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y ^ = a + bx

where a = y ¯ - b x ¯ and b = Σ ( x - x ¯ ) ( y - y ¯ ) Σ ( x - x ¯ ) 2 .

x ¯ and y ¯ are the averages of the x values and the y values, respectively. The best fit line always passes through the point ( x ¯ , y ¯ ) .

The slope b can be written as b = r ( s y s x ) where s y = the standard deviation of the y values and s x = the standard deviation of the x values. r is the correlation coefficient which is discussed in the next section.

Least squares criteria for best fit

The process of fitting the best fit line is called linear regression . The idea behind finding the best fit line is based on the assumption that the data are scattered about a straight line. The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the least squares regression line .

Computer spreadsheets, statistical software, and many calculators can quickly calculate the best fit line and create the graphs. The calculations tend to be tedious if done by hand. Instructions to use the TI-83, TI-83+, and TI-84+ calculators to find the best fit line and create a scatterplot are shown at the end of this section.

Third exam vs final exam example:

The graph of the line of best fit for the third exam/final exam example is shown below:

Scatterplot of the third exam scores by final exam scores and its line of best fit.

The least squares regression line (best fit line) for the third exam/final exam example has the equation:

y ^ = -173.51 + 4.83x
  • Remember, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship betweenthe variables, then it is reasonable to use a best fit line to make predictions for y given x within the domain of x -values in the sample data, but not necessarily for x -values outside that domain.
  • You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam.
  • You should NOT use the line to predict the final exam score for a student who earned a grade of 50 on the third exam, because 50 is not within the domain of the x-values in the sample data, which are between 65 and 75.

Understanding slope

The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

INTERPRETATION OF THE SLOPE: The slope of the best fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.

    Third exam vs final exam example

  • Slope: The slope of the line is b = 4.83.
  • Interpretation: For a one point increase in the score on the third exam, the final exam score increases by 4.83 points, on average.

Using the ti-83+ and ti-84+ calculators

    Using the linear regression t test: linregttest

  1. In the STAT list editor, enter the X data in list L1 and the Y data in list L2, paired so that the corresponding (x,y) values are next to each other in the lists. (If a particular pair of values is repeated, enter it as many times as it appears in the data.)
  2. On the STAT TESTS menu, scroll down with the cursor to select the LinRegTTest. (Be careful to select LinRegTTest as some calculators may also have a different item called LinRegTInt.)
  3. On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1
  4. On the next line, at the prompt β or ρ, highlight "≠ 0" and press ENTER
  5. Leave the line for "RegEq:" blank
  6. Highlight Calculate and press ENTER.

1. Image of calculator input screen for LinRegTTest with input matching the instructions above. 2.Image of corresponding output calculator output screen for LinRegTTest: Output screen shows: Line 1. LinRegTTest; Line 2. y = a + bx; Line 3. beta does not equal 0 and rho does not equal 0; Line 4. t = 2.657560155; Line 5. df = 9; Line 6. a = 173.513363; Line 7. b = 4.827394209; Line 8. s = 16.41237711; Line 9. r squared = .4396931104; Line 10. r = .663093591

The output screen contains a lot of information. For now we will focus on a few items from the output, and will return later to the other items.

  • The second line says y=a+bx. Scroll down to find the values a=-173.513, and b=4.8273 ; the equation of the best fit line is y ^ = -173.51 + 4.83 x
  • The two items at the bottom are r 2 = .43969 and r =.663. For now, just note where to find these values; we will discuss them in the next two sections.

    Graphing the scatterplot and regression line

  1. We are assuming your X data is already entered in list L1 and your Y data is in list L2
  2. Press 2nd STATPLOT ENTER to use Plot 1
  3. On the input screen for PLOT 1, highlight On and press ENTER
  4. For TYPE: highlight the very first icon which is the scatterplot and press ENTER
  5. Indicate Xlist: L1 and Ylist: L2
  6. For Mark: it does not matter which symbol you highlight.
  7. Press the ZOOM key and then the number 9 (for menu item "ZoomStat") ; the calculator will fit the window to the data
  8. To graph the best fit line, press the "Y=" key and type the equation -173.5+4.83X into equation Y1. (The X key is immediately left of the STAT key). Press ZOOM 9 again to graph it.
  9. Optional: If you want to change the viewing window, press the WINDOW key. Enter your desired window using Xmin, Xmax, Ymin, Ymax

Questions & Answers

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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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
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