,0000,0000,0000,,population saying 1, is within\Ntwo times-- remember, our best
Dialogue: 0,0:04:21.72,0:04:28.60,Default,,0000,0000,0000,,estimate of this right here is\N0.05 of a sample mean that we
Dialogue: 0,0:04:28.60,0:04:33.50,Default,,0000,0000,0000,,take is equal to 95.4%.
Dialogue: 0,0:04:33.50,0:04:40.77,Default,,0000,0000,0000,,And so we could say the\Nprobability that p is within 2
Dialogue: 0,0:04:40.77,0:04:46.65,Default,,0000,0000,0000,,times 0.05 is going to be equal\Nto-- 2.0 is going to be
Dialogue: 0,0:04:46.65,0:04:53.29,Default,,0000,0000,0000,,0.10 of our mean is equal to\N95-- and actually let me be a
Dialogue: 0,0:04:53.29,0:04:54.23,Default,,0000,0000,0000,,little careful here.
Dialogue: 0,0:04:54.23,0:04:58.42,Default,,0000,0000,0000,,I can't say the equal now,\Nbecause over here if we knew
Dialogue: 0,0:04:58.42,0:05:01.11,Default,,0000,0000,0000,,this, if we knew this parameter\Nof the sampling
Dialogue: 0,0:05:01.11,0:05:03.12,Default,,0000,0000,0000,,distribution of the sample\Nmean, we could
Dialogue: 0,0:05:03.12,0:05:05.25,Default,,0000,0000,0000,,say that it is 95.4%.
Dialogue: 0,0:05:05.25,0:05:06.28,Default,,0000,0000,0000,,We don't know it.
Dialogue: 0,0:05:06.28,0:05:09.05,Default,,0000,0000,0000,,We are just trying to find our\Nbest estimator for it.
Dialogue: 0,0:05:09.05,0:05:11.45,Default,,0000,0000,0000,,So actually what I'm going to\Ndo here is actually just say
Dialogue: 0,0:05:11.45,0:05:14.26,Default,,0000,0000,0000,,is roughly-- and just to show\Nthat we don't even have that
Dialogue: 0,0:05:14.26,0:05:17.51,Default,,0000,0000,0000,,level of accuracy, I'm going\Nto say roughly 95%.
Dialogue: 0,0:05:17.51,0:05:20.68,Default,,0000,0000,0000,,We're reasonably confident that\Nit's about 95% because
Dialogue: 0,0:05:20.68,0:05:23.88,Default,,0000,0000,0000,,we're using this estimator that\Ncame out of our sample,
Dialogue: 0,0:05:23.88,0:05:26.07,Default,,0000,0000,0000,,and if the sample is really\Nskewed this is going to be a
Dialogue: 0,0:05:26.07,0:05:26.95,Default,,0000,0000,0000,,really weird number.
Dialogue: 0,0:05:26.95,0:05:29.81,Default,,0000,0000,0000,,So this is why we just have to\Nbe a little bit more exact
Dialogue: 0,0:05:29.81,0:05:30.59,Default,,0000,0000,0000,,about what we're doing.
Dialogue: 0,0:05:30.59,0:05:31.89,Default,,0000,0000,0000,,But this is the tool\Nfor at least saying
Dialogue: 0,0:05:31.89,0:05:33.91,Default,,0000,0000,0000,,how good is our result.
Dialogue: 0,0:05:33.91,0:05:37.84,Default,,0000,0000,0000,,So this is going to\Nbe about 95%.
Dialogue: 0,0:05:37.84,0:05:46.50,Default,,0000,0000,0000,,Or we could say that the\Nprobability that p is within
Dialogue: 0,0:05:46.50,0:05:49.87,Default,,0000,0000,0000,,0.10 of our sample mean\Nthat we actually got.
Dialogue: 0,0:05:49.87,0:05:51.84,Default,,0000,0000,0000,,So what was the sample mean\Nthat we actually got?
Dialogue: 0,0:05:51.84,0:05:53.46,Default,,0000,0000,0000,,It was 0.43.
Dialogue: 0,0:05:53.46,0:05:59.55,Default,,0000,0000,0000,,So if we're within 0.1 of 0.43,\Nthat means we are within
Dialogue: 0,0:05:59.55,0:06:07.73,Default,,0000,0000,0000,,0.43 plus or minus 0.1 is\Nalso, roughly, we're
Dialogue: 0,0:06:07.73,0:06:11.85,Default,,0000,0000,0000,,reasonably confident\Nit's about 95%.
Dialogue: 0,0:06:11.85,0:06:12.87,Default,,0000,0000,0000,,And I want to be very clear.
Dialogue: 0,0:06:12.87,0:06:15.06,Default,,0000,0000,0000,,Everything that I started all\Nthe way from up here in brown
Dialogue: 0,0:06:15.06,0:06:17.61,Default,,0000,0000,0000,,to yellow and all this magenta,\NI'm just restating
Dialogue: 0,0:06:17.61,0:06:19.34,Default,,0000,0000,0000,,the same thing inside of this.
Dialogue: 0,0:06:19.34,0:06:22.49,Default,,0000,0000,0000,,It became a little bit more\Nloosey-goosey once I went from
Dialogue: 0,0:06:22.49,0:06:25.95,Default,,0000,0000,0000,,the exact standard deviation of\Nthe sampling distribution
Dialogue: 0,0:06:25.95,0:06:27.34,Default,,0000,0000,0000,,to an estimator for it.
Dialogue: 0,0:06:27.34,0:06:29.96,Default,,0000,0000,0000,,And that's why this is just\Nbecoming-- I kind of put the
Dialogue: 0,0:06:29.96,0:06:32.54,Default,,0000,0000,0000,,squiggly equal signs there\Nto say we're reasonably
Dialogue: 0,0:06:32.54,0:06:35.11,Default,,0000,0000,0000,,confident-- and I even got rid\Nof some of the precision.
Dialogue: 0,0:06:35.11,0:06:36.95,Default,,0000,0000,0000,,But we just found\Nour interval.
Dialogue: 0,0:06:36.95,0:06:39.24,Default,,0000,0000,0000,,An interval that we can be\Nreasonably confident that
Dialogue: 0,0:06:39.24,0:06:42.46,Default,,0000,0000,0000,,there's a 95% probability that\Np is within that, is going to
Dialogue: 0,0:06:42.46,0:06:45.19,Default,,0000,0000,0000,,be 0.43 plus or minus 0.1.
Dialogue: 0,0:06:45.19,0:06:48.46,Default,,0000,0000,0000,,Or an interval of-- we have\Na confidence interval.
Dialogue: 0,0:06:48.46,0:06:59.62,Default,,0000,0000,0000,,We have a 95% confidence\Ninterval of, and we could say,
Dialogue: 0,0:06:59.62,0:07:04.24,Default,,0000,0000,0000,,0.43 minus 0.1 is 0.33.
Dialogue: 0,0:07:04.24,0:07:08.71,Default,,0000,0000,0000,,If we write that as a percent\Nwe could say 33% to-- and if
Dialogue: 0,0:07:08.71,0:07:16.63,Default,,0000,0000,0000,,we add the 0.1, 0.43 plus\N0.1 we get 53%-- to 53%.
Dialogue: 0,0:07:16.63,0:07:20.89,Default,,0000,0000,0000,,So we are 95% confident.
Dialogue: 0,0:07:20.89,0:07:24.34,Default,,0000,0000,0000,,So we're not saying kind of\Nprecisely that the probability
Dialogue: 0,0:07:24.34,0:07:28.55,Default,,0000,0000,0000,,of the actual proportion is 95%,\Nbut we're 95% confident
Dialogue: 0,0:07:28.55,0:07:35.87,Default,,0000,0000,0000,,that the true proportion\Nis between 33% and 55%.
Dialogue: 0,0:07:35.87,0:07:37.83,Default,,0000,0000,0000,,That p is in this\Nrange over here.
Dialogue: 0,0:07:37.83,0:07:40.98,Default,,0000,0000,0000,,Or another way, and you'll see\Nthis in a lot of surveys that
Dialogue: 0,0:07:40.98,0:07:45.07,Default,,0000,0000,0000,,have been done, people will say\Nwe did a survey and we got
Dialogue: 0,0:07:45.07,0:07:55.18,Default,,0000,0000,0000,,43% will vote for number one,\Nand number one in this case is
Dialogue: 0,0:07:55.18,0:07:56.43,Default,,0000,0000,0000,,candidate B.
Dialogue: 0,0:08:02.42,0:08:04.45,Default,,0000,0000,0000,,And then the other side, since\Neveryone else voted for
Dialogue: 0,0:08:04.45,0:08:13.29,Default,,0000,0000,0000,,candidate A, 57% will\Nvote for A.
Dialogue: 0,0:08:13.29,0:08:15.46,Default,,0000,0000,0000,,And then they're going to\Nput on margin of error.
Dialogue: 0,0:08:15.46,0:08:17.75,Default,,0000,0000,0000,,And you'll see this in any\Nsurvey that you see on TV.
Dialogue: 0,0:08:17.75,0:08:22.35,Default,,0000,0000,0000,,They'll put a margin of error.
Dialogue: 0,0:08:22.35,0:08:24.92,Default,,0000,0000,0000,,And the margin of error is just\Nanother way of describing
Dialogue: 0,0:08:24.92,0:08:26.48,Default,,0000,0000,0000,,this confidence interval.
Dialogue: 0,0:08:26.48,0:08:29.33,Default,,0000,0000,0000,,And they'll say that the margin\Nof error in this case
Dialogue: 0,0:08:29.33,0:08:37.20,Default,,0000,0000,0000,,is 10%, which means that there's\Na 95% confidence
Dialogue: 0,0:08:37.20,0:08:41.92,Default,,0000,0000,0000,,interval, if you go plus or\Nminus 10% from that value
Dialogue: 0,0:08:41.92,0:08:42.51,Default,,0000,0000,0000,,right over there.
Dialogue: 0,0:08:42.51,0:08:44.85,Default,,0000,0000,0000,,And I really want to emphasize,\Nyou can't say with
Dialogue: 0,0:08:44.85,0:08:48.62,Default,,0000,0000,0000,,certainty that there is a 95%\Nchance that the true result
Dialogue: 0,0:08:48.62,0:08:52.18,Default,,0000,0000,0000,,will be within 10% of this,\Nbecause we had to estimate the
Dialogue: 0,0:08:52.18,0:08:55.03,Default,,0000,0000,0000,,standard deviation of\Nthe sampling mean.
Dialogue: 0,0:08:55.03,0:08:58.14,Default,,0000,0000,0000,,But this is the best measure\Nwe can with the information
Dialogue: 0,0:08:58.14,0:09:00.50,Default,,0000,0000,0000,,you have. If you're going to\Ndo a survey of 100 people,
Dialogue: 0,0:09:00.50,0:09:03.57,Default,,0000,0000,0000,,this is the best kind of\Nconfidence that we can get.
Dialogue: 0,0:09:03.57,0:09:05.60,Default,,0000,0000,0000,,And this number is actually\Nfairly big.
Dialogue: 0,0:09:05.60,0:09:08.56,Default,,0000,0000,0000,,So if you were to look at this\Nyou would say, roughly there's
Dialogue: 0,0:09:08.56,0:09:12.36,Default,,0000,0000,0000,,a 95% chance that the true\Nvalue of this number is
Dialogue: 0,0:09:12.36,0:09:15.10,Default,,0000,0000,0000,,between 33% and 53%.
Dialogue: 0,0:09:15.10,0:09:18.24,Default,,0000,0000,0000,,So there's actually still a\Nchance that candidate B can
Dialogue: 0,0:09:18.24,0:09:21.17,Default,,0000,0000,0000,,win, even though only\N43% of your 100 are
Dialogue: 0,0:09:21.17,0:09:21.92,Default,,0000,0000,0000,,going to vote for him.
Dialogue: 0,0:09:21.92,0:09:25.25,Default,,0000,0000,0000,,If you wanted to make it a\Nlittle bit more precise you
Dialogue: 0,0:09:25.25,0:09:26.77,Default,,0000,0000,0000,,would want to take\Nmore samples.
Dialogue: 0,0:09:26.77,0:09:28.39,Default,,0000,0000,0000,,You can imagine.
Dialogue: 0,0:09:28.39,0:09:31.71,Default,,0000,0000,0000,,Instead of taking 100 samples,\Ninstead of n being 100, if you
Dialogue: 0,0:09:31.71,0:09:35.31,Default,,0000,0000,0000,,made n equal 1,000, then you\Nwould take this number over
Dialogue: 0,0:09:35.31,0:09:37.80,Default,,0000,0000,0000,,here, you would take this number\Nhere and divide by the
Dialogue: 0,0:09:37.80,0:09:40.57,Default,,0000,0000,0000,,square root of 1,000 instead\Nof the square root of 100.
Dialogue: 0,0:09:40.57,0:09:43.25,Default,,0000,0000,0000,,So you'd be dividing\Nby 33 or whatever.
Dialogue: 0,0:09:43.25,0:09:48.55,Default,,0000,0000,0000,,And so then the size of the\Nstandard deviation of your
Dialogue: 0,0:09:48.55,0:09:50.76,Default,,0000,0000,0000,,sampling distribution\Nwill go down.
Dialogue: 0,0:09:50.76,0:09:53.35,Default,,0000,0000,0000,,And so the distance of two\Nstandard deviations will be a
Dialogue: 0,0:09:53.35,0:09:55.48,Default,,0000,0000,0000,,smaller number, and so\Nthen you will have a
Dialogue: 0,0:09:55.48,0:09:57.21,Default,,0000,0000,0000,,smaller margin of error.
Dialogue: 0,0:09:57.21,0:10:00.29,Default,,0000,0000,0000,,And maybe you want to get the\Nmargin of error small enough
Dialogue: 0,0:10:00.29,0:10:02.92,Default,,0000,0000,0000,,so that you can figure out\Ndecisively who's going to win
Dialogue: 0,0:10:02.92,0:10:04.34,Default,,0000,0000,0000,,the election.