# 11.4 The regression equation (modified r. bloom)

 Page 1 / 2
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? Table showing the scores on the final exam based on scores from the third exam.

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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Collaborative statistics: custom version modified by v moyle. OpenStax CNX. Nov 14, 2010 Download for free at http://legacy.cnx.org/content/col11238/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics: custom version modified by v moyle' conversation and receive update notifications? By Brooke Delaney By Dewey Compton By Zarina Chocolate By Jazzycazz Jackson By OpenStax By Edgar Delgado By By OpenStax By Mary Matera By Megan Earhart