Lesson 3 Getting Schooled Solidify Understanding

Learning Focus

Interpret data using linear models.

Consider questions and necessary data for further research.

How can correlation coefficients and linear regressions help us to understand the differences in men’s and women’s incomes?

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

Leo and Araceli are two students learning that statistics help people to analyze their world. In Making More $, Leo and Araceli noticed a difference in men’s and women’s salaries. Araceli thought that it was unfair that women were paid less than men. Leo thought that there must be some good reason for the discrepancy, so they decided to dig deeper into the Census Bureau’s income data to see if they could understand more about these differences.

First, they decided to compare the incomes of men and women who graduated from high school (or the equivalent) but who did not pursue further schooling. They created the scatterplot below, with the $x$ value of a point representing the average woman’s salary for some year and the $y$ value representing the average man’s salary for the same year. For instance, the year 2011 is represented on the graph by the point ( $17887, 30616$ ). You can find this point on the graph in the bottom left corner.

1.

Based upon the graph, estimate the correlation coefficient.

2.

Estimate the average income for men in this time period. Describe how you used the graph to find it.

3.

What is the average income for women in this time period? Describe how you used the graph to find it.

4.

Leo and Araceli calculated the linear regression for these data to be $y = 2.189 x - 6, 731.8$ .

What does the slope of this regression line mean about the income of men compared to that of women? Use precise units and language.

“Hmmmm,” said Araceli. “It’s just as I suspected. The whole system is unfair to women.” “No, wait,” said Leo. “Let’s look at incomes for men and women with bachelor’s degrees or more. Maybe it has something to do with levels of education.”

5.

Leo and Araceli started with the data for men with bachelor’s degrees or more. They found that the correlation coefficient for the average salary from 2000–2011 was $r = - .894$ .

Predict what the graph might look like, and draw it. Be sure to scale and label the axes and to put $12$ points on your graph.

The actual scatterplot for salaries for men with bachelor’s degrees from 2000–2011 is provided. How did you do?

6.

Both Leo and Araceli were surprised at this graph. They calculated the regression line and got $y = - 588.46 x + 69, 978$ . What does this equation say about the income of men with bachelor’s degrees from 2000–2011? Use both the slope and the $y$ -intercept of the line of regression in your answer.

Next, they turned their attention to the data for women with bachelor’s degrees or more from 2000–2011. Here are the data:

Year	Income for Women ($)
$2011$	$41,338$
$2010$	$42,409$
$2009$	$42,746$
$2008$	$42,620$
$2007$	$44,161$
$2006$	$44,007$
$2005$	$42,960$
$2004$	$42,539$
$2003$	$42,954$
$2002$	$42,871$
$2001$	$42,992$
$2000$	$43,293$

7.

Analyze the data for women with bachelor’s degrees by creating a scatterplot, interpreting the correlation coefficient and the regression line. For consistency with the men’s graph above, use $0$ for the year 2000, $1$ for the year 2001, etc. Draw the graph, and report the results of your analysis:

8.

Now that you have analyzed the results for women, compare the results for men and women with at least a bachelor’s degrees over the period from 2000–2011.

9.

Leo believes that the difference in income between men and women may be explained by differences in education, but Araceli believes there must be other factors, such as discrimination. Based on the data in this task and Getting More $, make a convincing case to support either Leo or Araceli.

10.

What other data would be useful in making your case? Explain what to look for and why.

Ready for More?

Determine a research question that you would like to pursue to determine more about the differences in men’s and women’s incomes.
Decide what data you need to answer your question.
Research and find the data. (The U.S. Bureau of Labor Statistics was the original source for the data in this task, and they have a lot more!)
Analyze the data using the statistical methods that we have learned so far.
Write a paragraph that summarizes your results.

Takeaways

Ways that data can be used to make questionable claims:

Lesson Summary

In this lesson, we compared two sets of data to draw conclusions about men’s and women’s incomes. We interpreted the meaning of the correlation coefficients, the slope of the regression line, and intercepts of the regression line. We used the data to make claims and challenged the claims of others.

Retrieval

1.

Find the correlation coefficient for the bivariate data set.

Shoe Size	$4$	$5$	$7$	$9$	$5$	$8$	$7$
Number of Siblings	$2$	$3$	$4$	$6$	$2$	$4$	$5$

2.

Does knowing a person’s shoe size determine how many siblings they will have?

3.

Why would using the data set provided possibly lead to incorrect conclusions?

Classify each function as linear, exponential, or quadratic.

4.

$g (x) = 5^{x - 2}$

A.

linear

B.

exponential

C.

quadratic

5.

$f (x) = x^{2} + 8$

A.

linear

B.

exponential

C.

quadratic

6.

$h (x) = - 7 x + 3$

A.

linear

B.

exponential

C.

quadratic