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YouTube transcript (HoqzIR8xj4s) — Transcript

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0:10[Music]
0:15hello thank you for watching and welcome
0:18to the next video in my series on basic
0:20statistics now as usual a few things
0:23before we get started number one if
0:25you're watching this video because you
0:26are struggling in a class right now I
0:29want you to stay positive and keep your
0:30head up if you're watching this it means
0:32you've accomplished quite a bit already
0:34you're very smart and talented and you
0:36may have just hit a temporary rough
0:38patch now I know with a right amount of
0:40hard work practice and patience you can
0:43get through it I have faith in you many
0:46other people around you have faith in
0:48you so so should you number two please
0:52feel free to follow me here on YouTube
0:54on Twitter on Google+ or on LinkedIn
0:58that way when I upload a new video you
1:00know about it and it's always nice to
1:02connect with people who watch my videos
1:04online the world is much too large and
1:07life is Much Too Short not to take the
1:09opportunity to connect with one another
1:12number two if you like the video please
1:14give it a thumbs up share it with
1:17classmates or colleagues or put it on a
1:18playlist cuz that does encourage me to
1:21keep making them for you on the flip
1:23side if you think there is something I
1:24can do better please leave a
1:26constructive comment below the video and
1:28I will try to take those ideas into
1:30account when I make new ones for you and
1:33finally just keep in mind that these
1:34videos are meant for individuals who are
1:36relatively new to Stats so I'm just
1:39going over basic concepts and I will be
1:42doing so in a very slow deliberate
1:44manner not only do I want you to know
1:47what's going on but also why and how to
1:50apply it so all that being said let's go
1:53ahead and get
1:56started so this video is the next in our
1:59series on hypothesis formulation and now
2:02finally hypothesis testing so up to this
2:06point we've talked about what a
2:07hypothesis is we talked about the no
2:10hypothesis we talked about the
2:11alternative hypothesis we talked about
2:14type one error and type two error with
2:17many many examples so all those were
2:20leading up to this very topic and that
2:22is actually conducting a hypothesis test
2:26now there are many types of hypothesis
2:29test but we're going to do the most
2:30simple in this video and that is where
2:33we have a single sample with a known
2:37Sigma or we have a single sample we are
2:41testing against a hypothesized mean and
2:44we are given Sigma which is the
2:46population standard deviation so we'll
2:50go over several distribution curves
2:52we'll talk about critical values and how
2:54that affects our Alpha level and things
2:56like that and then we will walk through
2:59two real world examples so all that
3:02being said let's go ahead and Dive Right
3:06In now it's very important to point out
3:08the hypothesis tests follow a very
3:11prescribed procedure now as usual it
3:14always starts with a well-developed
3:17clear research problem or analytical
3:21question if the problem is poorly
3:23thought out if what you're trying to
3:25accomplish is unclear then no amount of
3:29statistic is going to be able to solve
3:31that and it can actually make it worse
3:34so always think through what you're
3:36trying to find out at the problem
3:39stage now once you have that we always
3:42establish our hypothesis both the null
3:46and the alternative so remember the null
3:49and the alternative are complete
3:51opposites of each other and they must
3:54account for all possible
3:57outcomes then we determine the approach
3:59apprpriate statistical test and sampling
4:02distribution so as I said before there
4:05are many types of hypothesis tests so in
4:09this one we're going to be looking at
4:10the Z test in other ones we might look
4:12at the T Test and there are more still
4:16after that then of course the sampling
4:18distribution will depend on whether or
4:20not we have Sigma given to us or we know
4:23it or we have to estimate it so step
4:26three is always determine the
4:28appropriate statistical test and and the
4:29sampling
4:31distribution then we choose our type one
4:33error rate so what Comfort level do we
4:37have with making a type one error is it
4:415% 1% 10% again it will just depend on
4:46what our study asks for and what we are
4:49comfortable with it all also has to do
4:52with what level of type two error we are
4:55comfortable making because remember they
4:57are inversely related
5:00then we State our decision rule so in
5:03this case we're going to come up with a
5:04z statistic and then we will have to
5:07determine whether or not based on that Z
5:10statistic we're going to reject our null
5:14hypothesis or fail to reject our null
5:18hypothesis then and only then do we go
5:22out and gather our sample data so I know
5:25a lot of students I've worked with are
5:27really excited about going out and
5:29collecting data the very first thing but
5:32I always have to say no always form your
5:36research question or your analytical
5:38question first set up your hypothesis so
5:41you know what you're actually going at
5:43and then you choose your test your
5:46distribution your error rate decision
5:48rule Etc then go out and get your data
5:52so there is this impulse to want to go
5:54out and collect data first and then form
5:56the research question based on the data
5:59you collected
6:00no it's the other way around always form
6:02your question
6:04first now once we have the data we
6:07calculate our test statistics so in this
6:09case it will be the Z
6:11statistic now based on those test
6:13statistics we will state our statistical
6:17conclusion so we'll have a statistic to
6:19then compare to our decision Rule and
6:22then however our statistic compared to
6:24the decision rule will be our
6:27conclusion and then finally in the real
6:29world we can either make a decision or
6:32an inference based on that conclusion so
6:36it may be some research question in a
6:38journal we're looking at it may be a
6:40policy in our business we are looking at
6:43it may be some analytical work we are
6:45doing maybe in the financial industry or
6:47in the insurance industry or in the
6:49production industry whatever it might be
6:51so we finally get to the point where we
6:53can make a decision or some policy
6:56recommendation based on our conclusion
7:01now as I said there are really two types
7:04of these statistical tests there are
7:06ones where we know Sigma and ones where
7:08we don't so as with confidence intervals
7:11there are two types of single sample
7:13hypothesis tests when the population
7:16standard deviation Sigma is known or
7:18it's given to us and when the population
7:21standard deviation Sigma is not known
7:24and therefore we have to estimate it
7:26using S the sample standard deviation
7:30now when Sigma is known or given to us
7:33we use the normal standard or the Z
7:37distribution to establish the
7:39non-rejection region and the critical
7:43values in our sampling distribution so
7:46again we talked about that at Great
7:47length when we looked at type one and
7:49type two error rates so if you're still
7:51unsure what this concept is go back and
7:54look at those videos but when we know
7:56Sigma we're going to use the normal
7:58standard or the Z distribution to
8:01establish these regions now when Sigma
8:04is not known we will use the T
8:07distribution instead because remember
8:09the T distribution is a little bit
8:11shorter in the middle and it has a
8:13little bit more probability in the tails
8:16to account for that unknown or that
8:19estimation we're doing with the standard
8:22deviation of our
8:23population now some instructors in some
8:25books will indicate that using the Z
8:28distribution is acceptable anytime the
8:32sample size is 30 or greater whether or
8:35not you know Sigma or not now I prefer
8:39to go ahead and use the T distribution
8:42anytime I do not know Sigma now remember
8:46the reality is is that as sample size
8:49increases the Z distribution and the T
8:52distribution actually converge so it
8:55just depends on what your instructor or
8:57your book is asking you to do because
9:01the T distribution with its fatter Tails
9:04will actually change a little bit how
9:06the alpha level affects your critical
9:10values now it's always good to check the
9:12sample data for normality better safe
9:15than sorry so you might want to look at
9:17a histogram or a QQ plot or a PP plot of
9:21your sample data to make sure it's not
9:23skewed heavily in One Direction you
9:25don't have any really crazy outliers or
9:29whatever ever else that might be it's
9:30just always good to check your data for
9:35normality so remember what we're talking
9:37about here is the hypothesized versus
9:40the true mean so mu is the true mean of
9:45the population under analysis so if
9:48we're analyzing a
9:50population it actually has a real world
9:53true
9:55mean now mu sub Z is the hypo ized mean
10:00of the population under analysis so we
10:03might have some guess or some previous
10:07study or something else we are testing
10:09it against so we're testing two means
10:14we're testing our data's mean the actual
10:17population mean versus some hypothesized
10:20value we think it
10:22is so what we're asking here is the true
10:25mean the same as the hypothesized mean
10:29are they coming from the same
10:32distributions now we will test that
10:34question or this question using sample
10:36means of course and confidence intervals
10:39which we'll call critical regions here
10:42in a
10:44minute now let's just remind ourselves
10:46about the two-tailed test rejection
10:49region so here we have our two
10:51hypothesis as we had before and then in
10:54this case we're going to choose an alpha
10:56of
10:5705 so we have our distribution our
11:00sampling distribution that looks like
11:03this now remember what we're actually
11:05saying here with an alpha of
11:0805 we are saying that this blue area in
11:10the middle is
11:1295% now 95% of what well what we're
11:16saying is that 95% of our sample means
11:19that we would take should be within this
11:22blue region and then we risk 5% being
11:27outside that region
11:29now our hypothesized mean is set here in
11:32the middle and we call this blue region
11:35the non-rejection region and on the ends
11:38and the Tails those are both rejection
11:43regions now our Alpha in this case is
11:46spread evenly among both Tails so our
11:49Alpha of 05 we have 025 in the lower
11:52tail and 025 in the upper tail so it's
11:562.5% in the lower 2.5% in the upper tail
12:01now dividing the non-rejection region
12:03and the rejection region it's called the
12:06critical value it's kind of that
12:08boundary between the two now remember
12:11the critical value is determined by
12:13Alpha in this case 05 and if we are
12:16using the T or the Z
12:19distributions with an alpha of 05 and
12:22sigma Noone we would consult the Z table
12:25and find the corresponding zc scores for
12:28a two-tail test with the alpha of
12:3105 now when we do that we see that our Z
12:35critical values are negative
12:381.96 and positive
12:411.96 so that zcore is the boundary
12:45between the non-rejection region and the
12:48rejection region based off our Z table
12:52and our Alpha and again we're using the
12:54Z table because we know our Sigma
13:01now what if we change the alpha level to
13:030.10 so we had 05 now we have an alpha
13:07of 0.10 which is twice the previous
13:11Alpha now if you look at our taals
13:13something should be fairly obvious right
13:16off the bat our rejection regions are
13:20larger and our non-rejection region is
13:23smaller or
13:25narrower now we are saying that 90% of
13:28our sample means should be in the blue
13:31in the non-rejection region therefore
13:3310% would be in the tails in the
13:36rejection region either above or below
13:40so now we have an alpha divid two of
13:4305 so that's 5% in the lower tail and 5%
13:47in the upper tail now as far as critical
13:49values go are they going to become
13:52smaller or
13:54larger well they're going to become
13:56smaller because the critical values move
14:00inward we have less probability there in
14:02the middle so it has to move inward so
14:05our Z critical values are now netive
14:081.645 and positive
14:121.645 so what happens when our Alpha
14:15level
14:16increases so in this case we went from
14:1905 to
14:200.10 our non-rejection region gets
14:23smaller in the middle and the rejection
14:26regions in the tails get larger and of
14:29course our critical values move
14:34inward so finally let's look at what
14:36happens to our critical values when we
14:38use an alpha of
14:400.1 so in the previous slide we looked
14:42at an alpha of 0.10 so now this is
14:4501 so let's make some predictions about
14:47what's going to happen here well you
14:49notice that our non-rejection region in
14:51the middle is much wider there's much
14:54more area there in the blue now the
14:57reason that is is because we have to
14:58take the this 01 or 1% and divide it
15:02evenly among both Tails so we have 05 or
15:061 half of 1% in the lower tail and we
15:09have 05 or 1 half of 1% in the upper
15:13tail so what we're saying is that we
15:15expect 99% of our sample means to be in
15:18this non-rejection region in the blue
15:20region in the middle and of course in
15:23the tails that is our rejection region
15:25and we expect 1% of our sample means to
15:28either be in the upper or the lower
15:30rejection region now of course the whole
15:32point of these series of slides is to
15:34talk about what happens to our critical
15:36values so what's going to happen to our
15:38critical values with a very very small
15:41Alpha of
15:4201 it's going to get larger or
15:46smaller well the critical values are
15:48going to get larger so here we have plus
15:51or minus
15:552576 and those are by far the largest
15:59values or you can think of them as the
16:00widest values of all the alphas we have
16:03used so the overall point of these last
16:06few slides is look at the relationship
16:08between Alpha and the area of our
16:11non-rejection region in the middle our
16:13rejection regions on the ends and the
16:16effects of the critical value as it sets
16:19the demarcation or the boundary between
16:22these two
16:25regions so what are we really asking in
16:27sort of real World Language what we're
16:31asking is did our sample come from the
16:33same population we assume is underlying
16:38the null hypothesis so if we take a
16:41sample from a population to use in our Z
16:45statistic we want to make sure we're
16:47testing whether or not our sample came
16:50from the population we are hypothesizing
16:53it came from now if so then we expect
16:57our sample mean to be inside the
17:00critical region either 90% of the time
17:0295% of the time or 99% of the time
17:05depending on what we choose for Alpha
17:09that's what we are really asking is our
17:11sample mean from the same population we
17:15are hypothesizing it to be coming
17:20from so let's go ahead and look at the
17:22actual Z test for a single mean so here
17:26is our formula now it is comprised of
17:28xar which is the sample mean mu Sub 0
17:32which is the hypothesized population
17:34mean given in our problem Sigma is the
17:37population standard deviation again
17:38that's a given or unn to us in this case
17:41and N of course is the sample size as it
17:44always is now if you remember from the
17:46previous videos this denominator is a
17:49very special term it is the standard
17:53error of the mean which is another name
17:56for the standard deviation of of the
17:59sampling distribution so the standard
18:02error of the mean is the standard
18:04deviation of a distribution of many many
18:07many samples so you may see it written
18:11like this so Sigma subx bar is the same
18:16thing as it's written over here on the
18:19left they're both representations of the
18:22standard error of the mean so just
18:24wanted to show you both ways depending
18:26on whatever class you're in whatever
18:27book you're using you might see it
18:29either way now the question we are
18:31asking when we find this Z statistic is
18:36is this Z test value in the
18:39non-rejection region in the
18:41middle or is it in the rejection region
18:45in the Tails so one of the other taals
18:48depending on how our hypothesis is set
18:50up and that's what we're doing when we
18:53do a z
18:57test e

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