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Linear Regression

Site: Harrison
Course: Michigan Algebra I Sept. 2012
Book: Linear Regression
Printed by: Guest user
Date: Saturday, 23 November 2024, 1:43 PM

Description

Linear Regression

Table of contents

Introduction

Like correlation coefficients, linear regression analyzes the relationship between two variables, x and y. If the data on the scatter plot seems to represent a linear relationship, then linear regression can be used to find the line that best fits the data. If the data is curved, a line would not be the best equation to use. The goal is to find an equation that fits the data so that it can be used to predict other x and y values in the situation.


Line of Best Fit

When given some data that seems to have a linear relationship, it is possible to find an equation that best fits the data. Follow these steps to find the equation:

Step 1. Plot the ordered pairs and determine if the relationship between the x and y values looks linear.

Step 2. Draw a line that seems to best fit the data.

When drawing in a line, keep these guidelines in mind:

      • Have as many points below the line as above the line whenever possible.
      • Be sure the line is going in the same direction as the points. One way to check the direction is by drawing in the smallest rectangle possible around the points. The slope of the line should follow the same direction.
      • If possible, a point or points should be on the line but the above guidelines are more important.

Step 3. Use two points and find the slope of the line. Use the slope formula m = BestFit .

Step 4. Estimate the y-intercept or use a point and the slope found in Step 3 to write the equation. Substitute the slope (m) and the point (x1 , y1) into the point-slope form of an equation y - y1 = m (x - x1).


Example

Is there a relationship between the fat grams
and the total calories in fast food?

BestFitEx1-1

Step 1. Plot the ordered pairs and determine if the relationship between the x and y values look linear.

BestFitEx1-2

Example Continued

Step 2. Draw a line that seems to best fit the data.

BestFitEx1-3

Step 3. Use two points and find the slope of the line. Use the slope formula m = BestFitEx1-4.

Suppose the points are (9, 260) and (30, 530). You may choose different points. Now calculate the slope through the two points.
BestFitEx1-5

Step 4. Estimate the y-intercept or use a point and the slope found in Step 3 to write the equation. Substitute the slope (m) and the point (x1, y1) into the point-slope form of an equation y - y1 = m (x - x1).


Video Lesson

To see a video lesson on how to find a line of best fit by hand, select the following link:

Line of Best Fit

Interactive Activities

To investigate a linear regression line, select the following link (this link will also review correlation coefficient on the same graphs):

Linear Regression Investigation

To see what adding points to a graph does to a linear regression line, select the following link:

Linear Regression Interactive

Online Examples

The following link gives examples of data that result in the same correlation coefficient and same linear regression equation, yet are very different. Think about why this is.

Online Examples

Using a Calculator

To use a calculator to find the linear regression line or line of best fit, select the following link:

Calculator Tutorial

Guided Practice

To solidify your understanding of lines of best fit, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Practice

Weight and Drug Doses Worksheet

Bike Weight Worksheet

Linear Regression Worksheet

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Answer Keys

Weight and Drug Doses Answer Key

Bike Weight Answer Key

Linear Regression Answer Key

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.

Sources

Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project

Holt, Rinehart, and Winston, http://my.hrw.com/math06_07/ nsmedia/lesson_videos/msm3/player.html?contentSrc=6296/6296.xml (accessed 08/27/2010).

Holt, Rinehart, and Winston, http://my.hrw.com/math06_07/nsmedia/ homework_help/msm3/msm3_ch12_07_homeworkhelp.html (accessed 08/27/2010).

http://en.wikipedia.org/wiki/File:Anscombe.svg (accessed 08/27/2010).

Illuminations, "Linear Regression 1." http://illuminations.nctm.org/ ActivityDetail.aspx?ID=82 (accessed 08/27/2010).

Regents Prep, "Line of Best Fit." http://regentsprep.org/Regents/math/ ALGEBRA/AD4/linefitcalc.htm (accessed 08/27/2010).

University of South Carolina, "Regression Applet." http://www.stat.sc.edu/ ~west/ javahtml/Regression.html (accessed 08/27/2010).