YT YouTube2Text

YouTube2Text → Transcripts → YouTube transcript (Qa2APhWjQPc)

YouTube transcript (Qa2APhWjQPc) — Transcript

4,595 words · 634 segments · language en · Watch on YouTube

Open in transcriber Summarize with AI Download SRT/VTT

Full transcript

0:02[Music]
0:17hello thanks for watching and welcome to
0:20the next video in my series on basic
0:22statistics now as usual a few things
0:25before we get started number one if
0:27you're watching this video because you
0:28are struggling in a class right now I
0:31want you to stay positive and keep your
0:33head up if you're watching this it means
0:35you've accomplished quite a bit already
0:37you're very smart and talented but you
0:39may have just hit a temporary rough
0:41patch now I know with the right amount
0:43of hard work practice and patience you
0:46can work through it I have faith in you
0:49many other people around you have faith
0:51in you so so should you number two
0:55please feel free to follow me here on
0:57YouTube on Twitter on Google+ or on
1:01LinkedIn that way when I upload a new
1:03video you know about it and it's always
1:06nice to connect with my viewers online I
1:08feel that life is much too short and the
1:10world is much too large for us to miss
1:12the chance to connect when we can number
1:16three if you like the video please give
1:18it a thumbs up share it with classmates
1:21or colleagues or put it on a playlist
1:23that does encourage me to keep making
1:25them for you on the flip side if you
1:27think there's something I can do better
1:29please leave a con instructive comment
1:30below the video and I will take those
1:32ideas into account when I make new ones
1:36and finally just keep in mind that these
1:37videos are meant for individuals who are
1:40relatively new to Stats so I'm just
1:42going over basic concepts and I will be
1:45doing so in a slow deliberate manner not
1:49only do I want you to know what is going
1:51on but also why and how to apply it so
1:56all that being said let's go ahead and
1:58get started
2:02so this video is the next in our series
2:03about simple linear regression in our
2:06last two videos we talked about the very
2:08basics of regression and introduced
2:10other basic concepts like the algebra of
2:12lines and general patterns to look for
2:14on a scatter plot in this video we're
2:17going to learn about the fundamental
2:19Concept in linear regression the least
2:21squares method we will talk about how
2:24the least squares method relates to
2:26previous Concepts we have learned and
2:28then we will actually use the meth
2:29method to calculate the least squares
2:32line or the regression line this video
2:35will involve formulas and simple
2:38calculations while you may not have to
2:39find a simple regression line by hand
2:41very often it's not that difficult to do
2:44and this video will at least show you
2:46how it's done so you understand the
2:48underlying mechanics so if you are new
2:51to regression or are still trying to
2:53figure out exactly what it even is this
2:55video is for you so sit back relax and
2:59let's go ahead and get to
3:03work so in this video we will continue
3:06to use the previous problem we have used
3:08in other videos so I'll review it very
3:11quickly let's assume that you're a small
3:13restaurant owner or a very
3:15business-minded server or waiter at a
3:17nice restaurant here in the US and
3:20elsewhere tips are a very important part
3:23of a waiters pay most of the time the
3:26dollar amount of the tip is related to
3:29the amount of the total bill so there's
3:33dependency there as the waiter or owner
3:36you would like to develop a model that
3:38will allow you to make a prediction
3:40about what amount of tip to expect based
3:43on the bill therefore one evening you
3:46collect data for six
3:51meals now in the last video you only had
3:55the tip data but now you were able to go
3:58back and get the actual build data as
4:00well so now you are working with two
4:03variables that are matched pairs so you
4:06can see we have six meals over here on
4:08the right so on the left column we have
4:10the total bill and then on the right we
4:13have the tip so for our first meal or
4:16first table or whatever it was the total
4:18bill was
4:18$34 and the tip that went along with it
4:21was $5 and then the next meal the total
4:24was
4:25$18 and then the tip amount was $17 and
4:28so on and so
4:30forth now you want to know to what
4:33degree can the tip amount be predicted
4:37by the bill amount so in this case the
4:40tip is the dependent variable so we're
4:44sort of making a logical claim that the
4:47tip amount is dependent on the total
4:51bill amount and it's important to set up
4:54your variables this way it would not
4:56make sense to say the total bill amount
4:59is dependent on the tip amount that's
5:02reverse so we want to say what's
5:04actually true sort of in real life that
5:06the tip amount is dependent on the total
5:09bill amount so the tip amount is the
5:11dependent variable and the bill amount
5:14is the independent
5:18variable now in a previous video we
5:20looked at that situation where we only
5:22had the tip data and what we determined
5:25in that case where we only have the one
5:27variable the tip data
5:30all we could do was use its mean as the
5:33best predicted value So based on that
5:36data we found that the mean of the tips
5:38was
5:39$10 therefore our best prediction for
5:43the seventh meal would be $10 that's the
5:47best we could do so we went ahead and
5:50plotted our points and we noticed that
5:52we have a horizontal line of the tip
5:54amount of $10 that's the black dotted
5:57line and then we put all of our tips
5:59tips in relation to that line then we
6:02found the difference or the distance
6:04between the line and the tip then we
6:07squared that difference and then we
6:09added up all those differences and what
6:12that gave us are the squared residuals
6:15or the squared error and then we edit
6:18them up so the sum of the squared errors
6:20or the sum of the squared residuals was
6:23120 now when conducting simple linear
6:25regression with two variables we will
6:28determine how good the regression line
6:30fits the data by comparing it to this
6:33type literally in this case in this
6:36problem we're going to compare our
6:37regression solution to this line where
6:41we pretend the second variable doesn't
6:43even exist so remember beta sub one is
6:46our slope a horizontal line has a slope
6:50of zero so in this case the line we're
6:52looking at here our beta sub 1 is zero
6:56but the whole idea is that when we do
6:59Reg ression we're going to compare that
7:01model to this model and hopefully it's
7:05better and we'll talk about exactly what
7:07better means as we
7:09go so what exactly is the least squares
7:12method well it depends on this general
7:14idea called the least squares Criterion
7:16and it looks like this now it's kind of
7:19an ugly expression there but I'm going
7:20to pick it apart so we can understand
7:21exactly what each thing means now let
7:24start at the left what does Min n mean
7:28well it means minimum or
7:30minimization then we have the summation
7:32symbol there in the middle now we look
7:35over further to the right now we notice
7:36that we have two values in parentheses
7:38that we are subtracting so we're going
7:40to find the difference of two something
7:43we don't know yet and then we're going
7:45to square that difference so just put it
7:47all together we're going to find the
7:49difference of two things we're going to
7:51square that difference and then we're
7:54going to add them all together that's
7:55the summation and the goal is to
7:58minimize
7:59that
8:01sum so y sub I is the actual observed
8:05value of the dependent variable in this
8:08case it's the actual tip amount that
8:10occurred in the restaurant that's why
8:12it's the observed
8:15value now y hat sub I is the estimated
8:19or the predicted value of the dependent
8:21variable so this is the predicted tip
8:23amount based on our regression model so
8:28what are we going to do here we're going
8:29to have two values for every X on the
8:32graph we're going to have the actual
8:34observed tip and then we're going to
8:36have the tip the model predicted now
8:39those are not going to usually be the
8:41same there's going to be some difference
8:42between the two so we're going to find
8:44the difference between those two things
8:46we're going to square the difference and
8:47then add up all the differences and we
8:50want that sum to be as little as
8:52possible so in plain English the goal is
8:55to minimize the sum of the squared
8:58differences
9:00between the observed value for the
9:02dependent variable so the actual tip in
9:04the restaurant and the estimated or
9:06predicted value of the dependent
9:08variable that is provided by the
9:10regression line so think about this
9:13that's so we have a bill I don't know
9:17that's
9:18$50 and the patron at the restaurant
9:21left a tip of
9:24$5 but our regression line predicted a
9:27tip amount of $7
9:30.50 so we have a difference there our
9:32observed value was $5 our predicted
9:35value was
9:37$7.50 so we find the difference between
9:39those two and then we'll square the
9:41difference and then we'll do that for
9:43every point along the regression
9:47line now not only that but the sum of
9:49the squared residuals should be much
9:52smaller than when we use just the
9:54dependent variable alone so remember in
9:56that case the slope of the line was Zero
9:59the the predicted value for every point
10:00of X was $10 cuz we only had the tip
10:03data and that sum of squared residuals
10:06or sum of squared errors was
10:08120 so when we actually find these
10:11squared residuals using the regression
10:13line it should be a lot smaller than
10:18120 so let's walk through this step by
10:21step so step one is to do a scatter plot
10:24now it seems obvious but a lot of people
10:25just go in and do the math and don't
10:27actually look at the data so do a
10:29scatter plot of your data you can look
10:30at the general pattern you can look for
10:32any outliers anything that seems odd now
10:35you also want to make sure that your
10:36graph is scaled correctly so you can see
10:38down here on the bill amount I started
10:40at $20 that's because our smallest bill
10:43was $34 so if I went all the way down to
10:47zero that wouldn't make a whole lot of
10:48sense so I went ahead and did the scale
10:50from 20 to 120 same thing for the tip
10:53amount our smallest tip was $5 so I
10:57started that axis at four
10:59so you always want to make sure that
11:01your graph is set up proportionally so
11:04the scatter plot is not
11:08distorted so step two look for a visual
11:12line for a rough visual line now does
11:16this data seem to fall along a
11:20line well yes it does so we don't know
11:24if any one of these lines here is the
11:26actual regression line but in general
11:29the data points do fall along a line now
11:32what if they don't well if the data
11:34points are all scattered all over the
11:36place if they're like a Big Blob or if
11:39you ever seen sort of a shotgun blast
11:41against a Target the shot is everywhere
11:45then there would be no linear pattern
11:47you would actually stop in your
11:49regression there's not going to be a
11:51linear pattern in a blob of data points
11:54now some people will go ahead and do it
11:56because with computers it's very easy to
11:57do anyway but really it's a waste of
12:00time and it's technically not an
12:01appropriate test or appropriate
12:03technique to use when the data are just
12:06sort of randomly all over the
12:09place now step three correlation I would
12:12consider optional but I think it's a
12:15good thing to do because the correlation
12:18coefficient is involved in other things
12:20later in regression and also in multiple
12:22regression so you might as well go ahead
12:24and do it anyway so what is the
12:26correlation coefficient for our data
12:28here
12:30well in this case it's
12:320.866 now you obviously have to know
12:35what that means so in this case is
12:38relationship strong well yes it is a
12:42correlation coefficient of 866 indicates
12:45a strong positive linear relationship so
12:50it sort of gives evidence to our
12:51conclusion earlier that in fact there is
12:53a linear relationship between data
12:56points step four descriptive statis ICS
12:59and the centroid so over here on the
13:01right we can see that we have the bill
13:03column and the tip column the first
13:05thing we want to do is find the mean of
13:07each variable so our average bill amount
13:10or a mean bill amount was
13:12$74 for the tips it was $10 now what we
13:16can do is actually graph this on the
13:19graph so we'll take our mean of the
13:21bills of
13:22$74 then we'll take the mean of the tips
13:25which was $10 and we'll actually graph a
13:28point there now this point is very
13:31important and it's called the centroid
13:33and here's why it's important the best
13:36fit or the least squares regression line
13:39will or must pass through the centroid
13:44so whatever our regression line happens
13:46to be it has to go through the centroid
13:50which is comprised of the mean of the X
13:52variable and the mean of the Y variable
13:55and remember it takes two points to make
13:58a line so the centroid automatically
14:00gives you a point to work with and
14:02that's important as we go forward but
14:05always find the mean of each variable
14:07then we can plot the centroid on the
14:08graph knowing that our regression line
14:11must go through that
14:14point so let's go ahead and walk through
14:16the calculations now remember the
14:18general model so y hat sub i = b Sub 0
14:22plus b sub1 x sub I that's a lot of
14:26variables in there but all that means is
14:28this it's comprised of two parts so B
14:31sub one is the slope now the formula to
14:34find the slope is this over here on the
14:37right now that looks very complex but
14:41it's not as you can see we have things
14:43in there like xbar well that's the mean
14:45of the X variable we have Y Bar that's
14:47the mean of the Y variable so on and so
14:50forth it's not that hard to do it just
14:52takes a few steps and we'll walk through
14:54them but that's how we find the B sub1
14:57which is the slope of our regression
15:00line so in this case xar is the mean of
15:03the independent variable in this case
15:05the bill amount Y Bar is the mean of the
15:09dependent variable which in this case
15:10are the
15:12tips now X ofi is the value of the
15:15independent variable for a point and Y
15:17subi is the value of the dependent
15:19variable for a point so we have the mean
15:22of the independent variable we have the
15:24mean of the dependent variable and then
15:26X subi and Y subi are simp a pair of tip
15:31and meal
15:33data now The Intercept is the other
15:36component here so it's B subz now to
15:39find B subz we just take the Y Bar which
15:42is the mean of the dependent variable
15:45and then subtract the slope times the
15:47mean of the independent variable so we
15:49have to find B sub1 first because we
15:52will use that to find The Intercept so
15:55these are very simple calculations if
15:57you set them up in a table and Mak you
15:59walk through them step by step but there
16:01is no magic here the four things we need
16:04are things we already know the mean of
16:06both variables and then a point so in
16:10this case a dollar and a tip
16:14amount let's talk about how to calculate
16:16these so here's our B sub1 remember this
16:18is the slope of our regression line so
16:20to find the numerator we do these things
16:22for each data point we take the x value
16:25and subtract the mean of the X variable
16:28or the independent variable we take the
16:31Y value and subtract the mean of the Y
16:34variable in this case the tip amount so
16:37all we're doing is taking the difference
16:38between the mean and the actual data
16:41point for each
16:43variable and then we multiply those two
16:45things together and then we add up all
16:48the products now we're going to walk
16:49through an actual calculation so you
16:50actually see it in action but this is
16:52what we do now on the bottom for each
16:55data point we take the x value and
16:58subtract the mean of X then we Square it
17:01then add them up so you can see it's
17:03actually pretty simple just using the
17:05four things we already have the mean of
17:08each variable and then a point that's an
17:10x sub I and a y sub I now to find B subz
17:14which is The Intercept we use what we
17:16find for B sub one the slope and then we
17:19use the mean of Y and the mean of X do a
17:21simple calculation and then we have
17:25it so here is a table where we can
17:28actually walk through each step of the
17:30calculation so here on the left we have
17:32each meal 1 through six and then we have
17:34the dollar amounts for the bill and the
17:36tip so the bill of $34 had a tip of5 the
17:39bill of $18 had a tip of $17 and so on
17:42and so forth on the bottom of this
17:44columns you'll see that we have the mean
17:45of each variable so the mean of the
17:48total bills was $74 and the mean of the
17:50tips was $10 so the first thing we got
17:53to do in this next column is the bill
17:56deviation so we're going to take x sub I
17:59and then subtract the mean of
18:02X so it looks like this so let's walk
18:06through a couple of them so you can see
18:07where we actually get them so remember x
18:09sub I all that is is the x value over
18:12here in the total bill column so in the
18:14first case it's 34 then we subtract the
18:17mean of that column which is 74 so 34-
18:2374 is -40 let's go to the next one so
18:27108 - 74 is
18:3134 same thing 64 - 74 is -10 so on and
18:37so forth so each x value minus the mean
18:41now I'll do the same thing for the
18:44Y's so for the first one a tip amount of
18:47$5 minus the mean of $10 is -5 then we
18:52have 17 - 10 which is 7 11 - 10 which is
18:561 8 - 10 10 which is -2 and so on and so
19:01forth so each y value minus its mean now
19:05look at the next column what we going to
19:07do well we're going to multiply those
19:09two things together it's just the
19:11product of those two so -40 * -5 is 200
19:1734 * 7 is
19:20238 -10 * 1 is10 and so on and so forth
19:25now we need to find the sum of those
19:27things so we're going to add them all up
19:29and then when we do we have a value of
19:33615 now the next column what do we do
19:36we're going to squared what we found in
19:38the bill deviation
19:40column so -40 squared
19:44is600 34 squared is
19:481,156 -10 sared is 100 and so on and so
19:53forth then we're going to add all those
19:55up and that's 4,26
19:59very very simple if you just followed
20:02along step by
20:06step let's go ahead and calculate the
20:09slope of our regression line so remember
20:11here it is so B sub1 is equal to this
20:15fraction here that involves things we
20:16just found in the previous slide now
20:19I've colorcoded everything so you can
20:21see how everything relates to the actual
20:23formula so in the numerator you can see
20:26that the terms in that numerator are the
20:28same thing as the terms in the deviation
20:30products column so the sum of that
20:33column will be our numerator now in the
20:36denominator we can see that that term is
20:39the same thing as our bill deviations
20:41squared over here so the sum of that
20:43column will be the
20:45denominator b sub1 or the slope of a
20:48regression line is 615 /
20:534,26 and we get those from our columns
20:56over here on the right so we go ahead
20:58and do that division the slope of our
21:01regression Line is
21:0801462 now what about the Y intercept
21:11well here is our general formula now we
21:13know letter B sub one is or our slope is
21:16we just found it and here is the rest of
21:18the information we need we need Y Bar
21:20and xar which is just the mean of the
21:23bills and the mean of the tips so we go
21:25ahead and substitute everything in so
21:27the mean of the tips or the Y Bar is 10
21:31minus B sub 1 which is our slope which
21:34is up there at the top and then the mean
21:36of the X's or the mean of our total
21:38bills which is 74 so we go ahead and do
21:41all that out and we end up with an
21:44intercept of
21:470.81
21:5188 what is our regression line so here's
21:55a general formula B Sub 0 is our
21:57intercept we found that out.
22:018188 our slope is
22:0401462 now all we have to do is assemble
22:07it we got to put it all together and it
22:10looks like this or this so y hat subi
22:16equal
22:180.818 that's our intercept plus
22:220.146 2 x so that's our slope B sub1 up
22:27there at the top now now we could
22:28rearrange it and put the slope first so
22:3301462 x minus 0. 8188 for The Intercept
22:39so it doesn't matter how you rearrange
22:40it some software packages will display
22:43it one way some will display it the
22:45other way it really does not matter as
22:47long as you know what each section or
22:49each term in there
22:53means now I went ahead and did this in
22:56Microsoft Excel and here's what it gave
22:59us it gave us a y =
23:0301462 xus
23:070.823 now what was our
23:11calculation
23:1201462
23:14xus 8188 now notice there's a little bit
23:18difference due to rounding in the
23:20intercept and that's okay now I will say
23:23this that regression is very sensitive
23:25to rounding so it's always best practice
23:28to take your calculations out to four
23:30decimal places but here we have a slight
23:32difference due to rounding but our
23:35manual calculation that we just did is
23:37the exact same that Excel came up with
23:40there at the
23:41top so our slope is
23:4501462 and our intercept is. 8203 in
23:50excel's case or. 8188 in our hand
23:53calculation case which is close enough
23:55for
23:56me now look at our centroid remember I
23:59said our centroid has to fall on the
24:02regression line so in this case our
24:04centroid was 74 10 so a bill amount of
24:0874 and a tip amount of $10 well guess
24:11what it does so that disproves the point
24:14that our regression line is accurate it
24:16goes to our centroid and her hand
24:18calculation matched excel's
24:22calculation so how do we actually
24:24interpret our regression line so here it
24:27is so why hat subi and again remember
24:29that's predicted the Y hat means this is
24:32how we find predicted values 0.146 2x
24:36minus
24:380.818 what does that actually
24:41mean well here's what it means for every
24:45$1 the bill amount which is our X
24:48increases for every dollar the bill
24:51amount increases we would expect the tip
24:54amount to also increase by
25:0001462 or about
25:0315 so for every dollar the bill amount
25:06increases we expect or predict the tip
25:09amount to increase by 15 cents
25:14approximately now what does The
25:15Intercept mean if the bill amount is $0
25:19then the expected or predicted tip
25:21amount is
25:2382 well does that make any sense no and
25:27here's another important important thing
25:29The Intercept may or may not have any
25:32real meaning in real life so it may or
25:35may not make sense in this case it
25:37doesn't make any sense so it has to be
25:40part of our prediction equation or a
25:42regression equation but it does not
25:44really make sense in real life sometimes
25:47it does sometimes it doesn't it just
25:49depends on the problem but the important
25:51thing to get out of this slide is how
25:54the dependent variable changes in
25:57relation to one unit change in the
26:00independent variable so for every $1 the
26:02bill amount increases we expect the tip
26:04amount to increase by about 15 cents or
26:0901462 and that's because it's
26:14positive
26:16but is this regression line model any
26:20good well we don't know that yet and
26:22that will be the topic of our next
26:27video okay so we've reviewed sort of the
26:29heart or the core of simple linear
26:31regression which is the least squares
26:33method so we talked about how to put our
26:35data on a graph make sure it actually
26:38follows a linear relationship if it
26:40doesn't we should abandon the regression
26:42because it doesn't make sense to try to
26:43force a linear model on data points that
26:47are scattered all over the place we
26:49talked about how we should format our
26:51graph so it doesn't distort our data
26:53points as well we also talked about
26:56doing a correlation analysis to f figure
26:58out if a strong linear relationship
27:00exists between our two variables now
27:03again that's sort of optional but I do
27:05suggest doing it because that comes into
27:08play later on and also in multiple
27:10regression now once we have that we use
27:13the actual step-by-step method to
27:15calculate B sub1 which is the slope of a
27:18regression line and B Sub 0 which is The
27:21Intercept now when we calculated those
27:23we found out that by putting them
27:25together we generated our regression
27:28line now we can go ahead and graph our
27:31regression line and it better go through
27:33our centroid which is the mean of each
27:36variable as plotted as a distinct point
27:39on the graph but of course in the end we
27:42don't know if this regression line is
27:44any good we're going to compare it to
27:47the situation where we did not have an
27:49independent variable so remember in that
27:51case the best prediction we had for any
27:53tip was $10 so when we find the squared
27:56residuals using the regression line
27:58it had better be a lot less than
28:01120 otherwise a regression model is no
28:04better than just using the mean of the
28:06tips alone and that's the whole point
28:09we're going to compare a regression line
28:11to the situation where we're only using
28:13the mean of the dependent variable so
28:16we'll get to that in the next video
28:19[Music]

About this transcript

This page contains the full transcript of YouTube transcript (Qa2APhWjQPc) , generated from the public captions YouTube serves with the video. The transcript has 4,595 words across 634 segments, with the original timestamps preserved so you can click any line to jump to that moment in the embedded player.

What you can do with it

Use the transcript to take notes, quote the speaker, build a study guide, generate a summary with ChatGPT or Claude via the YouTube Summary tool, or export it as a timed subtitle file with YouTube to SRT. You can also re-open it in the transcriber to translate the transcript into 100+ languages.

Free YouTube transcript tool

YouTube2Text is a free YouTube transcript generator — no signup, no daily limit. Paste any YouTube link and get the full transcript instantly, with timestamps, click-to-jump, translation to 100+ languages, AI prompts for ChatGPT, Claude, and Gemini, and exports to TXT, SRT, VTT, or Markdown.