Lesson 2 Making More $ Solidify Understanding

Learning Focus

Model data with a linear function.

Use a linear model to analyze data.

How can we apply what we know about linear functions to statistics?

Technology guidance for today’s lesson:

Linear Regression, Correlation Coefficient, and Average for a Data Set: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Each year the U.S. Census Bureau provides income statistics for the United States. In the years from 1991 to 2005, they provided the following data in the tables. (All dollar amounts have been adjusted for the rate of inflation, so that they are comparable from year to year.)

Year	Median Income for All Men in Dollars
$2005$	$41, 196$
$2004$	$41, 464$
$2003$	$40, 987$
$2002$	$40, 595$
$2001$	$41, 280$
$2000$	$41, 996$
$1999$	$42, 580$
$1998$	$42, 240$
$1997$	$40, 406$
$1996$	$38, 894$
$1995$	$38, 607$
$1994$	$38, 215$
$1993$	$37, 712$
$1992$	$37, 528$
$1991$	$38, 145$

Median Annual Income for Men (in Dollars) from 1991–2005:

Year	Median Income for All Women in Dollars
$2005$	$23, 970$
$2004$	$23, 989$
$2003$	$24, 065$
$2002$	$23, 710$
$2001$	$23, 564$
$2000$	$23, 551$
$1999$	$22, 977$
$1998$	$22, 403$
$1997$	$21, 759$
$1996$	$20, 957$
$1995$	$20, 253$
$1994$	$19, 158$
$1993$	$18, 751$
$1992$	$18, 725$
$1991$	$18, 649$

Median Annual Income for Women (in Dollars) from 1991–2005:

1.

Estimate and draw lines to model each set of data.

2.

Describe how you estimated the line for men’s income. If you chose to run the line directly through any particular points, describe why you selected them.

3.

Describe how you estimated the line for women’s income. If you chose to run the line directly through any particular points, describe why you selected them.

4.

Write the equation for each of the two lines in slope intercept form.

a.

Equation for men:

b.

Equation for women:

5.

Use technology to calculate a linear regression for each set of data. Add the regression lines to your scatterplots.

a.

Linear regression equation for men:

b.

Linear regression equation for women:

6.

Compare your model to the regression line for men. What does the slope mean in each case? (Include units in your answer.)

7.

Compare your model to the regression line for women. What does the $y$ -intercept mean in each case? (Include units in your answer.)

8.

Compare the regression lines for men and women. What do the lines tell us about the income of men versus that of women in the years from 1991–2005?

9.

What do you estimate will be the median incomes for men and women in 2015?

10.

The Census Bureau provided the following statistics for the years from 2006–2011. With the addition of these data, what would you now estimate the median income of men in 2015 to be? Why?

Year	Median Income for All Men
$2011$	$37, 653$
$2010$	$38, 014$
$2009$	$38, 588$
$2008$	$39, 134$
$2007$	$41, 033$
$2006$	$41, 103$

Year	Median Income for All Women
$2011$	$23, 395$
$2010$	$23, 657$
$2009$	$24, 284$
$2008$	$23, 967$
$2007$	$25, 005$
$2006$	$24, 429$

11.

How appropriate is a linear model for men’s and women’s incomes from 1991–2011? Justify your answer.

Ready for More?

Look back over the analysis you have done in this task. Write three questions about men’s and women’s incomes that could be investigated. Then, think of what data would need to be gathered to investigate your three questions.

Takeaways

Line of best fit or linear regression:

Vocabulary

Lesson Summary

In this lesson, we modeled data with linear functions. We estimated our own lines of best fit and found linear regressions using technology. We interpreted the slope and $y$ -intercept of the regression line and compared two sets of data. We learned that using the linear model to predict outcomes beyond the available data can sometimes lead to incorrect conclusions.

Retrieval

Use the given number line for problems 1 and 2.

1.

Find the distance between $B$ and $F$ .

$B F =$

2.

Find the distance between $C$ and $E$ .

$C E =$

3.

What does it mean to find the average?

4.

Solve the system of equations: ${\begin{cases} - 4 x + 5 y = 20 \\ 2 x + 3 y = 23 \end{cases}$