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0:01[Music]
0:08hello thank you for watching and welcome
0:10to the next video in my series on basic
0:12statistics now since this is part two of
0:15a two-part video I'll keep this intro
0:17short but just keep in mind if you're
0:19watching this because you're struggling
0:21in a class stay positive and keep your
0:23head up you're smart and talented and
0:26you must have the confidence that you
0:27can work through it please feel free to
0:30follow me here on YouTube on Twitter on
0:32Google+ or on LinkedIn that way when I
0:35upload a video you know about it if you
0:38like the video please give it a thumbs
0:39up share it with classmates or
0:42colleagues or put it on a playlist that
0:44does encourage me to keep making them
0:45for you and just keep in mind that these
0:47videos are meant for individuals who are
0:49relatively new to Stats so I'm just
0:51going over basic concepts so all that
0:54being said let's go ahead and get
0:56started on part
0:58two
1:01so remember in part one of this video we
1:03talked about the conceptual background
1:05of the single sample T Test and we did
1:09that through the lens of the single
1:11sample Z test which we learned about in
1:14previous videos so remember the main
1:16difference is that in the Z test we know
1:20Sigma or Sigma is given to us and of
1:23course that's the population sard
1:25deviation in the T Test we are not given
1:30Sigma or we do not know it therefore we
1:33estimate it using the sample standard
1:35deviation so that's the primary
1:37difference between the two now from part
1:40one we walked away with two conceptual
1:44pillars the first one is that the alpha
1:47level affects the location of our
1:50critical values so if we have an alpha
1:52of
1:54say10 that's going to be a probability
1:57of 09 in the middle of the distrib
2:00distribution now if we go to an alpha of
2:0405 that's going to be
2:0695% probability in the middle so more
2:10well because we have more probability in
2:12the middle what has to happen to our
2:14critical values as far as their location
2:17well they have to move outward now let's
2:20say we go from an alpha of 05 to an
2:23alpha of
2:2501 so the probability is now .99 in the
2:28middle of the distribution
2:30then what has to happen to the location
2:32of our critical values well they're
2:34going to have to go out even further to
2:36accommodate that 99% area or 099
2:40probability so the alpha level affects
2:44the location of our critical values in
2:48our distribution that's the first
2:49important point the second important
2:52point is the difference between the Z
2:54distribution and the T distributions
2:57because remember every sample size has
3:00its own t distribution but in general we
3:04talked about the following
3:06idea for any given Z distribution if we
3:10take the T distribution of say a
3:12moderate sample size like 20 or 25 if we
3:16look at the critical values of that Z
3:19distribution as compared to the T
3:22distribution for a given Alpha what
3:24happens to the location of the critical
3:26values in the T
3:28distribution well again they move
3:30outward towards the tails ever so
3:33slightly and why is that well remember
3:36the T distribution has a little bit less
3:38probability in the middle a little bit
3:40more in the Tails so if we compare it to
3:44the Z distribution the critical values
3:47are going to be slightly outward to
3:50accommodate that sort of squishiness of
3:53the T distribution so these two things
3:56affect the location of the critical
3:58value the alpha level Lev and the T
4:02distribution as far as its sample size
4:04goes because in general as the sample
4:06size gets smaller the critical values
4:09are going to be pushed outward because
4:11there is more uncertainty in the
4:13sampling distribution now why am I
4:16telling you all this I spent three
4:17minutes telling you this why well the
4:20whole idea of hypothesis testing is
4:22about the location of the critical
4:24values so if our test statistic is on
4:27one side of the critical value or the
4:29other
4:30that completely changes what our
4:32conclusion is so you have to understand
4:35what's happening with the location of
4:36the critical values in terms of the
4:39alpha level and the characteristics of
4:42the T distribution because as we'll see
4:44coming up here in a few minutes it can
4:47really change the outcome of your
4:50problem so let's go ahead and take a
4:51look at our first
4:55example so this is the same problem I
4:58used in the Z test video but I changed
5:02it to adapt it for the T Test so we're
5:04not going to reinvent the wheel we'll
5:05just change a few things and use the
5:06same problems so let's say a report from
5:096 years ago indicated that the average
5:11gross salary for a business analyst was
5:1469,8 73 now since this survey is now
5:18outdated the BLS wishes to test this
5:21figure against current salaries to see
5:24if the current salaries are
5:25statistically different from the old
5:28ones so we're testing to see if the
5:31current population of salaries is
5:34equivalent to the old population of
5:37salaries now based on this sample we
5:39found a sample standard deviation of
5:4214,985 now again we do not know Sigma
5:46therefore we'll have to estimate it
5:48using this sample standard
5:50deviation now for this study the BLS
5:52will take a sample of 12 current
5:55salaries so look at what we have here we
5:58have a hypothesi ized population mean
6:01that's
6:0269873 we have a sample standard
6:05deviation
6:0714985 and we have n our sample size we
6:12almost have everything we need to
6:13conduct our
6:15test but of course we're not going to do
6:17that yet we're going to set up a proper
6:20hypothesis so our null hypothesis is
6:23that the current population of salaries
6:26is equivalent to the old population of
6:29salaries
6:30therefore the alternative is that the
6:32current population of salaries is not
6:35equivalent to the old population of
6:38salaries now this would be a two-tailed
6:40test the salaries could be higher or
6:43they could be lower I mean we're in a
6:45recession right now so it could they
6:47could be lower now since Sigma is
6:49unknown and N is small we'll use the T
6:52distribution and there is the T
6:54statistic formula over there on the
6:56right it looks very similar to the Z
6:59statistic
7:02formula so specify the type 1 error rate
7:05now this is my choice I'm going to
7:07choose an alpha of
7:0805 I could have selected an alpha 01 or
7:120.10 but I choose the middle ground and
7:14I'll select an alpha 05 that means I
7:17accept the possibility that I might make
7:19a type one error 5% of the
7:23time so State the decision Rule now
7:26remember we are using the T distribution
7:28so the location of our critical values
7:30and our regions will depend on our
7:33sample size and our Alpha so in this
7:36case our sample size is 12 therefore our
7:39degrees of freedom is 11 so we will
7:42consult our T table we'll find the
7:44column that is a two-tailed test at an
7:47alpha of
7:4705 then we'll find the row that is 11°
7:51of Freedom we'll find where those
7:53intersect and we find that we have a t
7:55value of
7:582.21 so our critical values are t plus
8:02or minus
8:062201 then we'll gather our data so we
8:09have 12 in our sample and we found out
8:12that our sample mean is 79
8:16180 so now we have everything we need to
8:19go ahead and figure out our test
8:24statistic so we have our sample mean of
8:2679 1880 our hypothesize pop relation
8:29mean of 69873 there's almost a $10,000
8:32difference there our sample standard
8:35deviation of
8:3614985 or sample size of 12 so we can go
8:40ahead and substitute all that into our
8:42formula so we have 79 180 minus
8:4669873 ided 14985 divid the RO of 12 so
8:50we just substituted everything in there
8:53and that generates a t statistic of
8:572.15 so that is our test
9:01statistic now we'll go ahead and put
9:03that in the context of our sampling
9:06distribution so our hypothesis there are
9:08the same we have our hypothesized mean
9:11they're in the middle our non-rejection
9:13region in blue and our rejection region
9:15in the brown so where does our test
9:17statistic fall well it's 2.15 so it
9:21falls right there now what do we notice
9:26since the test statistic is in the
9:28non-rejection region and not beyond the
9:32critical T value we fail to reject the
9:35null
9:36hypothesis so it's not out of the
9:39ordinary that this sample came from a
9:42population with a mean population mean
9:45of
9:4669873 as we
9:48hypothesized now it's in the upper area
9:52of our non-rejection region but that
9:54really doesn't make a difference it's
9:55either in that region or it's not so we
9:58just just happened to get a sample that
10:01fell here now we could have gotten a
10:04sample that fell in the same place on
10:06the other side so maybe minus
10:102.15 or we could have gotten a sample
10:12that fell right in the middle or we
10:14could have gotten a sample that fell in
10:16the rejection region that's the idea of
10:19a sampling distribution we expect
10:2395% of our samples to be in this blue
10:26region and ours just happened to be one
10:29of those now had it been a bit higher it
10:32would have been outside but that is just
10:35the idea of chance our sample happened
10:38to fall right there so we failed to
10:40reject the null
10:42hypothesis and remember our T value was
10:452.20 one so it would have had to have
10:49been beyond 2.20 one for us to actually
10:53reject the null
10:57hypothesis now let's do the same same
10:59problem but change the sample size so
11:02we're going to go from a sample size of
11:0412 to a sample size of 15 let's see what
11:07happens so everything else is the same
11:10sample means the same hypothesize means
11:13the same deviation the same just the
11:15sample size has changed so we'll go
11:17ahead and substitute all that into this
11:20formula now we have a t statistic of
11:252.41 so that is our test statistic let's
11:28go ahead and place it on the
11:31curve so here everything is the same so
11:34our degrees of freedom are now 14 cuz
11:36remember our sample size was 15 so our
11:39degrees of freedom is 14 now if we go to
11:42our T table we find that the critical T
11:44value is plus or minus
11:482.45 so that is the critical value there
11:50on the bottom now remember that for a
11:54degrees of freedom of
11:5611 it was plus or minus 2.2
11:592011 so when we increased the sample
12:02size what happened to the location of
12:05the critical T values well they moved
12:08inward ever so slightly but they did
12:11move Inward and why is that well we'll
12:15talk about that here in a second
12:16actually so our T value is
12:222.41 so look where it falls well it's
12:25now in the rejection region so you look
12:29over there on the left if T is greater
12:31than
12:322.45 we reject the null
12:36hypothesis so all we did here was change
12:39the sample size from 12 to
12:4415 and now we came up with a completely
12:47different conclusion and the question is
12:50why is
12:51that
12:52well the larger sample size decreased
12:57the standard error of the mean so the
13:00larger sample size decreased the
13:02standard deviation of this
13:05distribution it made it
13:08narrower so what ended up happening is
13:11it made our sample stand further out on
13:15its own it made it a little bit more
13:18likely to belong to a different
13:20population that does not overlap much
13:23with this population it created a
13:26separation between our sample xbar and
13:30our
13:31hypothesized sample mean so the larger
13:35sample size decreased the standard
13:37deviation the standard error of the mean
13:40of this distribution so it kind of like
13:43sucked in its middle It's Kind like when
13:45your pants don't fit so you suck in your
13:47belly that's kind of what happened to
13:49this distribution when we increase the
13:51sample size it pulled it inward towards
13:53the middle and that left our poor t
13:56statistic standing out there by itself
13:59self now also the larger sample size led
14:03to a higher degrees of freedom this
14:06brought more probability as well in
14:09towards the middle of the T distribution
14:11around our hypothesized population mean
14:15so being inside the non-rejection region
14:18this blue area is a bit more exclusive
14:21Club so you can think of our our sample
14:24mean is kind of like someone outside the
14:26bar who cannot afford to get in to hang
14:29out with the cool people so two things
14:31went on here and that's why we went over
14:33all that conceptual information the
14:36larger sample size decreased the
14:39standard error of this distribution so
14:41it kind of sucked in towards the middle
14:44also the larger sample size led to a
14:46higher degrees of freedom so that
14:49brought more probability in towards the
14:51middle as well so two things going on
14:53there and therefore our sample mean it's
14:59the same sample mean but this time it
15:01was left outside of the non-rejection
15:04region and all we did there was change
15:06the sample size from 12 to
15:1215 okay so example two Starbucks
15:15customer
15:16satisfaction so Starbucks is interested
15:19in assessing customer satisfaction in
15:21the Canadian city of Toronto
15:23Ontario to conduct the study Starbucks
15:26asked 25 customers in the city the
15:29following question compared to other
15:31coffee houses in Toronto would you say
15:34the customer service at Starbucks is
15:36much better than average that's a score
15:38of five better than average that's a
15:40score of four average a score of three
15:44worse than average a score of two or
15:47much worse than average a score of one
15:50and of course we call that a ler scale
15:53in
15:54research now we found that the mean
15:56rating was determine to be 3
15:593.5 also based on this sample the
16:02standard deviation was found to be 1.4
16:05so our sample mean was 3.5 or sample
16:08Center deviation was
16:131.4 so it set up our hypothesis so our n
16:16hypothesis is that the mean customer
16:19sentiment or feeling is three or less so
16:24average or lower therefore the
16:27alternative hypothesis is that customer
16:30sentiment is higher than average or
16:33greater than
16:35three now this will be a one-tailed test
16:38and we can tell that by the signs in our
16:42hypothesis and Starbucks remember is
16:45interested in a better than average
16:47rating so here's what they're kind of
16:48trying to do in this test they're
16:51setting up a n hypothesis that says oh
16:54our customer you know sentiment is
16:57average or lower
16:59and then we're going to collect some
17:00data and we're going to see whether or
17:03not our data support the idea that we
17:06can reject that null hypothesis if we
17:09can reject that null hypothesis then we
17:12can proceed to the alternative that says
17:14customer sentiment is better than
17:18average now since Sigma is unknown and
17:21our sample size is small we will use the
17:23T distribution so the T formula is over
17:26there on the right
17:29so we go ahead and specify the type 1
17:31error rate now I'm going to choose an
17:33alpha of 01 and that's just my choice
17:36for this one-tailed test then we'll
17:39State our decision rule remember our
17:41sample size is 25 therefore our degrees
17:44of freedom is 24 so we need to go to our
17:47T table find the column for a onet
17:51tailed with an alpha of
17:5301 find the row for 24° of freedom and
17:58we do do that we come up with a critical
18:00T value of
18:022.49 2 so if our test statistic is
18:06greater than 2.49 2 we will reject our
18:11null
18:13hypothesis now on the curve we can
18:15actually place it so remember with an
18:17alpha of 01 what we're saying is that we
18:20have a 1% chance that we're ruling to
18:22accept a 1% probability of committing a
18:26type 1 error so that's the 1 % in the
18:29upper tail and therefore 99% of our
18:32sample means should be in the blue so
18:35we'll place our critical value our T
18:37value right there so 2.4
18:4192 of course then we'll gather our data
18:44so we had a sample size of 25 and our
18:46mean was
18:513.5 now we can calculate our test
18:54statistic so our mean of 3.5 our
18:57hypothesized mean of of three sample
18:59standard deviation of 1.4 and sample
19:02size of 25 so we'll go ahead and
19:04substitute those numbers into our
19:06formula and we arrive at A T statistic
19:09our test statistic of
19:151.79 so now we have to place that in the
19:17context of our sampling distribution so
19:20the same hypothesis we have our mean was
19:233.5 our critical value was 2.49 2
19:28our test statistic was
19:321.79 so what happens since the test
19:35statistic is inside the non-rejection
19:38region and inside the critical value
19:41sort of inside of towards the mean we
19:44fail to reject the null hypothesis that
19:47customer satisfaction is at or below
19:50average okay so we fail to reject the N
19:54hypothesis it is higher so it's 3.5 it
19:57is higher
19:58but the sample is just one of many that
20:01could be inside this non-rejection
20:04region the sample mean would have had to
20:07have been much higher to actually
20:10surpass that critical value now the
20:13question is what would that sample mean
20:16have to be to surpass that critical
20:20value of 2.4
20:2592 so finding the hypothetical sample
20:27mean that aligns with our T critical
20:30value of 2.49 2 is actually fairly
20:33straightforward remember when we first
20:35found our T statistic the T was the
20:38unknown and our sample mean was what we
20:41found in our data analysis so now we're
20:43going to flip that a little bit on its
20:45head now we know the T value and we want
20:49to know the sample mean that corresponds
20:52with that critical T value so our xbar
20:56is now the unknown and our T is known so
21:00again we'll just use our handy algebra
21:02from many years ago to solve for xar so
21:05we go ahead and simplify our denominator
21:08so that is 28 we multiply both sides by
21:1128 so we have 698 = xarus 3 so we'll go
21:17ahead and simplify that and we have a
21:19sample mean of 3.
21:23698 so therefore any sample of size 25
21:28with a mean that's greater than
21:323.69 would lead to the rejection of the
21:35null hypothesis assuming the same sample
21:38deviation which is not all that likely
21:40but we'll assume and the same Alpha
21:43level so the sample mean of
21:483698 is the sample mean that would
21:50hypothetically fall right onto our T
21:54critical
21:57value now if we go ahead and look at our
21:59distribution again we can actually sort
22:00of write these in so where t equal 1.79
22:05that's the test statistic from our
22:06analysis our sample mean was of course
22:103.5 now the sample mean that would
22:12correspond with our critical value of
22:152.49 2 is
22:203.69 so you can see the difference
22:22between our sample mean and the mean
22:24that would have been required to reject
22:27the
22:31hypothesis So based on our Alpha of 01
22:35we know that 1% of our area is in the
22:36upper tail so that's the 1% past RT
22:40critical of
22:432492 right there now in the P Value
22:47method we ask how much area or
22:49probability is above our test statistic
22:52so we're interested in how much area is
22:56to the right of our test
22:59statistic Now using a t table we can
23:01tell that the probability is between 05
23:05and 025 so we kind of eyeball it between
23:08two known now in Excel we can find it
23:11exactly Excel gives us a P value of
23:15043 now those are that's greater than 01
23:19so we know that we cannot reject the
23:22null
23:24hypothesis now this is often referred to
23:27as the observe D significance level so
23:30again we can find the P value here to
23:32actually show that we cannot reject the
23:35null because the area to the right of
23:38our T statistic is
23:410.43 but our Alpha is
23:4401 so therefore if we wanted to reject
23:47the null it would have to be less than
23:5301 now how did I find that in Excel so
23:56remember our degrees of freedom are 20
23:5724 our T value is
24:001.79 so we just use this formula in
24:03Excel 2010 it's
24:06t.d. RT that stands for right tail it
24:09saves us from having to subtract from
24:11one so 1.79 is our T value and 24 is our
24:16degrees of
24:17freedom so you can actually use the
24:20function tool in Excel it's a little F
24:23ofx button you can press and it'll tell
24:25you exactly what to put in so our T
24:28value is 1.79 and our degrees of freedom
24:31were 24 so if we look where it actually
24:34solves it the formula result is
24:38.43 and that's how I came up with the P
24:41value of
24:43043 in the previous
24:47slide okay so that wraps up our two
24:50examples on single sample T tests and
24:53the T distribution so just a reminder of
24:55the process so you can do it correctly
24:57ly always start with a well-developed
24:59clear research problem or question no
25:03fancy stats will solve a bad research
25:05problem so always hone in always clarify
25:09your problem before you start establish
25:11your hypothesis both null and
25:14alternative determine the appropriate
25:16statistical test and sampling
25:18distribution and again this depends on
25:20really two things do you know Sigma or
25:23do you not know Sigma is your sample
25:26size below 100 or not so again that will
25:30determine whether or not you're using
25:32the Z distribution and Z test or the T
25:35distribution and T Test choose your type
25:381 error rate again that's up to you then
25:41State your decision rule So based on the
25:43statistical test and distribution you're
25:45using and your error rate you will come
25:48up with a decision rule that says if my
25:50T value Falls here I cannot reject the
25:53null hypothesis if it falls here I have
25:56to reject the null hypothesis
25:58then gather your sample
26:00data once you have your sample data
26:02calculate the test statistic once you
26:05have the test statistic you compare that
26:07to your decision Rule and then State
26:09your
26:10conclusion once you have the conclusion
26:12you can then make some sort of real
26:15world recommendation or some real world
26:18publication whatever your application
26:20may happen to
26:23be okay so that wraps up part two of our
26:26video on the single sample T Test and I
26:29really hope you have a firm grasp of how
26:32all this comes together so we talked
26:34about the T Test as compared with the Z
26:36test we talked about what happens when
26:38you maybe even change one small
26:40parameter like the sample size in that
26:42case we got to completely different
26:43conclusion by changing the sample size
26:46by three so you can see that these some
26:49of these tests are very sensitive to
26:51their parameters and I really want you
26:53to understand how these things affect
26:56the overall test the alpha level
26:58the sample size and things like that so
27:01just keep in mind if you're watching the
27:02video cuz you're struggling in a class
27:04stay positive and keep your head up if
27:06you're watching this it means you've
27:07accomplished quite a bit you're smart
27:09and talented and you can get through it
27:11please feel free to follow me here on
27:13YouTube on Twitter on Google+ or on
27:15LinkedIn that way when I upload a new
27:17video you know about it and it's always
27:20nice to hear from people who watch my
27:21videos online life is Much Too Short The
27:24World Is much too large for us not to
27:26take the opportunity to connect with one
27:28another if you like the video please
27:30give it a thumbs up share it with
27:31classmates or colleagues or put it on a
27:33playlist and finally just keep in mind
27:35the fact that you're on here trying to
27:36learn trying to improve yourself as a
27:38student or as a business person or
27:41whatever else you may be doing that's
27:43what really matters I firmly believe if
27:46you have the right learning process in
27:48place the results will take care of
27:50themselves so thank you very much for
27:52watching and look forward to seeing you
27:54again next
27:56time
28:01[Music]
28:06oh

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