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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
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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
1:08I'll feel that life is much too short
1:10and the world is much too large for us
1:12to miss the chance to connect when we
1:14can number three if you like the video
1:17please give it a thumbs up share it with
1:20classmates or colleagues or put it on a
1:22playlist that does encourage me to keep
1:24making them for you on the flip side if
1:27you think there's something I can do
1:28better please leave a instructive
1:30comment below the video and I will take
1:32those ideas into account when I make new
1:35ones and finally just keep in mind that
1:37these videos are meant for individuals
1:39who are relatively new to Stats so I'm
1:42just going over basic concepts and I
1:45will be doing so in a slow deliberate
1:48manner not only do I want you to know
1:51what is going on but also why and how to
1:55apply it so all that being said let's go
1:58ahead and get started
2:02so this video is the next in our series
2:04about simple linear regression in our
2:06last video we talked about the very
2:08basics of regression first we discussed
2:12what residuals are and then we learned
2:14about the sum of squares of those
2:16residuals lastly we talked about how a
2:19regression model is actually a
2:21comparison to a special model where the
2:24independent variable does not even exist
2:27and we just use the mean of the
2:29dependent
2:30variable so in this video we will talk
2:33about some of the nuts and bolts of
2:34regression I will introduce some basic
2:37terminology and Concepts that will carry
2:39you through your work using regression
2:42we will talk about how regression is
2:44related to the algebra of lines and
2:47discuss General patterns to look for on
2:49Scatter Plots now there are no formulas
2:52or calculations in this video the goal
2:54is to help better prepare you for
2:56interpreting regression analysis when it
2:59is performed if you are new to
3:01regression or are still trying to figure
3:03out exactly what it even is this video
3:06is for you so sit back relax and let's
3:09go ahead and get to
3:13work so simple regression is part of a
3:16special area of Statistics called by
3:19variate statistics of course by variate
3:21means two variables now in our previous
3:25videos we talked about correlation and
3:28we also talked about a Nova now if
3:31correlation and an NOA got together and
3:34had a child a very scary thought I know
3:38that child would be simple linear
3:41regression because regression shares
3:43trades with correlation and it shares
3:46traits with Anova now the sharing of all
3:49of those traits will become more
3:51apparent as we get further into
3:53regression but for now just going to
3:55take my word for it now correlation and
3:58regression also have something else in
4:00common and that is that those data
4:03points are plotted on a scatter plot or
4:06a coordinate plane so for both
4:09correlation and regression we have
4:10variable one and we have variable two
4:13and then we plot where each variable
4:15meets and we get a data point now of
4:17course our samples will have more than
4:19one data point so we'll populate it with
4:21some more points now the relationship
4:23between variable one and variable two is
4:25special in this case dealing with
4:27regression so we can say it like this
4:30the value of one variable is a function
4:33of the other variable so in this case
4:36the value of variable two is a function
4:39of the value of variable
4:41one we can say it like this the value of
4:44y is a function of X so y equal F ofx so
4:50we will either choose randomly or be
4:52given a value for x we will put that
4:55into some function and then it will
4:57generate a value for y
5:00but the value for y is always dependent
5:03on the value of
5:05x so we can say it like this the value
5:08of the dependent variable in this case Y
5:11is a function of the independent
5:13variable in this case the variable X now
5:16remember in regression the whole idea is
5:19to form a line through our data points
5:22that minimizes the residual sum of
5:25squares but before we can really learn
5:27about that we have to quickly review the
5:29basic
5:30of the algebra of
5:33lines so I know you don't really want to
5:35go back to algebra but you'd be
5:37surprised how many students I've worked
5:38with over the years that take stats
5:40they're sophomore junior year of college
5:42they haven't had algebra in five six
5:44seven years and they forget some of this
5:47stuff so just on one slide going to
5:49quickly review the algebra of lines now
5:52the most important thing to remember in
5:54this case is the slope intercept form of
5:57a line because that is basically
6:00how a regression line is stated so
6:03remember that y = mx plus b is the slope
6:07intercept form of a line and it's called
6:09that because it has two real components
6:12the slope and The Intercept so X in this
6:17case is some random variable that we
6:18either Choose Or we are given in the
6:21problem m in the slope intercept form is
6:25the slope of the line This represents
6:28you've also probably seen it written as
6:30rise over run but X is the random
6:33variable and M is the slope now B is the
6:37Y intercept is where the line crosses
6:39the Y AIS so wherever our line is on
6:43this graph wherever it crosses the Y AIS
6:46that is our Y
6:47intercept now the Y intercept has a
6:50couple of special properties the Y
6:52intercept is where x equals z so if you
6:57think about all the points on the Y AIS
6:59they all sound like this 02 05 010 0 -2
7:0705 Etc so it's always where the value of
7:11x is zero so every point on the Y AIS
7:16looks like this 0 comma y now let's take
7:20an example so we'll have this equation y
7:23= 2x + 3 now look at that equation and
7:26then superimpose it on top of the slope
7:29inter intercept form of a line we can
7:31learn a lot about this equation the
7:33first thing we can learn is that the
7:35slope is two so we have 2X + 3 over here
7:39on the left we have mx + b so 2 is where
7:42the m is therefore the slope is 2 now if
7:46we write it as a fraction that's 2 over
7:49one because we don't write the one when
7:50we write whole numbers like that so in
7:53terms of rise over run it's 2 over
7:56one now let's find the Y inter intercept
8:00so we know that the Y intercept is
8:02wherever X is zero so we can substitute
8:050 in for x and then calculate the value
8:08of y so Y = 2 * 0 + 3 so y = 3 so we
8:14have the point of The Intercept is the
8:16point
8:1703 so maybe it's about right there where
8:20the dot is now my graph here is not
8:23proportionally correct so don't go off
8:26how the graph looks just sort of take
8:28into account what I'm writing on the
8:30graph now of course to graph a line we
8:32have to have at least two points let's
8:35go ahead and figure out a second point
8:36and luckily this is easy to do we can
8:39just pick any random value for x and
8:41then calculate y so let's choose x = 1
8:44so we'll substitute one in where X is
8:47and we'll go and calculate the value of
8:48y so 2 * 1 is 2 + 3 so 2 + 3 is 5 so the
8:54second point on our graph will be the
8:57point 15 now we can go ahead and graph
9:01our
9:01line now remember our slope was 2 over 1
9:05so if this were graphed properly we
9:08could go up two and over one now
9:12remember they're both positive so two is
9:15upward and one is to the right because
9:17they're both positive and we would end
9:19up at another point on our line so this
9:23is a basic overview of the algebra of
9:25lines because regression lines when you
9:28actually do them in a stat software
9:30package come out in this
9:35form now they don't come out in exactly
9:37that form but they are related so let's
9:40go ahead and show that relationship so
9:42the slope intercept is y = mx + b now
9:46the overall regression model for the
9:48entire population we are considering
9:51looks like this y = beta Sub 0 plus beta
9:56sub 1 x + e so don't freak out this is
10:01essentially the same thing you see over
10:04on the left now remember as in many
10:07areas of stats there is this very large
10:10overall population that we really don't
10:13know everything there is to know about
10:16it because maybe we can't so therefore
10:18we have to use sample data to estimate
10:22it now if we had the population
10:25available to us this is what the
10:27regression model would look like
10:30now how does it relate to over here on
10:31the left so I color coded it as well so
10:34you can see the relationship so beta Sub
10:360 is the Y intercept of the population
10:40parameter so you can see on the left
10:42it's B and over here on the right it's
10:45beta Sub
10:470 Now Beta sub one is the same thing as
10:50the slope of the population parameter so
10:53beta sub 1X over here on the right
10:56corresponds to MX over here on the left
11:00and then e is the error term in
11:03regression it's the Unexplained
11:05variation in our y variable so I don't
11:09want to get ahead of myself here but the
11:11beta Sub 0 + B beta sub 1X will explain
11:16part of the variation and then what's
11:19left will be error or unexplained
11:22variation now when we actually do simple
11:25linear regression we write it like this
11:27the expected value of y equal beta Sub 0
11:32+ beta sub 1 x now all you really need
11:36to know at this point is that this is
11:38pretty much the exact same thing as y =
11:41mx plus b up here on the top left so
11:46what is the expected value of y well
11:49it's the mean or the expected value of y
11:53for a given value of x so whatever X we
11:57choose or or given
12:00the expected value of y is where we
12:02expect that to intersect in our graph
12:06now again this will become more apparent
12:07as we go but this is how we actually
12:10write the simple regression
12:14model now the expected value is the mean
12:18so what does that actually well mean so
12:22if we have a coordinate plane here where
12:23we have X and Y AIS and we pick a value
12:27for x so maybe here sort of in the
12:30middle well that corresponds to a value
12:33of y and that's our data point maybe
12:35right
12:37here but that's the expected value of y
12:41it's really not that simple there's
12:44actually a distribution of Y's for that
12:48given X so remember our regression model
12:51is not going to be perfect so any
12:54expected value of y we come up with is
12:57at best going to be an approximation
13:00so when we say the expected value what
13:01we mean is that it's the mean of a small
13:05distribution for that y so it can be
13:09fairly narrow like this you can see that
13:12our distribution of y's Falls in a
13:14narrow band or it can be the exact same
13:18point so this exact same point with the
13:22expected value of y could be much
13:25wider like that now obviously in
13:28regression we want to have the situation
13:30like is over here on the left so we want
13:33to have the expected value of our y's to
13:35be in sort of a narrower range a narrow
13:38distribution than we do over here on the
13:41right but just remember the expected
13:44value of y is really the mean the mean
13:48of a distribution of those y values now
13:51this will become more important as we
13:53get into more complex topics about
13:55regression but I just want to point out
13:57that the value of y isn't really a point
14:01it's the mean of a distribution around
14:03the
14:06Y's now regression lines can take three
14:09General forms so here we have our
14:11expression up here on the left or
14:13equation up here on the left so expected
14:15value of y = beta Sub 0 + beta sub 1 * X
14:20so we can have the first case so it
14:22looks like this the expected value of y
14:25equal beta Sub 0 + 0 * X well what is
14:32the zero in this case well that's where
14:34the beta sub one is so what is that
14:37saying well that's saying that the slope
14:40is zero but if we go ahead and multiply
14:44this out we end up with just expected
14:46value of y equals beta Sub 0 because
14:50everything to the right of that is 0
14:53because of the multiplication so on our
14:55graph if our slope is zero it looks like
14:59like this the line is flat that is a
15:02line with the slope of zero so the slope
15:06b sub1 or beta sub one is
15:10zero the second type might look like
15:12this expected value of y equal beta Sub
15:160 plus beta sub 1 x now notice that the
15:20slope the beta sub 1 is positive so it
15:25looked like this so the slope beta sub 1
15:28is is a positive so our line goes from
15:31the lower left upward to the right now
15:35the other General type looks like this
15:37the expected value of y = beta Sub 0
15:40minus beta sub 1 * X in this case our
15:44slope the beta sub one is negative so it
15:48might look like this where our line goes
15:51from the top left down to the lower
15:53right so again these are General
15:55patterns that are based on the sign and
16:00the value of beta sub 1 because beta sub
16:031 is the same thing as the slope so mx +
16:07b this is beta sub 1 * X beta sub 1 is
16:12the slope so our lines might sort of
16:14form a general pattern based on the
16:16value of beta sub
16:20one so if we actually knew the
16:23population parameters beta Sub 0 and
16:25beta sub 1 we could use the simple
16:28linear regression equation that looks
16:30like this that we already saw but in
16:33reality we almost never have the
16:34population parameters therefore we will
16:37estimate them using sample data when
16:39using sample data we have to change our
16:41equation a little bit it looks like this
16:45now the Y with the little thing on top
16:48is pronounced y hat and it is the point
16:52estimator of the expected value of
16:56y so y hat is the mean
16:59value of y for a given value of x so
17:04just like other cases in stats where
17:06we're using sample data we write the
17:08equations a bit differently so all we do
17:10is use lowercase b where we have the
17:13betas in the top equation and of course
17:15we have y hat versus the expected value
17:18of y now in functional terms they act
17:22the same way we just write it this way
17:25to acknowledge that we are using sample
17:28data
17:31so here is our same graph but now we
17:34write it with Y hat so in this case
17:36we're using sample data but the basic
17:39idea is the same so y hat is the mean of
17:44the expected values of Y for any given
17:47value of x otherwise it's the exact same
17:53concept so let's look at this in the
17:55context of a problem now this data or
17:58this graph is from our first video where
18:01we had the dollar amount of tips like a
18:04waiter or waitress received in a
18:06restaurant but we didn't have the
18:08matching amount of the actual bill so
18:11all we had was the tip amount we only
18:13had the one variable we only had the
18:16dependent variable at that so the best
18:19way we could do is make a graph based on
18:21the mean of the tip data which was $10
18:25so then we went through and found the
18:26residuals and squared those and then
18:28added them up so with the sum of squared
18:30errors or squared residuals of
18:33120 so in this case what's the slope of
18:37this line well it's zero because the
18:40independent variable doesn't even exist
18:43so the slope of this line is zero
18:46because we're only using the values of
18:48the dependent
18:49variable so when conducting simple
18:52linear regression with two variables we
18:54will determine how good the regression
18:56line fits the data by comparing ing it
18:59to this type where we pretend the second
19:02variable does not even exist so we
19:05covered that in the previous video but
19:07the important thing to apply in this
19:09video is that the slope of this example
19:13the slope of the special type of
19:16situation is zero so this is when beta
19:19sub one is zero so any regression line
19:23we come up with we will always be
19:26comparing it to the situation a where
19:29the slope is zero and the most extreme
19:33case where the slope is zero is where
19:35the dependent variable is where the
19:37independent variable does not even exist
19:42so in this situation the value of y hat
19:46is 10 for every value of x so we can
19:51actually do this algebraically so y Hat
19:53= B Sub 0 + B sub 1 * X but the slope 0
19:59so we insert that in where B sub1 is so
20:02we're left with Y hat equals B Sub 0 but
20:07what is B Sub 0 well it's 10 so y hat
20:12equals 10 for every point along the x
20:15axis so for the second tip it's $10 for
20:19the fifth tip it's $10 for the fourth
20:21tip it was $10 so we're always comparing
20:23our regression model to this example
20:26where beta sub one is Z or the slope is
20:33zero so let's go ahead and do a quick
20:35exercise where we can match an equation
20:37with the general regression model so we
20:39have an equation that's y hat equal 0.3
20:43minus 3.3x then we have y hat = 48 +
20:487.8x and then y hat = 14.87%
20:58our slope beta sub one is at or near
21:01zero in the second example our slope is
21:04positive so beta sub 1 is positive and
21:07our last one beta sub one is negative so
21:11let's go back up to our first equation
21:13what is the value of the slope in our
21:16first equation well it's
21:183.3 so it's negative so which one does
21:21that go
21:22with that General shape so if we graph
21:26this equation it probably look something
21:27like the bottom right right now what
21:29about our second equation what is the
21:32slope there well it's positive
21:357.8x so it would look maybe like the
21:38second example here at the bottom now
21:40what about our third one our slope is
21:44.014 well that's very very close to zero
21:47it's not exactly zero but it's close so
21:50if we graph that line it would probably
21:52look something like that so again
21:55depending on the value or the sign of
21:58the slope we can have a general idea
22:01what the regression line will look
22:05like let's go ahead and get ready for
22:08conducting the least squares which we
22:09will do in the next video now remember
22:12in the first video in regression we only
22:15had the dollar amount of the tip that
22:17the waiters or waitresses received we
22:19didn't have the meal amount so now we're
22:22going to go ahead and add that
22:24in so for the Bill of $34 I had a tip of
22:27$5 for a bill of $18 I had a tip of $17
22:32and so on and so forth now when we go
22:34ahead and graph those on a scatter plot
22:37it looks like this over here on the left
22:40so you can see where each data point
22:41Falls with relationship to the bill
22:43amount and the tip amount now remember
22:46what we're saying here is that the tip
22:49dollar amount depends on the original
22:54bill amount that's what we think that's
22:56what we're sort of hypothesizing so in
22:59general a lower bill will result in a
23:02lower tip and a more expensive bill will
23:05result in a higher tip that's sort of
23:08our hypothesis so if we graph these on
23:11the scatter plot we can look and it
23:13appears that there's some sort of linear
23:15relationship that goes from left to
23:18right so if we drew a line it would
23:21probably start the lower left and go to
23:23the upper right and most importantly
23:28if that
23:29line reduces the residual sum of squares
23:34significantly from the model where we
23:36only use the mean of the tips of
23:38$10 then that's when we will say that a
23:41regression model is good sort of an equ
23:44qualitative sense now quantitatively
23:46we'll be able to figure that out more
23:48concretely but generally if the residual
23:51sum of squares is a lot less using our
23:54regression line we come up with using
23:56the least squares method then we know
23:59that our regression line is much better
24:01than the example where we only use the
24:03mean tip
24:06amount okay so that wraps up our second
24:08video in our linear regression series
24:11and again here I wanted to review a
24:12couple things sort of the algebra of
24:14regression lines some general model
24:17equations that you will see going
24:18forward and you will see in your own
24:20work and then some general graph
24:22patterns that you can look at look at
24:23the equation and try to figure out what
24:25the graph means and vice versa so again
24:28this just sort of lays the groundwork
24:29for the actual calculations that will'll
24:31start in the next video where we will
24:33actually calculate the least squares
24:35method and figure out our regression
24:37line by hand
24:42[Music]

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