UNIT 2.5 USING LINEARUNIT 2.5 USING LINEARMODELSMODELS

Warm Up

Write the equation of the line passing through each pair of passing points in slope-intercept form.

1. (5, –1), (0, –3) 2. (8, 5), (–8, 7)

Use the equation y = –0.2x + 4. Find x for each given value of y.

3. y = 7 4. y = 3.5

1. Fit scatter plot data using linear models with and without technology.

2. Use linear models to make predictions.

Pa. Dept of Educ. Math Standards and Anchors:

2.6.A2.C/E

2.7.A2.C/E

M11.E.1.1.2

4.1.1

4.2.1

4.2.2

Objectives

regressioncorrelationline of best fitcorrelation coefficient

Vocabulary

Researchers, such as Researchers, such as anthropologists, are often anthropologists, are often interested in how two or interested in how two or more measurements are more measurements are related.related.(ex. Climate, Size, (ex. Climate, Size, Species)Species)

The study of the The study of the relationship between two relationship between two or more variables is or more variables is called called regressionregression analysis.analysis.

Scatter plots like those below are helpful in determining the strength and type of a relationship between variables.

Correlation is the strength and direction of the relationship between the two variables.

Try to have about the same number of points above and below the line of best fit.

Helpful Hint

If there is a strong linear relationship between two variables, a line of best fit, or a line that best fits the data, can be used to make predictions.

The correlation coefficient r is a measure of how well the data set is fit by a model.

Example 1: Meteorology Application

Albany and Sydney are about the same distance from the equator. Make a scatter plot with Albany’s temperature as the independent variable (the “x” coordinate). Name the type of correlation. Then sketch a line of best fit and find its equation.

o

o

•••••• ••

•

Step 1 Plot the data points.Step 2 Identify the correlation.

Notice that the data set is negativelynegatively correlated...as temperature rises in Albany, it falls in Sydney.

Example 1 Continued

•••

Step 3 Sketch a Line of Best Fit

Step 4 Identify two points on the line.

For this data, you might select (60,52) and (40,61).

Step 5 Find the slope of the line that models the data.

Step 6 Use either point-slope or slope intercept form to develop the equation

An equation that models the data is y = –0.45x + 79

y – y1= m(x – x1)

y – 52 = –0.45(x – 60)

y = –0.45x + 79

Example 1 Continued

y= mx + b

52 = -.45(60) + b

52 = -27 + by – 52 = –0.45x + 27

52 + 27 = b

y = –0.45x + 79

61 – 5240 - 60

9 -20

-.45

y = –0.45x + 27 + 52 79 = b

Check It Out! Example 1

Make a scatter plot for this set of data. Identify the correlation, sketch a line of best fit, and find its equation.

•

••••

•••••

Step 1 Plot the data points.

Step 2 Identify the correlation.

Notice that the data set is positively correlated…as time increases, more points are scored

Check It Out! Example 1 Continued

Step 3 Sketch a Line of Best Fit

Step 4 Identify two points on the line.

For this data, you might select (20, 10) and (40, 25).

Step 5 Find the slope of the line that models the data.

Using point-slope form.

y = 0.75x - 5

y – y1= m(x – x1)

y – 10 = 0.75(x – 20)

y = 0.75x – 5

Check It Out! Example 1 Continued

Using slope-intercept form.y= m(x) + b

10= .75(20) + b

10= 15 + b -5= by – 10 = 0.75x – 15

Step 6 Develop the equation:

You can use a graphing calculator tograph scatter plots and lines of best fit.

Find video tutorial linksfor both of these

calculator functionsSCATTER PLOTS and LINES OF BEST FIT

in MoodleAlgebra 2Chapter 2

Example 2: Anthropology Application

Anthropologists can use the femur, or thighbone, to estimate the height of a human being. The table shows the results of a randomly selected sample.

••••

•• •

•

a. Make a scatter plot of the data with femur length as the independent variable (which means “y”).

Example 2 Continued

b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem.

Example 2 Continued

c. A man’s femur is 41 cm long. Predict the man’s height.

Substitute 41 for l.

The height of a man with a 41-cm-long femur would be about 173 cm.

h ≈ 2.91(41) + 54.04

The equation of the line of best fit is h ≈ 2.91l + 54.04. Use it to predict the man’s height for a 41cm femur.

h ≈ 173.35

Example 2 Continued

Check It Out! Example 2

The gas mileage for randomly selected cars based upon engine horsepower is given in the table.

•••••

•••••

Check It Out! Example 2 Continued

a. Make a scatter plot of the data with horsepower as the independent variable.

The scatter plot is shown on the right.

b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem.

Enter the data into lists L1 and L2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4:LinReg. The equation of the line of best fit is

y ≈ –0.15x + 47.5.

Check It Out! Example 2 Continued

c. Predict the gas mileage for a 210-horsepowerengine.

Substitute 210 for x.

The mileage for a 210-horsepower engine would be about 16.0 mi/gal.

y ≈ –0.15(210) + 47.50.

The equation of the line of best fit is y ≈ –0.15x + 47.5. Use the equation to predict the gas mileage. For a 210-horsepower engine,

y ≈ 16

Check It Out! Example 2 Continued

A line of best fit may also be referred to as a trend line.

Reading Math

Worksheet 2.7A

Lesson Quiz: Part I

Use the table for Problems 1–3.

1. Make a scatter plot with mass as the independent variable.

Lesson Quiz: Part II

2. Find the correlation coefficient and the equation of the line of best fit on your scatter plot. Draw the line of best fit on your scatter plot.

Lesson Quiz: Part I

Use the table for Problems 1–3.

1. Make a scatter plot with mass as the independent variable.

Lesson Quiz: Part II

2. Find the correlation coefficient and the equation of the line of best fit on your scatter plot. Draw the line of best fit on your scatter plot.

r ≈ 0.67 ; y = 0.07x – 5.24

All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.