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0:08[Music]
0:18hello thank you for watching and welcome
0:20to the next video in my series on basic
0:22statistics now as usual a few things
0:24before we get started number one if
0:27you're watching this video because you
0:28are struggling in a class right now I
0:30want you to stay positive and keep your
0:32head up if you're watching this it means
0:34you've accomplished quite a bit already
0:36you're very smart and talented and you
0:38may have just hit a temporary rough
0:40patch now I know with the right amount
0:42of hard work practice and patience you
0:45can get through it I have faith in you
0:48many other people around you have faith
0:50in you so so should you number two
0:54please feel free to follow me here on
0:55YouTube on Twitter on go+ or on LinkedIn
1:00that way when I upload a new video you
1:03know about it and it's always nice to
1:05connect with people who watch my videos
1:07online life is Much Too Short The World
1:10Is much too large for us not to take the
1:13opportunity to connect with each other
1:15when we can number three if you like the
1:18video please give it a thumbs up share
1:21it with classmates or colleagues or put
1:23it on a playlist cuz that does encourage
1:25me to keep making them for you on the
1:27flip side if you think there is
1:29something I can do better please leave a
1:31constructive comment below the video and
1:33I will try to take those ideas into
1:35account when I make new ones for you and
1:39finally just keep in mind that these
1:40videos are meant for individuals who are
1:42relatively new to Stats so I'm just
1:45going over basic concepts and I will be
1:48doing so in a very slow deliberate
1:50manner not only do I want you to know
1:53what's going on but also why and how to
1:56apply it so all that being said let's go
2:00ahead and get
2:02started so this video is the next in our
2:05series on hypothesis testing now in the
2:09videos leading up to this one I talked
2:11about the fact that there are really two
2:13types of single sample hypothesis tests
2:17in the first type we know or are given
2:21Sigma which is the population standard
2:24deviation in the second type we do not
2:27know Sigma it's not given to us and
2:30therefore we have to estimate it using
2:32our sample standard deviation now we're
2:36going to be using two different tests in
2:38two different distributions for each
2:41case when we know Sigma we use the Z
2:46distribution and we do single sample Z
2:49tests when we do not know Sigma we use
2:53the T distribution and conduct single
2:56sample T tests so this video is really
3:00about the second type cuz I covered the
3:02Z distribution and Z tests in the
3:05previous videos leading up to this one
3:07so I'm going to divide this video into
3:10two parts in the first part we'll talk
3:12about the conceptual background of the
3:15single sample T Test and we will examine
3:19that in a compare and contrast fashion
3:22with the Z test we used in the previous
3:26video in part two we will actually work
3:29two problems sort of two real world
3:32problems where we will set up and
3:35conduct single sample T tests and then
3:38make some conclusions based on our
3:40results so let's go ahead and get
3:43started on part
3:46one now I mentioned all this in the
3:49previous video but it's worth going over
3:51again and that is the idea that when
3:54done correctly hypothesis tests follow a
3:58fairly standard procedure procedure now
4:01we always start with a well-developed
4:03clear research problem or analytical
4:06question no amount of fancy statistics
4:10is going to save a very bad very poorly
4:14developed research problem or analytical
4:18question so a lot of the heavy work is
4:21done before any hypothesis is set up
4:25before any decision rule is set up
4:28before any data is collected CED you got
4:30to have a very clear problem up front
4:34otherwise everything else is kind of
4:37just a
4:38waste now once the problem is set up
4:41properly then we establish our
4:43hypothesis both the null and the
4:46alternative now remember the null and
4:48the alternative are complete opposites
4:51of one another they have to account for
4:54all
4:55possibilities and the equal sign in some
4:58form or another always goes with the
5:01null so if you remember those three
5:03rules setting up your hypothesis should
5:06be fairly
5:08straightforward now once the hypothesis
5:10is set up then we determine the
5:12appropriate statistical test and
5:15sampling distribution so this is where
5:17we choose whether or not we're going to
5:19be using a z test and the Z distribution
5:21or a T Test and the T distribution and
5:26again that depends on whether or not we
5:29know
5:30Sigma or not or as I'll mention later if
5:34our sample size is over 100 I always go
5:37ahead and just use the Z distribution
5:41regardless then we choose our type one
5:43error rate so remember this is Alpha so
5:46we can choose an alpha of 05 an alpha of
5:5001 or 0.10 it just depends so you choose
5:55a type 1 error rate based on your
5:57comfort level of making a type one error
6:01I usually stick with 05 it's kind of the
6:04middle ground between type 1 and type
6:06two error but other people will choose
6:0901 really it's just up to you whatever
6:11field you're working in or whatever line
6:13of business you work in there tends to
6:15be sort of a standard tradition for
6:18choosing that type 1 error
6:20rate now once we have the error rate and
6:23the appropriate statistical test and
6:25distribution we can State our decision
6:28rule So based on the error rate and the
6:31statistical distribution we will able to
6:34set up critical
6:35values now once we calculate our test
6:39statistic based on where it falls in
6:42relation to that critical value we will
6:45either failed to reject the null
6:48hypothesis or reject the null hypothesis
6:52but we go ahead and set up that decision
6:54rule up front and get it out in the open
6:57get it down on paper then we go ahead
7:00and gather our sample data but we always
7:03do step six Gathering the sample data
7:07after steps 1 through five so we do not
7:10want to go out and gather sample data
7:12first because sometimes we tend to
7:15massage the problem to match the data we
7:18think we obtained so we always set up
7:21our problem first our hypothesis our
7:23tests and everything up front then we go
7:26out and gather our data once we have our
7:29data that we calculate our test
7:30statistics so we will have a t test
7:34statistic or a z test
7:36statistic and then we will compare that
7:39to our decision rule so where does our
7:42test statistic fall in relation to our
7:45non-rejection region or rejection region
7:48which is the same way of saying where it
7:49falls in relation to our critical values
7:53So based on that we can refer to our
7:55decision Rule and then State our
7:58statistical decision based on this very
8:01sound very logical
8:04process now once we have our conclusion
8:07we can go ahead and make our decisions
8:08or inferences in real life based on that
8:11conclusion so it's a finding we can put
8:14in a journal article or it may be a
8:16finding we can take to our uh boss at
8:19work and make some sort of policy
8:22recommendation based on this conclusion
8:25but we only do that once we are sure
8:27steps 1 through 8 have been followed
8:30properly and our conclusion is
8:35sound now remember as I said before
8:37there are really two types of single
8:39sample hypothesis test where we know
8:42Sigma or where we do not so as with
8:45confidence intervals there are two types
8:48when the population standard deviation
8:49Sigma is known or given and when it's
8:52not and therefore we have to estimate it
8:55using the sample standard
8:57deviation when Sigma is known we we use
8:59the standard normal distribution or the
9:01Z distribution to establish the
9:03non-rejection region and critical
9:05values when Sigma is not known we use
9:09the T distribution instead because it
9:12accounts for that
9:14uncertainty now remember that in the T
9:16distribution every sample size has its
9:19own t distribution with n minus one
9:23degrees of freedom so there is no single
9:26T distribution like there is the Z
9:28distribution so we have a sample size of
9:3120 we will have a t distribution with 19
9:35degrees of
9:37freedom now some instructors in books
9:39will indicate that using the Z
9:41distribution is acceptable anytime the
9:43sample size is 30 or
9:46larger now I prefer to use the T
9:49distribution anytime Sigma is unknown
9:53and the sample size is under
9:56100 now if the sample size is greater
10:00than 100 regardless I always use the Z
10:04distribution and this is a more
10:06conservative approach so I tend to use
10:08the T distribution anytime Sigma is
10:11unknown and the sample size is under 100
10:14if it's over 100 I always use the Z
10:16distribution and the Z
10:18test now it's always good to check the
10:20sample data for normality is your sample
10:23data skewed to one side or the other
10:26does it have extreme outliers in it it
10:29so it's always just good to do a check
10:32of your sample to make sure it fits
10:34relatively close to
10:39normality let's keep in mind the big
10:41picture here what are we actually doing
10:43what we're saying is that there are some
10:45hypothesized population mean out there
10:48that we're looking at and then we're
10:50going to go out and collect data to see
10:53if the actual population mean matches
10:57the hypothesized
10:59population mean so mu is the true mean
11:04of the population under analice as it
11:06exists like out in the real
11:09world now mu sub Z is the hypothesized
11:13mean of the population under analys or
11:16what we think it is prior to testing it
11:20against actual data so is the true mean
11:24the same as the hypothesized mean for
11:28this popul ation and of course we'll
11:30test that question using sample means
11:32and confidence intervals and things like
11:37that now we're going to go over some
11:39curves so I want to show you the
11:40difference between the Z distribution
11:43and Z test and the T distribution and T
11:47Test and we'll do it using different
11:49Alpha levels because I want you to see
11:52the relationship really between three
11:54things what happens to our critical
11:56value when we change the alpha level
11:59what happens to the critical values when
12:01we change from the Z distribution to the
12:04T distribution and finally what happens
12:06when we change both so let's go ahead
12:09and take a look at a few curves the
12:11first one we'll look at is the
12:12two-tailed Z test that we looked at in
12:15the previous video so our hypotheses are
12:18the same our Alpha level is going to be
12:2105 so we're go and use that one and here
12:24is our sampling distribution so remember
12:27this distribution is a bunch of sample
12:30means and we put them in their own
12:32distribution and we come up with the
12:34sampling
12:35distribution and the hypothesized mean
12:38is there in the middle now we call the
12:41blue region in the middle the
12:42non-rejection region and in the Tails
12:45those are our rejection
12:48regions so in this case since our Alpha
12:51is 05 we have 025 in the lower tail and
12:55025 in the upper tail so 2.5% in the
12:59lower tail 2.5% in the upper tail for
13:02the rejection regions now the boundary
13:05between the two is called the critical
13:07value and this is a common one so with
13:10an alpha
13:1205 and sigma known we would consult the
13:14Z table and find the corresponding zc
13:17scores for a two-tail test at Alpha of
13:2105 and this is a very common thing in
13:24statistics so I would just recommend
13:26that you commit this one to memory so so
13:29it's plus or minus
13:321.96 so those critical values are 1.96
13:37standard errors or standard deviations
13:40of the sampling distribution away from
13:43the mean there of zero so plus or minus
13:461.96 using Alpha of
13:4905 for the Z
13:53test so let's go ahead and look at the T
13:56distribution so this going to be a
13:57two-tailed t test rejection region I'm
14:00going to choose a sample size of 20 cuz
14:02remember every T distribution is unique
14:05it depends on the sample size and
14:06therefore the degrees of freedom so we
14:09have our same hypothesis the same Alpha
14:11level so the curve looks pretty much the
14:13same we have our hypothesized mean there
14:16in the middle our non-rejection region
14:18in the blue our rejection regions on the
14:21ends sort of in the brown color we have
14:232.5% in the lower tail and 2.5% in the
14:27upper tail and of course we have
14:29critical values that separate the two
14:31regions now what do you think's going to
14:33happen to the number of the critical
14:36value now that we're using the T Test
14:40will it stay the same will it move
14:43inward will it move
14:46outward well what we can do is we can
14:48look this up in the T table and I'll
14:50show you how to do that here in a minute
14:52so with the alpha of 05 and degrees of
14:55freedom of 19 we'll locate the critical
14:58values in the T table and when we do
15:00that we come up with a t of plus or
15:03minus
15:062.93 now remember in the previous slide
15:10what was the Z critical values for the
15:13same 95% region well it was plus or
15:17minus
15:181.96 so what happened to the critical
15:22values they moved
15:25outward now why is that the blue region
15:28is still
15:2995% but why do they move outward that's
15:33because we're using the T distribution
15:35remember the T distribution depending on
15:37sample size of course but in general it
15:39has a little bit less probability in the
15:41middle a little bit more in the Tails so
15:46if you can think of putting your hand on
15:47the top of this distribution and pushing
15:50downward so you kind of push down and it
15:52squishes out on the ends it takes our
15:55critical values with it
16:00now looking up TA tabls in the book so
16:04let's look up the T table when we do not
16:06know Sigma and we have the sample size
16:07of 20 so we have degrees of freedom of
16:0919 and Alpha of
16:120.5 so here's a typical T table located
16:15in the front or back of your stats book
16:17and we're going to look up the T
16:18statistics we had in the previous slide
16:22so the first thing we going to do is we
16:23find the column in this case that says
16:26052 taals so you can see that that in
16:29the bottom of that box cuz we had a
16:30two-tail test with an alpha of
16:340.5 then we'll find our degrees of
16:36freedom so in this case it's 19 then we
16:40look where those intersect and you'll
16:41see we have a value of
16:442.0
16:4693 so that's where our T values of plus
16:49or minus
16:512.93 actually came
16:57from now let's do something a bit
17:00different so we're going to look at the
17:02Z test again but now we're changing the
17:05alpha level so now we're going from an
17:07alpha of 05 to an alpha of
17:11010 in this
17:13slide so the regions on the ends the
17:16rejection regions are now 5% each so
17:2005 now our critical values are going to
17:24change are they going to go inward or
17:27are they going to go outward well you
17:29can probably tell just by looking at
17:31this curve that the critical values
17:33moved
17:34inward so we can look that up in the Z
17:37table again in our stats book so the Z
17:41is now plus or minus
17:441.645 well why is that now we don't have
17:4995% probability in the blue we only have
17:5290% in the blue region therefore our
17:55critical values have to move inward
17:59because we literally have less area in
18:01that blue region so it kind of pulls our
18:04critical values Inward and of course we
18:07have more probability in the rejection
18:08regions in the Tails so our Z values
18:11moved inward so remember when our Alpha
18:14level gets larger so from 05 to
18:190.10 our area in the middle shrinks and
18:23therefore our critical values move
18:25inward towards the middle
18:32now let's do the same thing so same
18:35hypothesis same Alpha of 0 one0 but this
18:38time we're going to use the T Test with
18:40the same sample size of 20 so 5% in each
18:45tail now our critical values what do you
18:47think is going to happen as compared to
18:49the last
18:52slide well they're going to move outward
18:56so remember in the last slide it was
18:58plus or minus
19:011.645 for the Z distribution but now
19:05it's t is plus or minus
19:081729 so a bit further outward and
19:12actually about that much further outward
19:14and why is that again it's because we're
19:18using the T distribution instead of the
19:21Z distribution we have a little more
19:24probability in the Tails so we kind of
19:27push down on the Curve it pushes it out
19:29on the ends and it takes the critical
19:32values along with it again just ever so
19:35slightly but it is an important
19:41amount so just some General T
19:43distribution patterns a smaller sample
19:46size means more sampling error now this
19:50sampling error due to a small sample
19:52size means a higher probability of
19:55extreme sample means and this should
19:58sort of makes sense remember if my
20:00population is a million people or a
20:02million things if I go out and only
20:05sample five that's not as good as
20:07sampling 75 if I only sample five then
20:11I'm more likely to get some extreme
20:14value out in the sampling distribution
20:17because it's less representative of the
20:19overall population that's sampling error
20:22now more probability in the Tails means
20:26the center hump of the T distrib dist
20:28bution must come downward a bit again
20:31imagine putting your hand on the top of
20:33the T distribution and pushing down so
20:35you push down a bit and it pushes more
20:38probability into the Tails so this
20:41process sort of squishes the
20:43distribution slightly downward and
20:45outward thus taking the critical values
20:49along for the ride so going from the Z
20:52distribution to the T distribution
20:55literally pushes the middle of the
20:57distribution down
20:59pushes the Tails outward and takes the
21:01critical values with it assuming the
21:05alpha is the same and of course remember
21:08there'll be a different T distribution
21:09slightly for each sample size but in
21:12general that's what's happening so given
21:14the same Alpha and Sample standard
21:17deviation a smaller sample size will
21:19push the critical values further outward
21:21in the Tails due to the uncertainty
21:24associated with a small sample size so
21:28again with a very small sample size
21:30there will be a lot of probability in
21:32the Tails of the T
21:34distribution as the sample size
21:36increases the probability in the Tails
21:39decreases so the distribution begins to
21:41sort of Squish in and upward till
21:44finally when we get to a sample size of
21:46say around 100 the Z and the T
21:49distribution are basically the same so
21:53that's why I use the T distribution
21:55anytime it's under 100 because once you
21:57reach a 100 they are basically
22:02indistinguishable so let talk about the
22:04the T test and then we'll wrap up part
22:06one so the T test for a single mean is
22:10very similar to the Z test all we do is
22:13substitute t for Z and S for
22:18Sigma so xbar is a sample mean mu subz
22:22is the hypothesized population mean s is
22:25the sample Center deviation and of
22:28course n is the sample
22:30size now remember that this denominator
22:34is a very special thing it is the
22:36standard error of the mean which is the
22:38standard deviation of the sampling
22:40distribution basically it's the standard
22:42deviation of the curves we just spent 10
22:45minutes looking at now it could be
22:48written like this so you might see it as
22:51s subxbar but these mean the same thing
22:55the one on the left is just sort of the
22:56expanded version of the one there on the
22:59right so what we're asking is is this
23:02test value in the non-rejection region
23:06or in the rejection region based on a t
23:09distribution with n
23:11minus1 degrees of
23:16freedom okay so that wraps up part one
23:19of our single sample T Test video so
23:23remember this was just about the
23:24conceptual background and I want you to
23:26really get in your mind a couple of
23:28things
23:29how does alpha change the critical
23:32values in a distribution so all us being
23:35equal a larger Alpha will bring the
23:38critical values inward because there's
23:40less area in the middle say 90% if the
23:43alpha goes to 05 then there's 95%
23:46probability in the middle if it goes
23:48down further then there's 99%
23:50probability in the middle so to a
23:52account for those increasing areas in
23:54the middle of the distribution the
23:55critical values have to go out W so you
23:59always keep that in mind also keep in
24:01mind how the distribution is different
24:04between the Z test and the T Test the T
24:07test because it has more probability in
24:09the Tails and a little bit less in the
24:11middle it will tend to pull out the
24:14critical values ever so slightly again
24:17depending on the sample size so it's all
24:20about how these two things Alpha and the
24:24characteristics of the T
24:26distribution affect the critical values
24:29because remember the entire basis of
24:32hypothesis test is how our test
24:34statistic relates to that critical value
24:38if you don't understand what influences
24:40that critical value then you really
24:41don't understand what's going on with
24:43the test itself so that's why we did
24:45this part one conceptual background so
24:48just a few reminders if you're watching
24:50the video because you're struggling in
24:52the class stay positive and keep your
24:54head up you're very smart and talented I
24:57have faith in you so to other people
24:59associate you please feel free to follow
25:01me here on YouTube on Twitter on Google+
25:03or on LinkedIn that way when I upload a
25:06video you know about it and it's always
25:08nice to hear from you and finally just
25:10keep in mind that the fact that you were
25:12on here trying to learn trying to
25:13improve yourself that's what really
25:15matters I firmly believe that if you
25:17have the right learning process in place
25:20the results will take care of themselves
25:22so thank you very much for watching I
25:24look forward to seeing you again in part
25:26two where we work to full example
25:29problems
25:36[Music]

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