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Briefing #11 - PSYC/SOC-270-580 — Transcript

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0:07Oh, statistics scholars, Dr. McNeal
0:09here. Welcome back.
0:12Today we're going
0:14to introduce you to
0:17a brand new statistical procedure. What
0:20else is new, right?
0:22Analysis of variance or ANOVA. ANOVA
0:26stands for analysis AN of O variance VA.
0:32Now we've been doing a lot of
0:33mathematics over the last several weeks,
0:35but I want to continue to remind you
0:38that
0:39statistics is the study of language.
0:42It's not the study of math.
0:44You know by this time that there is
0:47statistics or there is math involved in
0:49statistics, but it's not complicated
0:52math. It's not big math. It's not much
0:54more than adding and subtracting,
0:56multiplying, dividing, and exponents and
0:59square roots. You've gotten much more
1:01comfortable doing all of that over the
1:05past what 11 12 weeks.
1:08Statistics hasn't gotten more
1:11complicated.
1:13It's gotten a little bit more complex,
1:15but not more complicated.
1:17We've calculated mean, standard
1:19deviations, standard errors, Z scores, T
1:24statistics for one sample, paired
1:26samples, and independent samples,
1:29confidence intervals, and effect sizes.
1:32And I want to make sure that we haven't
1:34lost sight of something very important.
1:37Now I know we've been sort of jumping in
1:40doing a lot of calculations of problems,
1:44but really in statistics none of the
1:46work that we do begins with a formula.
1:50It actually begins with a research
1:52question.
1:54So as scholars, we're interested in
1:56something that's actually happening in
1:58the world, something that we don't
2:00understand, something that we're curious
2:03about.
2:04And so we develop a research question
2:07because we want to learn something.
2:09Then we decide how to collect the
2:12evidence that might help us answer that
2:14question.
2:16Only after we've done all of that do we
2:19turn to the statistical analyses. And I
2:22want to
2:23begin to show you that moving forward
2:26because you have enough statistics
2:28underneath your belt that now we're
2:30going to move into the world of
2:32research. Remember, this course is um
2:35statistics for the psychological
2:37sciences.
2:40So today we're going to begin with a set
2:42of numbers and
2:45No, actually
2:46instead of starting with a set of
2:49numbers, I'm going to ask you
2:52No.
2:53Today, instead of starting with a set of
2:55numbers and asking you to calculate
2:58something, we're going to begin where
3:00researchers actually begin
3:02with a question.
3:05And I'm also committed to doing
3:06something else today
3:09that follows up in what I did last week.
3:12I'm going to show you all the mistakes I
3:13make.
3:15I'm not going to clean it all up in the
3:17edit so that it's nice and pretty. I
3:19want you to hear
3:20the effort that it takes.
3:23Now, I've worked out the scenario that
3:25I'm going to present to you today, but
3:28I might make a mistake along the way.
3:31And in fact,
3:33if I make a mistake, I'm going to offer
3:35you
3:3710 points for every mistake I make.
3:40How's that?
3:42So
3:44let's begin.
3:45Let's imagine that I'm a researcher.
3:48Well, we don't really have to imagine
3:50that because I am a researcher.
3:52But let's imagine that you are a
3:53researcher with me. And we're interested
3:56in children and their parents.
4:00More specifically, we're interested what
4:03parents do to get their children to go
4:07to bed.
4:08Because as many of you know, parents
4:10have different strategies.
4:12Some parents have a set bedtime.
4:158:00 means
4:178:00.
4:19Other parents negotiate with their
4:20children. Maybe tonight it's 8:00,
4:23tomorrow it's 8:30.
4:25There's a little discussion involved.
4:28And still other parents allow their
4:30children to decide
4:31when they're ready to bed, when they're
4:33ready to go to bed. I don't know who
4:34those parents are, but they exist.
4:36There are three different approaches.
4:39And and we're curious about something.
4:42Does the bedtime strategy parents use
4:45relate to how long it takes their
4:47children to fall asleep?
4:49That's our research question. In fact,
4:52I'm going to go to the map
4:54and start
4:55populating the map
4:59in a different way.
5:01We'll start with, again,
5:04research question.
5:07Uh does the bedtime
5:13strategy
5:17parents use
5:21relate to how long
5:27uh
5:28that it takes their children to go to
5:29bed.
5:40That's a legitimate question.
5:42Now, we may already have some idea about
5:46what we're going to find, but that's not
5:48how research works. We have to
5:52go to the research with an open mind
5:56to discover something.
5:59Now, we're going to find out instead of
6:04getting information or getting data from
6:06the textbook. Let's show you what it
6:09really what really happens.
6:11We would recruit parents
6:14so that we can find out
6:17what they do.
6:18But before we recruit anybody,
6:21before we collect a single observation,
6:25we need to establish our hypotheses.
6:27Remember, we've been doing that since
6:29chapter seven.
6:31So the first step is again our null
6:32hypothesis.
6:34Uh
6:35our null hypothesis states that there's
6:37no difference among the population mean.
6:39So in other words,
6:40there is
6:42no difference
6:47in
6:48children
6:51sleep time
6:54based on
6:58parents'
7:01strategy.
7:04There's no difference.
7:06Now for an ANOVA, that's different than
7:08we've done with one sample, paired
7:12samples, and independent samples, we're
7:15actually working with three or more
7:19uh
7:20three or more
7:21samples or three or more categories. So
7:23how we write that mathematically is mu
7:26one
7:27equals mu two
7:29equals mu
7:30three. That corresponds to the three
7:34uh strategy strategies that we just said
7:36that parents use. Parents can either use
7:39a kids go to bed when they decide, kids
7:42negotiate, or kids adhere to a strict
7:44bedtime. So that's mu one, mu two, and
7:46mu three.
7:49So we're saying in other words that no
7:50matter
7:52what the bedtime strategy parents use,
7:54the average time it takes uh for
7:56children to fall asleep is the same.
7:59Now our alternative hypothesis is a
8:02little bit different than what we've
8:03done up till now.
8:06Because we're calculating the ANOVAs by
8:08hand, we can't say for sure
8:12that we can find one or the other. So
8:15all we really say for anova
8:18uh when we're doing it by hand is at
8:21least one group is different. At least
8:27one parent's
8:30strategy
8:33is different.
8:36We can't say
8:38um
8:40that it's a tail test to the right or to
8:42the left. In fact, we're doing what's
8:44called a one way anova.
8:49We can only look in one direction.
8:52So again, notice here we're not just
8:54comparing two groups. We have three
8:55groups and that's why we need an anova.
8:59Our t tests have prepared us to compare
9:01means between two conditions, but anova
9:04allows us to examine differences among
9:06three or more groups in a single overall
9:10analysis.
9:11And again,
9:12here's something else to remember. If
9:15we reject the null hypothesis, we can
9:17conclude that at least one population
9:19mean difference differs, but the overall
9:23anova doesn't tell us which group
9:26uh which groups are significantly
9:28different from one another.
9:31For that, we would need uh additional
9:33procedure, something called a post hoc
9:36test
9:37uh which some of you may encounter when
9:39working with a program called SPSS.
9:42And that program is part of of a class
9:45called psych social 271. I actually
9:48teach that class as well. I'm teaching
9:50it this semester too. It just started
9:52three weeks ago.
9:54But for now, we're just concentrating on
9:56that first question.
9:57Are all of these three populations,
10:01their population means, are they equal?
10:04We've got our research question.
10:06We've established our hypotheses.
10:09Now we need participants. So I developed
10:12a recruitment flyer. Let's take a look
10:14at it.
10:16We're conducting a hypothetical study at
10:19Southwestern College involving scholars
10:21from Psych Social 270 sections 501 and
10:25508, which you see right here.
10:28We want to recruit 15 parents.
10:31parents whose children decide when they
10:33go to bed,
10:35five parents who negotiate bedtime with
10:37their children, and five parents who use
10:39a set bedtime.
10:41We're not telling parents to change
10:43their bedtime strategies. We're simply
10:44recruiting parents
10:46who already use these approaches. And
10:49we're going to ask them to observe their
10:51child for one night
10:53and
10:54tell us how long it takes the child to
10:57fall asleep. The measurement is fairly
11:00straightforward.
11:01How many times, again, does it take the
11:03child to fall asleep after getting into
11:06bed?
11:08We can work out exactly how parents
11:09report that information in a detailed
11:13uh in a more detailed research study,
11:15but
11:16sort of imagine this. You put the child
11:18to bed, the parents put the child to
11:19bed, and then they set a timer
11:22every 5 minutes and they go into the
11:25bedroom and if the child is asleep in
11:27the first 5 minutes, they say 5 minutes.
11:30The child isn't asleep, they come back
11:32uh at 10 minutes if the child is asleep,
11:34then they say 10 minutes and on and on.
11:37For our purposes, we're going to assume
11:39that we have a consistent,
11:41reasonable, accurate way, just like I
11:43just described, to collect that data.
11:45So again, let's say we recruit our 15
11:48families and they return
11:51with their observations. This is exactly
11:54the way research is done. We would have
11:58a recruitment flyer, we would have some
12:01recruitment strategy, we would have an
12:03idea of how many participants or
12:07respondents we need
12:09and we would get their information and
12:12move forward.
12:14So we collected data from the five
12:19parents from each group and now we have
12:21data. Let's take a look at our first
12:24group.
12:25These are our parents who allow their
12:27children to decide when they're going to
12:30bed.
12:31Their children took 25 30 35 40 and 45
12:36minutes to fall asleep.
12:39Notice that every child
12:41or notice that not every child had the
12:43same result.
12:44So even though the parents used the same
12:46bedtime strategy, the children are
12:48different.
12:50That's actually what's important.
12:52Now let's take a look at our second
12:54group.
12:55These are parents who allow their
12:57children to negotiate bedtime.
13:00Their results are 20 25 30 35 and 40
13:05minutes. Again,
13:07there's variation.
13:09Some children fall asleep relatively
13:10quickly,
13:11others take a little bit longer.
13:15And finally our third group. These are
13:17parents who have a consistent bedtime.
13:20Their results are 10 minutes, 15
13:23minutes, 20, 25 and 30 minutes.
13:27So now we have three groups,
13:29five observations in each group and a
13:32total of 15 observations.
13:35So we have our data set
13:38and now we're ready to investigate our
13:40research question.
13:44Now before we begin our calculations, I
13:47want you to to think about the three
13:49groups
13:51and look at these three groups.
13:54We're going to calculate the mean for
13:55each one
13:57and by now you should be able to do that
13:59almost in your sleep, right?
14:02Let's go over to the map and do that.
14:10Group one
14:12was the group uh
14:15the child goes to sleep when they uh are
14:17ready.
14:19Child ready.
14:23We're going to call that N one.
14:25And they went to bed at
14:2825 30
14:3035 40 and 45 minutes.
14:35Group two
14:36was the child negotiates.
14:41We're going to call that N two.
14:44And they went to bed 20 25 30 35
14:50and 40 minutes.
14:52Group three
14:55child strict.
14:59It's N three. They went to bed at 10 15
15:0420 30
15:07uh 25 sorry.
15:08There we go, my first mistake.
15:10And 30. 10 15 20 25 and 30. They're all
15:15five apart.
15:16I did that for a reason so that you
15:18could see how ANOVA's work more easily
15:22without
15:23sort of worrying about the calculations.
15:25But in the real world
15:28uh the data wouldn't necessarily be this
15:31nice and clean.
15:33So we need a mean for each group.
15:37Mean group one, mean group two
15:40and mean of group three.
15:42Now
15:43we could calculate this by to adding 25
15:4730 35 40 and 45 and dividing it by five.
15:51But because of how uniform this data set
15:54is, it's going to be just the middle
15:56number and it'll be the middle number
15:58for all three sets.
16:01Right? You don't need me to calculate
16:03that, do you?
16:05You don't, do you?
16:06You trust me?
16:07Okay.
16:09So the mean for group one is 35.
16:12The mean for group two is 30.
16:14And the mean for group three is 20.
16:20Now look at those means.
16:2335 30 and 20.
16:26That reminds me of something.
16:29If you're a child
16:31in the 70s,
16:33maybe even the 80s,
16:37one of these things certainly looks
16:39different from the others.
16:42But which one?
16:44Remember that song
16:45from Sesame Street?
16:51>> One of these things is not like the
16:54other. One of these things just doesn't
16:57belong. Can you tell which thing is not
17:01like the other before I finish my song?
17:08Now look very carefully because one of
17:10these things just doesn't belong here.
17:13Is it the apple
17:15or the uh the ice cream cone or the
17:17mitten
17:18or the hamburger?
17:22Well, if you guessed this thing is not
17:25like the other, if you guessed this
17:27thing just doesn't belong, if you
17:30guessed this thing is not like the
17:34other, well then
17:38you're absolutely correct because all of
17:41these things, the apple,
17:44the ice cream cone, and the hamburger
17:46are things you can eat.
17:47But uh
17:49did you ever try eating a mitten?
17:54>> The child decides has a mean of 35
17:57minutes. The negotiated mean has
18:0030 minutes and the strict time has a set
18:03time of mean of 20.
18:06We can see that the sample means are
18:08different, but here's the problem. Just
18:10because our sample means are different
18:12doesn't necessarily mean that our
18:14population means are different.
18:17Remember, we're working with samples and
18:19samples naturally vary.
18:22So,
18:23how do we determine whether
18:25these differences are statistically
18:27significant?
18:29That's where the ANOVA comes in.
18:32Now, look at all 15 observations.
18:36I'd like you to look more carefully at
18:37what we've collected. I want you to
18:39notice that there are two types of
18:42variation.
18:43First, we have variation between the
18:47groups,
18:48between the groups.
18:52They're not all the same.
18:54But we also have variation within the
18:57groups.
18:59So, this is within,
19:02and this is between.
19:06Looking at the child decides group, even
19:08though those parents follow the same
19:11bedtime strategy,
19:13one child takes 25 minutes to fall
19:16asleep, while another takes 45 minutes.
19:19And that brings us to the central idea
19:21of ANOVA.
19:23ANOVA stands for analysis of variance.
19:27And we've been studying variability all
19:30semester.
19:32You've calculated standard deviations.
19:35You've used standard errors. You've
19:36worked with the variability of samples
19:40to help you calculate t statistics.
19:43Now, we're going to make variability the
19:47centerpiece of the analysis. We're going
19:50to compare two sources of variation,
19:53variation between the groups
19:56and variation within the groups.
20:01To help you think about that, I'd like
20:02you to think about
20:06an old-fashioned radio.
20:09Some of you may remember
20:11turning the dial, trying to find a
20:13station.
20:15What did you hear?
20:16Perhaps first.
20:25Boy,
20:27Maybe some music.
20:29That's a signal. And the question
20:31becomes, can we distinguish the signal
20:35from all that noise?
20:38Well, that's a useful way of thinking of
20:41or thinking about an ANOVA. The
20:43variation between the group represents
20:46our potential
20:48Sorry. The variation between the groups
20:50represents our potential signal. The
20:52variation within the groups represents
20:55our background noise.
20:57And if the groups are truly different,
21:00we'd expect the variation between we
21:03Well, if the groups are truly different,
21:05we expect the variation between their
21:07means to be relatively large compared to
21:10the variation we would ordinarily
21:12observe among individuals within those
21:15groups.
21:16An ANOVA gives us a way to compare those
21:21two sources of variation.
21:26Signal,
21:31between group variation, differences
21:34among groups, noise within group
21:37variation, differences among
21:40individuals.
21:42And this is important. We're not simply
21:45asking which raw sums of squares is
21:48larger.
21:50We're going to adjust each source of
21:51variation
21:53for its degrees of freedom.
21:55And that gives us something called a
21:57mean square.
21:59And you'll see exactly how that works.
22:01So up till now, we've used a Z
22:04table, Z chart. We've used a T chart for
22:08a T test. And now we're going to use an
22:11F chart.
22:12To do that, we're going to use something
22:14called an ANOVA
22:16table. We've used
22:19a sum of squares table, sum of squares
22:22chart. And we're going to do something
22:23very similar to that called the ANOVA
22:26table.
22:27Before we do that, however, let's
22:29organize what we have.
22:32We're going to put our
22:34uh sort of
22:35mise en place on the
22:38on the map.
22:39We need to know
22:41how
22:42much we have in terms of the data set.
22:44So we could write it one of two ways. We
22:46could say N1 equals five
22:50because it does.
22:52N2 also equals five and N3 equals five.
22:57Or we could say N1 equals N2 equals N3
23:01equals five.
23:03That is the
23:05more
23:07streamlined way to do it. Now if these
23:10uh N sizes were different, we would have
23:12to say them different. But since they're
23:14all five, we can say N1 equals N2 equals
23:17N3 equals five.
23:19That's lowercase N.
23:22Now we have 15 observations altogether
23:25and we use the capital N for the whole
23:29data set.
23:31N equals 15.
23:34And now we have three groups. We denote
23:37the groups with a new symbol called K.
23:42You've probably seen that in other math.
23:45We have three groups, one, two, three
23:47and so
23:48K equals three. That's it.
23:52Now
23:53we need to establish our degrees of
23:56freedom.
23:58And we do that by setting up the ANOVA
24:02table.
24:04The ANOVA table is a little bit like
24:06Sudoku.
24:08I think I've said that in other
24:11past lectures or briefings.
24:15It's like you have a little bit of
24:17information and use some of the
24:18information that you have to create
24:20other information.
24:22So I'm going to set up the ANOVA table
24:25and then we'll start doing the
24:27calculations.
24:28The calculations, I'm just going to say
24:30this.
24:33The calculations seem like they're
24:35pretty gnarly, but they're not. It's
24:37fairly simple, fairly straightforward.
24:41So,
24:42the ANOVA table consists of these
24:45elements, the source of the variation,
24:50the sum of squares of the variation, the
24:52degrees of freedom,
24:54the mean square, and the F statistic.
24:59All right. So, I'm going to put this off
25:01to the Oh, let me finish filling it out.
25:04We have here are the different sources.
25:05We have a source of between
25:09the groups. That's one source of
25:10variation.
25:11And we have another source within the
25:14groups.
25:15So, this is between,
25:17this is within.
25:18So, we also have total.
25:21And so, let's set this off to the side.
25:25And you know the second step is to
25:27figure out the critical value, right?
25:30For ANOVA, we figure out the critical
25:32value a little bit different,
25:36differently.
25:37We need the degrees of freedom between
25:40over the degrees of freedom within.
25:44All right. So,
25:46there are very simple formulas. You
25:49don't have to memorize them. You don't
25:51have to
25:52look for them. You just have to follow
25:55the formulas that are listed in the
25:57chapter.
25:58So, it's a very simple formula.
26:01For the degrees of freedom between, the
26:03formula is K - 1.
26:08And for the degrees of freedom within,
26:10it's N - K.
26:16All right. We just just discovered that
26:18K is 3.
26:20So, it's 3 - 1
26:23or 2.
26:25And we discovered that N was 15. So, N -
26:29K or 3
26:3115 - 3
26:34is 12.
26:35So, we're looking for 12 two over 12.
26:39And
26:40to the ANOVA table, the F
26:44to figure out what critical value that
26:46is.
26:48Okay, here is our F table.
26:51You'll see that we have the numerator
26:55and the denominator. So, the numerator
26:58is across the top.
27:00The denominator is down the side. So,
27:02we're looking for two
27:04across the top, right?
27:06And 12.
27:10So, we
27:13can only do an ANOVA
27:17by hand at the alpha level of 0.05.
27:22And we can only do one tail. We can only
27:23do one direction. So,
27:26that's why you don't see the alpha
27:29values. So, we're looking for a critical
27:31value of 3.88.
27:34Let's head back over to the map.
27:383.88.
27:41We don't have to Well, we could.
27:45Draw curves and one, two, three.
27:49That's much further in.
27:51Uh
27:54the curve moved off the
27:57the the
27:59paper here.
28:02And that's really just to see where the
28:04critical value is. So, if we get a
28:05number that is greater than 3.88, we're
28:08going to reject the null hypothesis.
28:11And if not, we're going to fail to
28:13reject the null hypothesis.
28:17So, we have to also now
28:20fill in the degrees of freedom between
28:22and within that we just figured out, two
28:24and 12.
28:26So, two is for
28:28between. And 12 is for within.
28:32And for total, which I forgot the out
28:34there, is 2 + 12 that equals 14 degrees
28:39of freedom.
28:41Now we need to figure out the sum of
28:42squares between, the sum of squares
28:44within, the sum of squares total,
28:48the mean square between, the mean square
28:51within,
28:52and then the F statistic that will tell
28:55us either to reject or fail to reject.
28:58So I'm going to put a number one here, a
29:00number two here, a number three here,
29:03four, five, and then six because
29:07we're going to fill those out, right?
29:10So let's first work on the sum of
29:12squares
29:14between.
29:19Sum of squares between has a formula
29:21that
29:22looks a little crazy, but it's not.
29:24It is the sum of
29:27NJ
29:30times MJ - MG squared.
29:36Now I know what you're thinking,
29:38where the hell did the J come from?
29:41Let me show you what that is.
29:43And what did the G come from? Let me
29:45show you what that is.
29:49Let's work on first
29:50MG, which stands for
29:53uh
29:55my brain just stopped. It stands for
29:58the mean grand, the grand mean. There we
30:01go.
30:02The G stands for grand and M stands for
30:04mean.
30:05So the grand mean.
30:07And all it's doing is taking these uh
30:11all of these
30:13in the data set and getting, adding them
30:15all together and dividing them by 15.
30:21I'm going to do it with the calculator
30:22very very quickly.
30:24I'm going to enter the data into the
30:28calculator's memory.
30:30Second, reset,
30:33guess,
30:36and then clear again.
30:38Second data.
30:40We want one variable
30:42and then data. So we 20 25 30 35 40 is
30:46the first.
30:4725
30:4930
30:5135
30:5240
30:5445
30:55You're seeing me go press two times
30:57because I'm moving beyond
31:00the calculator asking is there going to
31:02be
31:03more than 125. So since there's a
31:05frequency of one, I'm just hitting it
31:09to move beyond it.
31:10So I did the first group
31:12I have all the way up to five. So now
31:13I'm ready for the second group. 20
31:1825
31:1930
31:2135
31:2340 Ooh.
31:25And see I messed that out. I only need
31:27one of those.
31:30And 40.
31:31Let me go back and check my data.
31:34Press data. Go back in. 25 30 35 40 45.
31:39That's the first group.
31:4220 25
31:44Ah, I need one of them.
31:47Let me go back.
31:49That's what happened.
31:51So
31:52I'm going to stop stop and start over.
31:55Reset.
31:57I promised you you would see errors.
31:59Here you are.
32:01Second data.
32:03One variable. Let me slow down.
32:05Data.
32:0720 5
32:1230
32:1435
32:1740
32:1845
32:20Okay, now I'm ready for the second. 20
32:2311 5
32:2530
32:2735
32:2940
32:31Ready for the third.
32:3310
32:3515
32:3720
32:3925
32:41and 30.
32:44All right, check the data.
32:4725
32:4830
32:4935 40
32:5145. That's the first I checked to make
32:54sure there was a one frequency in
32:55between.
32:5720
32:5825
33:0030
33:0135
33:0340 right?
33:0510
33:0715 20
33:0925 30
33:13There we go. It was 15.
33:15All right, so now I hit stat variable.
33:19It tells me that the n size is 15 which
33:22confirms.
33:23And I move over to find out the x bar
33:25which is the mean.
33:27So the grand mean is
33:3128.
33:3433.
33:36Let's stop with the second decimal as
33:39we've done so far.
33:42So that's the grand mean.
33:46So
33:48n
33:49j means do group one. So
33:54group one n and the group one mean
33:57and then the grand mean.
34:00Group two n
34:02mean
34:04for group two and then the grand mean.
34:07The n for group three
34:09the m for group three and the grand
34:12mean. So here's what it actually looks
34:15like. This j just means stay in
34:20every group. So it's the sum of
34:23five which is the first n, right? N one
34:27is five.
34:28Mean the first mean
34:31is 35.
34:3335
34:34minus the grand mean
34:3628
34:3833. When we subtract that, we're going
34:41to square it.
34:44That's
34:45group one.
34:46Now we're going to do group two.
34:48So, the n is again for group two is also
34:51five.
34:52But the mean for group two is 30.
34:5630
34:57The grand mean is still 28 33 squared.
35:01Then we're going to do it for group
35:02three. The n is still five.
35:06But the mean for group three is 20.
35:1020 - 28 33 squared.
35:15Now, all that's happening is we're
35:17staying n1 m1 n2 m2 n3 m3.
35:25That's all that's happening.
35:26We're getting the sum of all that. So
35:28we're going to subtract this 35 subtract
35:3128 33 from 35 whenever we get from for
35:35that result
35:37for for that difference, we're going to
35:38square it and then we're going to
35:39multiply it by five.
35:41Do that same thing again. Subtract 28 33
35:43from 30. Whatever that difference is,
35:46square it, then multiply that by five.
35:48Do the same thing.
35:50Subtract 28 33 from 20, which is going
35:52to be a negative number.
35:54Then square it, which is going to get
35:55rid of the negative number and then
35:57multiply it by five.
35:59So I could do that in the calculator
36:02very, very quickly by entering in into
36:05the data. Let me show you that. Let's
36:07clear the data again. Second reset.
36:11Yes.
36:13Clear.
36:14We're going to enter three calculations
36:16and the calculator is going to resolve
36:19this for us.
36:22But let's do it by ourselves so we can
36:23see.
36:2535 -
36:2728
36:30.33
36:32is 6.67
36:37squared
36:41times five
36:45is two two, two.
36:49Oh, it's here.
36:51Six.
36:54Let me go back.
37:0035 minus 28
37:0433
37:066.67
37:086.67
37:10We want to square that
37:12and we want to multiply it by five.
37:15Right?
37:16Square it
37:18and then multiply it by five
37:21equals two,
37:23two, two,
37:25four,
37:26four.
37:28Right? Now watch when I enter it into
37:30the calculator just like that, I could
37:32enter five
37:34parentheses
37:3735 minus 28.33
37:40square.
37:42I just entered that whole calculation
37:47and I should get 222
37:49.44.
37:51I did.
37:52And now since I want to add all of this
37:54up, I'm going to add each one of these
37:57into the calculator just like I just did
38:00for the
38:03data values
38:04and have the calculator give us the
38:06whole total.
38:08Again, make sure that the
38:11memory is cleared.
38:14Second data Well, one variable data.
38:18So this is going to be X1, this is going
38:21to be X2, this is going to be X3. So all
38:24I do now is
38:28five parentheses 35 minus 28 .33
38:33squared.
38:35Move down. I only want one of those.
38:38Now do the same thing for X2. Five
38:40parentheses 30
38:42minus 28 33 squared.
38:45One those.
38:48Five.
38:51Parentheses 20 - 28 33 squared.
38:58Now, if I go back
39:00see what it has for X 1.
39:05This is what the answer is for X 2.
39:08This is the answer for X 3.
39:10If I press stat variable, I can get the
39:12sum of that. We can see that the N is
39:15three cuz there are the three. We don't
39:17need the mean. We actually need to keep
39:20going
39:21to get the sum.
39:23That is the sum of X.
39:25So the sum of squares between is 583
39:3233.
39:34So if we go back
39:36and enter that as number one, 583
39:4133.
39:43So we have our sum of squares
39:45between.
39:47Now the formula for sum of squares
39:50within
39:52is
39:54just as simple. It's just there's much
39:56more of it.
39:58The formula is
40:00the sum of X IJ
40:03- MJ squared.
40:07So essentially what this means is every
40:09X within its set
40:12- the mean of that set squared.
40:18So
40:21if you look at it, it's tedious, but
40:24it's super easy.
40:26It looks like this.
40:28Group one.
40:32X I, this is X 1, X 2, X 3, X 4, X 5
40:37within group one. This is X 1, X 2, X 3,
40:40X 4, X 5 within group two.
40:431 2 3 4 5 within group three. And these
40:46are the means for each of those groups.
40:49So it it goes like this.
40:5125 which is X 1 within group 1 minus the
40:56group the mean for group 1 which is 35
41:00squared plus
41:02X 2 which is 30 within group 1 30 minus
41:0535
41:07squared plus
41:0935 which is X 3 minus 35
41:13squared plus
41:1540 minus 35
41:17squared plus
41:1945 minus 35
41:22squared plus
41:24Now you can see I've just done 25 minus
41:2635 30 minus
41:2925 minus 35 plus squared plus 30 minus
41:3235
41:34squared plus 35 minus 35 we're just
41:36doing that again and again and again.
41:40So for group 2
41:42you can see that it starts with 20 so 20
41:46minus 30 squared plus 25 minus 30
41:49squared and we would just keep going
41:51like that.
41:53So 20
41:55minus 30
41:57squared plus
41:5925 minus 30
42:02squared plus
42:0530 minus 30
42:07squared plus
42:0935 minus 30
42:11squared plus
42:1240
42:13minus 30
42:15squared
42:16plus
42:18and in 3
42:21that would be
42:2310 minus 20 squared plus
42:2810 minus 20 squared plus
42:3215 minus 20
42:34squared plus
42:3620 minus 20
42:38squared plus
42:3925 minus 20
42:42squared plus
42:4330 minus 20
42:45squared.
42:48Now
42:49even though that looks gnarly it is
42:51super simple. Because
42:5425 minus 35 is 10.
42:5810 squared is 100.
43:0230 minus 35 is 5.
43:065 squared is 25.
43:0935 minus 35 is 0. 0 squared is 0.
43:1440 minus 35 is
43:17uh
43:185.
43:225 squared
43:2325.
43:2545 minus
43:2635 is 10.
43:2810 squared is 100. Look, 100, 200, 225,
43:34250.
43:35So this first line is 250.
43:39Super easy, right? Look at this.
43:4110 minus 20 minus 30
43:44-10. Actually, this should have been
43:46negative, but it doesn't really matter.
43:4825 minus 35 is -10, but -10 squared is
43:52still 100.
43:5420 minus 30 is -10.
43:58-10 squared is 100. So it's really the
44:00same pattern. 25 minus 30
44:04is -5.
44:065 -5 squared is 25. So I gave you a
44:09nice, neat
44:11pattern which you can easily see.
44:19Only because this data set fell the way
44:22that it fell, or I designed it this way
44:24so that you could see the calculation is
44:27very easy.
44:29Um it wouldn't be as straightforward
44:31with different values, but it's still
44:34the same process. You take x minus the
44:38mean of that uh that group and then
44:40square it. So 250 + 250 + 250 3 250s is
44:477
44:4850. So the sum of squares
44:51within
44:53is 750.
44:55Go back to our ANOVA.
44:58750.
45:00So now the total,
45:02sum of squares total is sum of squares
45:04between plus sum of squares within.
45:07That's the total.
45:10583.33
45:13+ 750
45:171,333.33.
45:25See?
45:26Not that hard.
45:28Now
45:29we need the mean square uh between and
45:32the mean square
45:33uh
45:34within.
45:36So
45:40to get the mean square between, it's the
45:43sum of square between divided by the
45:46degrees of freedom between. So you just
45:48divide these.
45:51583.33
45:53divided by 2
45:55equals
45:57291.66
46:005 so 67 291.67.
46:05We do the same thing for the mean square
46:07within. It is the sum of square within
46:11divided by degrees of freedom within or
46:14750 divided by 12
46:1962
46:21.5.
46:24And now we need the F statistic. We just
46:27divide the mean square between from the
46:31mean square within or 291.67
46:35divided by 62.5.
46:37And if this number is greater
46:40than uh our F critical of 3.88 we're
46:44going to reject the null hypothesis.
46:49291.67
46:51divided by 62.5
46:544.666
46:56or 4.67.
46:59Look at that.
47:004.67 is definitely in the rejection zone
47:03because it's greater than 3.88.
47:07So we reject the null hypothesis.
47:10At least one parent's group strategy is
47:15different.
47:16Creates a different uh sleep time for
47:18children.
47:20We don't know which one.
47:22At least one group is different, but we
47:23can't tell which one.
47:26And just like we had uh a little bit of
47:29a different process for ANOVA for
47:32calculating the test statistic, we have
47:34a different strategy for calculating the
47:36effect size.
47:38It's called eta
47:40squared.
47:43The eta squared's formula is very
47:46simple. It's the sum of squares between
47:49divided by the sum of squares total.
47:52We know what those values are.
47:55The sum of squares between is 583.33.
47:59The sum of squares total is 133
48:011,333.33.
48:06So 583
48:09.33
48:11divided by 1,333
48:14.33
48:22583.33
48:24divided by 1,333
48:26.33
48:30We get
48:31.437
48:34so 0.4
48:374
48:38which means
48:3944%
48:41of the variation within the data set
48:44is being determined by
48:49which group you're in.
48:51And the ANOVA
48:55or eta squared just like Cohen's d, we
48:58use 0.2,
49:010.5,
49:020.8 for small, moderate, and large. We
49:06use a slightly different scale for
49:11at a square.
49:13Just want to check that before I tell
49:15you.
49:19Uh my brain remembers that it's 0.01,
49:230.09,
49:25point 25.
49:28Let's go check though.
49:31It's a lot of stuff in my brain.
49:55I need to add a squared.
50:02I'm still in chapter 10.
50:06Got to let that go.
50:08Let's go to chapter 11
50:11for anovas.
50:13There we are.
50:20While we're here, let me just stop and
50:22show you again the sum of squares
50:24between formula
50:28and the sum of squares within formula.
50:33Sum of squares total. This is really the
50:36same thing. You don't have to compute it
50:39this way. You can just add sum of
50:40squares between the sum of squares
50:42uh within, but here's the formula for it
50:45as well. This is a more simple formula.
50:48You could perform the calculation, but
50:50since you already have the sum of
50:51squares between and the sum of squares
50:53within, just add them together.
50:55Here is what you just saw me work out in
50:58the anova table.
51:00Sum of squares between, sum of squares
51:02within, sum of squares total.
51:04The degrees of freedom for between was k
51:06- 1.
51:08Within was n - k or 2 over 12 and this
51:12is just these two added up together. So
51:142 + 13 is 14.
51:20And uh
51:22we went to the F table to get 2 over 12
51:26for the critical value which was 3.88.
51:30Here we see that the mean square is just
51:32sum of squares between divided by
51:34degrees of freedom between.
51:36Right? So that's how we got that so far.
51:40I was trying to find
51:42eta squared.
51:49There we are.
51:52I was right.
51:54.01 is small.
51:57.09 is medium and .25 is large. We got
52:02.44 which is large. But it's also
52:05telling us how much is being driven by
52:10how much variation is being driven by
52:12the
52:13belonging to a particular group. And
52:15having .44 means that 44%
52:20of the variation in the sample is being
52:23driven by whether you are in group one,
52:27group two, or group three, which is a
52:28large effect.
52:31So all we have to do now is
52:33write out the decision, right?
52:35Let's head back over to the map.
52:39Oh, one more thing. I actually want to
52:40calculate the standard deviation for
52:44each one. But I want to show you
52:45something that I think may clear up
52:50standard deviation.
52:52You notice that this data set is pretty
52:55uniform, meaning it varies by five.
53:00And they vary this way by five. So they
53:04vary by five this way within.
53:07But they vary by uh there's a
53:09difference. This was varying by five,
53:11but this one's varying by ten. So, you
53:13can perhaps guess that this group is
53:17different than the others, but we can't
53:19know that without a post hoc test.
53:22We need the standard deviation for each
53:24group. And I think you're going to see
53:26something interesting because this is a
53:28unified or uniform data set about the
53:31standard deviation.
53:33We'll use the calculator again to uh
53:37calculate the standard deviation of
53:39data set one or N one. Let's clear
53:43the data.
53:44Second and reset.
53:46Yes.
53:47Clear.
53:48Second and data.
53:50One
53:52and one variable and data. So, it's
53:54ready for X one. Let's do
53:58uh group one. 25 30 35 40 45.
54:0225
54:0430
54:0635
54:0840
54:1045 just so that you can see.
54:12Stat variable will give us the data.
54:15So, the N size is five. We know that.
54:19The mean or X bar is 35.
54:23We know that.
54:25The standard deviation is 7.905
54:28or 7.91.
54:357.91.
54:38And would you guess that that would also
54:41be the standard deviation for group two?
54:45Let's see.
54:47Let's clear the data.
54:57Let's do 20 25 30 35 40.
55:0020
55:0225
55:0430
55:0545
55:0740
55:10N is five.
55:12The mean is thirty.
55:15Look at the standard deviation.
55:177.91
55:20Now, why is that?
55:23Because it doesn't matter
55:27that the means are different because
55:30once we take the mean out,
55:33they're all varying by five.
55:36That doesn't happen in real life and
55:39real data, but it helps you see that
55:42what we're talking about when we're
55:43talking about the standard deviation
55:45because that came up a lot
55:47in the radio, the call-in radio show,
55:50and it comes up
55:52I as I read through your comments in
55:54your group decision, but your group
55:56discussion,
55:57we can see that because there is uniform
56:01variation, the standard deviation for
56:04each of these groups, even though the
56:05means are different, are all going to be
56:077.91.
56:12We need that when we're reporting the
56:14decision.
56:16So,
56:18four.
56:19Based on
56:22the collected data
56:28from
56:30parents,
56:32we want to tell
56:33who's reading, anyone who's reading
56:34this, what the N size was. Capital N was
56:3815. That tells
56:40the reader that our entire data set was
56:4215.
56:45Parents using
56:49uh
56:54child ready
56:57for bed,
57:01that mean was
57:0435.
57:06The standard deviation Oh, here.
57:10N one is 5. We need to know that.
57:14M1
57:15is 35
57:18and standard deviation 1 is 7.91.
57:23That's the first one. Ciao. Ready for
57:25bed? Then child negotiates.
57:33N2 is 5.
57:36M2 is
57:3930
57:41and standard deviation 2 was
57:44you guessed it 7.91.
57:46And
57:49strict bedtime.
57:55N3 is 5. M3 was 20
58:00and SD 3 was
58:037.91.
58:05There is
58:08sufficient
58:11evidence
58:14to reject
58:16the null
58:18hypothesis.
58:23And ANOVA
58:25revealed
58:28that
58:29effect
58:32was
58:34large.
58:35And then report the F statistic. F
58:392
58:40and
58:4212.
58:43Those are our degrees of freedom, 2 and
58:4512
58:48equals
58:494.67.
58:51That was our F obtained. And because we
58:54rejected the null hypothesis, the
58:56probability is less than 0.05 because we
58:59are using an alpha of 0.05, which is the
59:02only kind of alpha for an ANOVA that we
59:04can do by hand.
59:07And the eta squared is 0.44.
59:11This
59:15that
59:16approximately
59:2144%
59:23of the variation
59:28is
59:29um
59:31driven
59:33by
59:35the
59:36group
59:39parent style.
59:43Meaning either
59:45child ready, child negotiation, or child
59:48strict bedtime.
59:51Okay.
59:53Take a breath.
59:55I can't believe I did all that. I made
59:57some mistakes. I did that all in one
59:59take. I'm not going to
1:00:00edit anything out.
1:00:03Let's just take a quick review of what
1:00:04we've done. We started out with our
1:00:07question.
1:00:09Does the bedtime strategy parents use
1:00:11relate to how long it takes their
1:00:12children to go to bed? That's the
1:00:14question.
1:00:16We started out with a question
1:00:17and we then started out with our
1:00:20hypotheses if there is no difference in
1:00:23children's sleep time based on parent
1:00:25strategy because we realized that we
1:00:27were going to examine three different
1:00:30parental strategies.
1:00:31Parent allows the child to go to bed
1:00:33when they want to
1:00:35or the child negotiates or the child
1:00:38goes to bed at a strict bedtime.
1:00:42So in the real world, we would go out
1:00:44and recruit
1:00:46data. We would recruit participants. We
1:00:48would collect data from people who have
1:00:51the information because we have the
1:00:52question.
1:00:54You saw the recruitment flyer. We were
1:00:56looking for five in each group. We
1:00:58weren't asking parents to change their
1:01:00strategy. We're just telling asking them
1:01:02to agree if they use one of those three
1:01:06strategies.
1:01:07And we got
1:01:09five parents in each strategy.
1:01:12And
1:01:14we continue with our well, our null
1:01:16hypothesis again is that there's no
1:01:18difference in children's sleep time
1:01:19based on parent strategy, mu 1 equals mu
1:01:222 equals mu 3 which is there is no
1:01:24difference in the population.
1:01:27Our alternative hypothesis is at least
1:01:29one parent's strategy is different,
1:01:31which is all we can do. We don't even
1:01:33have a mathematical way to express that.
1:01:37We got our data from each of the groups,
1:01:4125 30 35 40 45 and that follows through
1:01:45for child ready child negotiate no,
1:01:48child negotiate
1:01:49negotiated strategy or negotiated
1:01:52bedtime and child strict bedtime.
1:01:55We used that to create the mean for each
1:01:59group, 35 30 and 20 which we could see
1:02:02easily. We could have added those up,
1:02:04divided them by five, but we would have
1:02:05gotten the same thing.
1:02:08We acknowledge
1:02:10our data by putting the
1:02:13all the stuff that we need in these
1:02:15on the map. N 1 equals N 2 equals N 3
1:02:18which happens to be the way it is for
1:02:20this data set, but they don't all all
1:02:23have to be the same data just like an
1:02:26independent sample t test.
1:02:28So we say N 1 equals N 2 equals N 3 and
1:02:31they all equal five.
1:02:33Capital N is the whole data set of 15
1:02:38and K stands for how many groups we
1:02:40have.
1:02:42We used
1:02:44the all of the data in the 15 the 15
1:02:49data values in the data set to create
1:02:51the grand mean of 28.33.
1:02:56We also went back and figured out the
1:02:58standard deviation and because they all
1:03:01vary by five, we got the that
1:03:04they vary around the mean by 7.91
1:03:09minutes.
1:03:12Then we got introduced to the ANOVA
1:03:14table which is like the sum of squares
1:03:16table. We're going to use a table like
1:03:17that again in
1:03:19uh chapter 12 for correlations
1:03:22next week and chapter 13 for linear
1:03:25squares
1:03:27or linear regression, sorry.
1:03:30So we need force sum of squares, degrees
1:03:32of freedom, mean square, and the F
1:03:35statistic.
1:03:36Between variation, between group
1:03:38variation, within group variation, and
1:03:40total is the source.
1:03:43We found out what our degrees of freedom
1:03:45were by using the critical value
1:03:48formula, degrees of freedom between over
1:03:50degrees of freedom within. That formula
1:03:52is K - 1 over N - K.
1:03:56K is three, three groups, 3 - 1 is two.
1:04:00N - K, there are 15 in the data set,
1:04:02there are three group, 15 - 3 is 12. We
1:04:07went to the F
1:04:09table to find 2 over 12 and we got 3.88
1:04:12as our critical value. We drew out the
1:04:14curve, but you don't really need to draw
1:04:16it out.
1:04:19And we placed that in our ANOVA table
1:04:22two
1:04:23for between group uh between degrees of
1:04:26degrees of freedom between and for 12
1:04:28for degrees of freedom within, and 2 +
1:04:3112 is 14.
1:04:33We then had to figure out the sum of
1:04:35squares between, the sum of squares
1:04:37within, and then we added that to find
1:04:39the sum of squares total.
1:04:42How we figured out the sum of squares
1:04:43between, it's the sum of NJ times MJ -
1:04:48grand mean squared.
1:04:51So we just plugged in
1:04:52five, five, five because those are uh
1:04:55this is N1, this is N2, this is N3. They
1:04:58happen to all be the same number, but
1:05:00they could be different numbers.
1:05:03So uh N1 could have five, N2 could have
1:05:0712, and N3 could have 18. So this would
1:05:10be 5, 12, and 18 instead of 5, 5, and 5.
1:05:14Then we put in the mean for each group
1:05:18and we subtracted the grand mean and
1:05:20squared it.
1:05:22I showed you how we just simply
1:05:24subtracted the grand mean from the group
1:05:27mean, squared it, and then multiplied it
1:05:30by n, which happened to be five.
1:05:32When we added that all up, the sum of
1:05:34squares between was
1:05:3658.
1:05:37583.33.
1:05:39We plugged it in there.
1:05:41And then we figured out the sum of
1:05:43squares within.
1:05:45That formula is the sum of X IJ - MJ,
1:05:50which is really just every X - the
1:05:53individual mean for that group squared.
1:05:57When we take this, you got to see that
1:05:59this is a pretty simple formula. It's
1:06:01just tedious.
1:06:0325 - 35 is -10. -10 squared is 100.
1:06:0930 - 35 is -5 squared. -5 squared is 25.
1:06:1635 - 35 is 0. 0 squared is 0. 40 - 35
1:06:235. 5 squared 25.
1:06:2545 - 35 10. 10 squared is 100. You saw
1:06:29that pattern. So it was just 100
1:06:32100
1:06:33100 25 25 or 100 or 220 250 for each
1:06:38line. When we add them all up, the sum
1:06:40of squares within was 750.
1:06:44We put that in our sum of squares
1:06:46within. We added 583.33
1:06:51to 750 and we got 1,333.33
1:06:54as the sum of squares total.
1:06:58We already put the degrees of freedom
1:06:59between and within.
1:07:01And we used them to figure out the mean
1:07:03square between and within. So mean
1:07:06square between is sum of squares between
1:07:09divided by degrees of freedom between or
1:07:11583.33
1:07:13divided by 2, which is 291.67.
1:07:17The mean square within is sum of squares
1:07:20within divided by degrees of freedom
1:07:22within or 750 divided by 12 which is
1:07:2562.5.
1:07:27F statistic is mean square between
1:07:30divided by mean square within which is
1:07:324.67.
1:07:34That is beyond our F critical. We reject
1:07:39the null hypothesis. They are not the
1:07:41same. At least one parent's strategy is
1:07:44different, but we can't tell which one
1:07:46it is without a post hoc test.
1:07:49So then we figure out the effect size
1:07:52using eta squared. This is eta squared.
1:07:57This is the eta um character in the
1:07:59Greek alphabet. It looks like a kind of
1:08:01a cursive N with a little flip up.
1:08:04Eta squared. That formula is sum of
1:08:06squares between divided by sum of
1:08:08squares total
1:08:10or
1:08:11583.33
1:08:12divided by 1,333.33
1:08:16which is what we did here. We got 0.44
1:08:19which means 44% of the variation that
1:08:22exist in the data set is being driven by
1:08:25whether you belong in one of these
1:08:27groups
1:08:29which is a strong
1:08:32or large effect.
1:08:340.01
1:08:350.09 and 0.25 is small, moderate, and
1:08:40large.
1:08:41And then finally we write out the
1:08:45decision. Based on the collected data
1:08:47from parents N equals 15 using child
1:08:50ready for bed
1:08:52N 1 is 5 M 1 is 35
1:08:56standard standard deviation 1 is 7.91.
1:08:59You know that's going to be 7.91 for
1:09:00every group.
1:09:02Child negotiates N 2 equals 5 M 2 equals
1:09:0530. So they're different, right?
1:09:07Standard deviation 2 is 7.91 and strict
1:09:10bedtime N 3 equals 5 M 3 equals 20 set
1:09:14standard deviation 3 is 7.91.
1:09:17There is sufficient evidence to reject
1:09:19the null hypothesis.
1:09:21A null an ANOVA sorry and revealed that
1:09:24the effect was large.
1:09:26Report the F statistic F 2 comma 12
1:09:30which are are are
1:09:32degrees of freedom.
1:09:34And because we rejected the null
1:09:35hypothesis, the probability is less than
1:09:380.05. If we fail to reject the null
1:09:40hypothesis it it
1:09:42fail to reject the null hypothesis it
1:09:44would be probability is greater than
1:09:460.05.
1:09:47And the eta square is 0.44 0.44. This
1:09:51means that approximately 44% of the
1:09:53variation is driven by the group parent
1:09:56style.
1:09:57All right.
1:10:00Now
1:10:02that might have seemed like a lot.
1:10:05I'm going to give you extra credit
1:10:07points
1:10:09or extra assignment points cuz I don't
1:10:10give extra credit. Extra assignment
1:10:12points for doing one of for doing this
1:10:16data set exactly the way I've written it
1:10:19out.
1:10:20I want to continue to stress that if you
1:10:22see me write all of this out
1:10:25you write all of it out.
1:10:27I want to see everything so that I can
1:10:29follow what you're doing so that you can
1:10:32get full credit. You won't get full
1:10:35credit if I don't see
1:10:38um
1:10:39you writing out the full strategy. And
1:10:43for this problem
1:10:45I want to see the standard deviation for
1:10:48the data set.
1:10:507.9 I want to I want to see that on your
1:10:52calculator.
1:10:54And
1:10:56I want you to write out the full
1:10:58decision. Everything that I write is
1:11:00what I want you to write.
1:11:04And there you have it.
1:11:06We just computed our first ANOVA.
1:11:09And again, remember the Sesame Street
1:11:10question. One of these things is not
1:11:13like the other.
1:11:15Well, our ANOVA
1:11:17lets us or tells us at least one of
1:11:20these population means is different. But
1:11:23we still don't again know exactly which
1:11:25one. That's a question for for analysis.
1:11:27Here's what I want you to remember.
1:11:29Statistics doesn't begin with a formula.
1:11:33Statistics begins with a question,
1:11:35curiosity.
1:11:37The formula is simply tools
1:11:40to help us
1:11:41investigate the things that we're
1:11:43curious about.
1:11:44And ANOVA gives us a way to find a
1:11:47signal in all that noise.
1:11:51That's chapter 11. I'm looking forward
1:11:53to our
1:11:57live lecture this week.
1:12:00We're going to explore this a little bit
1:12:02further and I'll be able to answer any
1:12:04questions and sort of set you up for
1:12:06chapter 11 exercises. Having said that,
1:12:09as always,
1:12:10I'm super excited to see
1:12:13all the statistics you find this week.
1:12:16And can you believe we are only just a
1:12:18few chapters left?
1:12:20Chapter 12, chapter 13, chapter 14, and
1:12:23then
1:12:24we have our final exam. So our time
1:12:26together is coming to an end. And I hope
1:12:29what you can say is that
1:12:32even though you walked into this course
1:12:33with
1:12:35a pretty severe
1:12:37math wound,
1:12:40that
1:12:42you practiced,
1:12:44you focused,
1:12:46and you
1:12:47made a big, significant, maybe even
1:12:50statistically significant
1:12:53dent in reducing that anxiety, that math
1:12:56wound, and finding that you are,
1:13:00I don't know,
1:13:03a rock star.
1:13:04I'll see you this week.

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