College Algebra with Corequisite Support 2e

# 4.3Fitting Linear Models to Data

College Algebra with Corequisite Support 2e4.3 Fitting Linear Models to Data

### Learning Objectives

In this section, you will:

• Draw and interpret scatter diagrams.
• Use a graphing utility to find the line of best fit.
• Distinguish between linear and nonlinear relations.
• Fit a regression line to a set of data and use the linear model to make predictions.

### Corequisite Skills

#### Learning Objectives

• Plot points in a rectangular coordinate system (IA 3.1.1).
• Find an equation of the line given two points (IA 3.3.3).

#### Objectives: Plot points in a rectangular coordinate system (IA 3.1.1) and find an equation of the line given two points. (IA 3.3.3)

In this section we will be plotting collections of data points and looking for patterns in these data sets. A scatterplot is a collection of points plotted on the same coordinate system. When trying to fit a function to a data set it is important to note if there is a pattern to the data set and whether that pattern is linear or nonlinear. If the dependent variable increases as the independent variable increases, we call this a positive association. If the dependent variable decreases as the independent variable increases, we call this a negative association.

### Try It #1

Plot points in a rectangular coordinate system, then find a line through two of the data points.

A precalculus instructor is looking at a random sample of students to see if there is a relationship between the number of hours spent working in a homework platform for a given chapter, and the score for the chapter exam.

 Hours spent doing homework, x 10 8 0 13 21 11 5 9 18 Exam score, y 72 68 38 80 93 76 62 71 85

Plot each of the data points on a coordinate system below. You may either plot the points by hand or using a graphing utility. Be sure to label your x and y axes.

Observe any patterns in the data points. Do you think the association between the variables is positive or negative? Is the pattern linear or nonlinear?

What would you suggest to a friend enrolled in this course based on the data set you graphed?

Choose two points that seem to represent the general pattern in the data set. Write these points as ordered pairs below.
$(,)(,)$
$(,)(,)$

Find the slope of a line passing through these two points. Interpret its value in terms of the variables being measured.
$m=y2-y1x2-x1=m=y2-y1x2-x1=$

Use point-slope form or slope intercept form to write the equation of the line passing through these data points.
$y–y1=m(x–x1)y–y1=m(x–x1)$ or $y=mx+by=mx+b$

Write this equation in slope-intercept form.
$y=mx+by=mx+b$

Rewrite this equation using function notation.
$f(x)=f(x)=$

This equation is a linear model. Sketch the line on the graph created in part a.

Use this mathematical linear model to predict the exam score for a student who spent 15 hours working on this chapter in their homework system. Show your work below.

##### Practice Makes Perfect
1.

The data below shows the relationship between the mass of an automobile (measured in kg) and the fuel efficiency of the car (measured in miles per gallon) for 7 automobiles.

 Mass (kg), x 1305 1150 1925 1628 1506 1452 1835 Fuel Efficiency (MPG), y 27 28 15 24 23 25 19

Draw a scatter plot (by hand or using a graphing utility) for the data provided being sure to label your axes.

Does the data appear to be linearly related? Is the association between the variables positive or negative?

Choose two points that seem to represent the general pattern in the data set. Write these points as ordered pairs below.
$(,)(,)$
$(,)(,)$

Write the equation of the line passing through the points you listed in part c. in slope intercept form. Show your work below.

Use the linear function you found in part d. to predict the fuel efficiency of an Audi A5 Quattro whose mass is 1610 kg.

2.

The data set below shows the relationship between the number of hours worked and the tips received by Nyla, a server at Pi Pizzeria.

### Footnotes

• 5Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010
• 6Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.
• 7For example, http://www.shodor.org/unchem/math/lls/leastsq.html
• 8http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html
• 9Based on data from http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/2014.