# 11.4 The regression equation (modified r. bloom)

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This module provides an overview of Linear Regression and Correlation: The Regression Equation as a part of R. Bloom's custom Collaborative Statistics collection col10617. It has been modified from the original module m17090 in the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean. This module now includes instructions for finding and graphing the regression equation and scatterplot using the LinRegTTest on the TI-83,83+,84+ calculators.

## Understanding the regression equation

Data rarely fit a straight line exactly. Usually, you must be satisfied with rough predictions. Typically, you have a set of data whose scatter plot appears to "fit" a straight line. This is called a Line of Best Fit or Least Squares Line .

A random sample of 11 statistics students produced the following data where $x$ is the third exam score, out of 80, and $y$ is the final exam score, out of 200. Can you predict the final exam score of a random student if you know the third exam score?

The third exam score, $x$ , is the independent variable and the final exam score, $y$ , is the dependent variable. We will plot a regression line that best "fits" the data. If each of youwere to fit a line "by eye", you would draw different lines. We can use what is called a least-squares regression line to obtain the best fit line.

Consider the following diagram. Each point of data is of the the form $\left(x,y\right)$ and each point of the line of best fit using least-squares linear regression has the form $\left(x,\stackrel{^}{y}\right)$ .

The $\stackrel{^}{y}$ is read "y hat" and is the estimated value of $y$ . It is the value of $y$ obtained using the regression line. It is not generally equal to the observed $y$ from data.

The term $y-\stackrel{^}{y}$ is called the residual . It is the observed $y$ value − the predicted $\stackrel{^}{y}$ value. It can also be called the "error".It is not an error in the sense of a mistake, but measures the vertical distance between the observed value $y$ and the estimated value $\stackrel{^}{y}$ . In other words, it measures the vertical distance between the actual data point and the predicted point on the line.

If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for $y$ . In the observed data point lies below the line, the residual is negative, and the line overestimates that actual data value for $y$ .

In the Figure 2 diagram above, ${y}_{0}-{\stackrel{^}{y}}_{0}={\epsilon }_{0}$ is the residual for the point shown. Here the point lies above the line and the residual is positive.

$\epsilon$ = the Greek letter epsilon

For each data point, you can calculate the residuals or errors, ${y}_{i}-{\stackrel{^}{y}}_{i}={\epsilon }_{i}$ for $i=\text{1, 2, 3, ..., 11}$ .

Each $\epsilon$ is a vertical distance.

For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points. Therefore, there are 11 $\epsilon$ values. If you square each $\epsilon$ and add, you get

$\left({\epsilon }_{1}{\right)}^{2}+\left({\epsilon }_{2}{\right)}^{2}+\text{...}+\left({\epsilon }_{11}{\right)}^{2}=\stackrel{11}{\underset{\text{i = 1}}{\Sigma }}{\epsilon }^{2}$

This is called the Sum of Squared Errors (SSE) .

Using calculus, you can determine the values of $a$ and $b$ that make the SSE a minimum. When you make the SSE a minimum, you have determined the points that are on the line of best fit. It turns out thatthe line of best fit has the equation:

#### Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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