1. Home
  2. College algebra
  3. Linear functions
  4. Fitting linear models to data

4.3 Fitting linear models to data  (Page 3/14)

<< Chapter < Page
  College algebra     Page 3 / 14
Chapter >> Page >

Given data of input and corresponding outputs from a linear function, find the best fit line using linear regression.

  1. Enter the input in List 1 (L1).
  2. Enter the output in List 2 (L2).
  3. On a graphing utility, select Linear Regression (LinReg).

Finding a least squares regression line

Find the least squares regression line using the cricket-chirp data in [link] .

  1. Enter the input (chirps) in List 1 (L1).
  2. Enter the output (temperature) in List 2 (L2). See [link] .
    L1 44 35 20.4 33 31 35 18.5 37 26
    L2 80.5 70.5 57 66 68 72 52 73.5 53
  3. On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier, with technology we obtain the equation:
T ( c ) = 30.281 + 1.143 c
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Will there ever be a case where two different lines will serve as the best fit for the data?

No. There is only one best fit line.

Distinguishing between linear and nonlinear models

As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the correlation coefficient    , which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a ”diagnostic on” selection to find the correlation coefficient, which mathematicians label as   r   The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.

We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of   r   and the graph of the data, [link] shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.

Correlation coefficients values range from -1.0 - 1.0. Collections of dots representing an example of each kind of correlation coefficient are plotted underneath them. The closer to 1.0 the more the points are grouped tightly to form a line in the positive direction. The closer to -1.0 the more the points are grouped tightly to form a line in the negative direction. The closer to 0 the points are very scattered and do not form a line. Several shapes are displayed at the bottom row, none of which are lines, but all of them have values of 0.
Plotted data and related correlation coefficients. (credit: “DenisBoigelot,” Wikimedia Commons)

Correlation coefficient

The correlation coefficient is a value,   r , between –1 and 1.

  • r > 0   suggests a positive (increasing) relationship
  • r < 0   suggests a negative (decreasing) relationship
  • The closer the value is to 0, the more scattered the data.
  • The closer the value is to 1 or –1, the less scattered the data is.

Finding a correlation coefficient

Calculate the correlation coefficient for cricket-chirp data in [link] .

Because the data appear to follow a linear pattern, we can use technology to calculate   r   Enter the inputs and corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation coefficient,   r = 0.9509.   This value is very close to 1, which suggests a strong increasing linear relationship.

Note: For some calculators, the Diagnostics must be turned "on" in order to get the correlation coefficient when linear regression is performed: [2nd]>[0]>[alpha][x–1], then scroll to DIAGNOSTICSON.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Fitting a regression line to a set of data

Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask