HELLO!!! Mr. Tarrou again. Thank you very much for watching my videos, for liking them,
and subscribing. I am really having a good time starting these video lectures for you.
Today we are going to look at and continue our discussions in Statistics of scatter plots.
Now again, my notes, these long lectures, are just an outline of the concepts you need
to know. Please make sure you read your textbook thoroughly, look at those examples, and do
your homework. Use these as a guidelines as "Hey, I should know all these facts." Then
refer to these notes later as you start to study for your exams. The other video I just
did, we were looking at scatter plots. This was our first bi-variate quantitative graph
where the x and y axis were numerical values. We talked about how the x axis was the explanatory
variable and the y axis was the response variable and we made scatter plots. We talked about
how you would look at those and describe the form and direction and the strength of whatever
pattern that may have happened to show up. We talked about correlation which is "r",
the strength and direction of a linear relationship. R squared and we also talked about lurking
variables. If you want to take a look at that video, please do. Well now we are going to
say HEY I looked at these scatter plots and they looked like they are straight, or at
least we think that those patterns we are seeing in the plots are linear... straight.
Well if you have a linear pattern in a scatter plot, going up or down, we can make a regression
line or a Least Squares Regression Line (that is what I am talking about today) to put through
those points and use that Least Squares Regression Line to make estimates of y values. We will
use the x values (independent variable) to make estimates of y values (dependent variable)
Where we use the x values to make predicted y values and basically allow that line to
model the actually data and make predicted y values in places where we don't have any
real values of x. What are some properties and things you guys need to know about regression
lines? The form is y hat equals a plus b x. and Y hat is read as the predicted y value.
So when you see an actual point it will be (x,y) and when you see (x, y hat) you will
know that y hat is a predicted y value or one that came from the regression line. A,
and you might know from y equals mx plus b the slope intercept form of a line, when you
look at a line y=mx+b, the constant and this case "a" is going to be the y intercept. Now
that is the place the line crosses the y axis when x is equal to zero. Our stats problems
are all real applications and the variables of x and y can take on any multitude of values.
They are not all going to be centered around the origin so the value of "a", the y intercept,
in these application problems don't have a real life interpretation, however b does.
"B" is the slope and it always has a real interpretation. Please read those examples
in your textbook about how to interpret slope. This is really important when you get to the
end of the class and you start doing linear regression t-tests your full understanding
of what slope is and being able to interpret it is a big deal. Well what is slope? The
idea of slope being rise over run is great for when you are drawing a graph and basic
algebra but it is not good for word problems and application problems. The real definition
of slope is the average change in y per one unit of change in x. In your word problems
when you trying to interpret what slope is, if you just take out the y and replace it
with whatever it is about.. blood pressure, or SAT math scores, or whatever, just take
the y out and replace it with whatever your response variable is in your word problem.
The x is whatever, if I am talking about GPA compared to future SAT math scores, then the
slope is the average change in the SAT math score per one unit of change in their GPA.
So slope is the average change in y per one unit of change in x. SO, over here we have
a little sketch. I have got one, two, three, four points and I am drawing a regression
line through those points. Now let me talk about why we call the regression line the
Least Squares Regression Line. In algebra, and honestly this is all I did in this drawing,
when you did scatter plots in Algebra 1 and Algebra 2 maybe you looked at some scatterplot
and just put a ruler down and drew a line through the points. Then you just used two
points to find the slope and find an equation of that fitted... Line of Best Fit. Well that
is all fine but with the help of calculators and computers we can properly place that line
through those points. How we place that line through the points is hinted by the name Least
Squares Regression Line. If you have a scattering of points clearly the line which is perfectly
straight is not going to hit many of the points. It might not hit any of the points. You are
always going to have these vertical distances between the actual data points, in this case
there are four of them, and the regression. Those vertical distances that I have highlighted
in green, if you are watching on a computer with the high definition turned on of 720p,
those vertical distances are called residuals. We use our x values our explanatory variables
to predict the response variable of y. So every time you make a prediction you might
have some error. Now I just pointed to an actual data point. Here I have an actual point
of (x,y) and then up here I have a predicted point that is from the regression line of
(x, y hat). Again that y hat is a predicted y value. Well there is an error there. You
cannot help to make a mistake or an error with your predictions. That vertical distance
is a big deal and it is called a residual. That vertical distance is calculated by doing
y minus y hat. If your residual is negative this y hat value is bigger than the actual
y value of the point, then when you do y minus y hat you get a negative difference. So when
your points are below the line your residuals are negative and when your points are above
the line like these two your residual is going to be positive. So some residuals are positive
and are negative. Well these vertical distances if we add them always come out to be zero,
kind like the distances between all your individual data points and the mean when you are finding
standard deviation. Those distances between your data points and the mean is always zero.
So the regression line has something to do with averages. Every regression line goes
through the point mean of x mean of y. This is another important property of regression
lines. If these residuals always add up to zero, then we can't just add them all up because
we would always get an answer of zero. So what we do is we square them and the sum of
those squared residuals needs to be as small as possible. When that sum get as small as
possible then your Least Squares Regression Line is in the proper place. So there is your
basic definition of Least Squares Regression Line, how it is placed in a scatter plot,
how you read it, let's move on!!! When you take the AP exam they are going to try and
ask you questions that basically from being completely dependent on your calculator. Here
we have one of those examples. Here are a couple of formulas that you will find on your
formula sheet when you take the AP Stats exam. Find the regression line with descriptive
statistics. The slope of a regression line is equal to correlation value, the strength
and direction of your linear relationship, times the standard deviation of y over the
standard deviation of x. A which is your y intercept is is equal to the mean of y minus
b times the mean of x. Here is a reminder off of your formula sheet that every regression
line goes through the point mean of x mean of y. What is the equation of the regression
line of y on x, this is standard language in your textbook, for data with the following
characteristics. The mean of x is ten, the standard deviation of x is two, the mean of
y is thirty, the standard deviation of y is twelve, and the correlation coefficient is
.33 so we have a fairly weak linear relationship. First we find out what the slope is. That
is the correlation value r times the standard deviation, so you will have to read your problem
and recognize what is going to be on your x and y axis or what is your explanatory and
response variable, over the standard deviation of x which is two. That comes out to be b
is equal to .33 times 6 and .33 as a fraction is close to one third so six divided by three
is about 2. Ok, now I have got my slope of our regression line and "a" is equal to the
mean of y which is thirty minus b, which we just found to be approximately two, times
the mean of x which I gave to you as a value of ten. That means that a is going to be thirty
minus twenty or ten. My regression line going through a scatter plot with these characteristics
would be y hat equals a plus bx. This is not a question that you do tremendously often
in the textbook but it is a very very good question for the AP exam or a chapter test
because I can ask you for a regression line and I am preventing you from just tippy tapping
all the data in the calculator and doing a stats calculation automatically. I am making
sure that you know the concepts and the formulas and not just depending completely on your
calculator. Pay attention to this question even though I know that may seem relatively
basic. This "y on x" is also language that you may see in your word problem that will
help you identify what should should be the x axis and what should on the y axis because
it is always y on x. WHOOO!!! Moving on. I have three minutes left. I will do my last
bit of notes in a separate video. Properties of Regression Lines one more time. They all
pass through the point mean of x mean of y. The sum of the residuals is always zero. The
Least Squares Regression Line is placed so the sum of the residuals squared is as small
as possible. I just said that over there. It is not a resistant measure because you
are using means in the calculation of the regression lines so it is not resistant to
normal outliers or influential points. Remember normal outliers are outliers that are in the
vertical direction and influential points will tend to be outliers in the horizontal
direction and they have a lot more influence over the slope and y intercept of the regression
line. WE DON'T WANT OUTLIERS EVER. Influence outliers are worse though. This repeats what
I just said, regression lines are not resistant. Regression lines requires we have an explanatory
and response variable. This means that if you a regression line and you flip the x and
y axis for some reason. Maybe you switch which variable you are predicting, we always predict
the y values. If you do that your x and y axis are going to change, your slope is going
to change, and your y intercept is going to change. You don't have to have a clear explanatory
variable but what is on the x axis and what is on the y axis will determine the slope
and y intercept. If you switch those around of course they are going to change, hence
this statement. However, as you will see in one of my calculator videos, when you switch
the x and y axis that is not going to change the strength of the linear relationship so
where the slope and y intercept will change with the regression line, r and r squared
the correlation coefficient and the coefficient of determination will stay the same. R and
r squared does not care what you set as the x and y axis. I do have a minute thirty seconds
left. So I know this is quick, but you can pause at any point and take these notes down
and rematch this. Here is some Minitab output that has some numbers that umm we are not
going to learn what they mean until chapters fourteen, or whatever book you have that does
linear regression t-tests, but I want to talk about how to read this part of the Minitab
output so you can write the equation of a regression line from computer output. Some
AP exams questions have had this skill as part of one of the questions. The regression
equation is y hat is equal to a plus bx. What do we call a number when it is off by itself?
A constant. Well the coefficient of the constant is 7.4 so a comes out and 7.4 goes in its
place. The second number or the second line of that Minitab output will be about what
is on the x axis. What is the coefficient of the GPA? So we are using GPA to predict
your SAT math scores. The slope, the coefficient of the x variable, is 12.3. And BAM!!!, I
just looked at some Minitab output and from that wrote the equation of a regression line!
I DID FINISH! WHOO!!! BAM!!! Have a great day:)