1. Tuesday December 10 Make a scatter plot Estimate a line of best fit
Hours
GPA 4
2.9 5
3.3 11
3.9 1
2.2 15
4.1 2
1.8 10
4.6 6 7 3 9
4.5
Make a scatter plot
Estimate a line of best fit
Write a rule
Interpret the slope and y intercept
Find each residual
Find your upper and lower boundaries

2. You may recall that to find the equation of the LSRL, your calculator minimized the sum of the squares of the residuals. The smaller the sum of the squares, the closer the data was to the line of best fit. However, the magnitude of the sum of squares depends on the units of the variables being plotted. Therefore the sum of squares cannot be compared between different scatterplots—it is not a good way to compare the strength of various associations. Today you will investigate a better way to describe the strength of an association, by using the correlation coefficient.

3. 6-67. CORRELATION COEFFICIENT
The correlation coefficient, r, is a measure of how much or how little data is scattered around the LSRL.
That is, if you have already plotted the residuals and decided that a linear model is a good fit, the correlation coefficient, r, is a measure of the strength of a linear association. The correlation coefficient does not have units, so it is useful no matter what the units of the variables are.
This investigation will help you learn more about the correlation coefficient r.

4. Select any three points in the first quadrant.
Use your calculator to make a scatterplot and find the LSRL. Make a sketch of the scatterplot in your notebook, and record the value of the correlation coefficient r. Your teacher will show you how to use you calculator to find r.
Continue to investigate different combinations of three points and graph each of their scatterplots. Work with your team to discuss and record all of your conclusions about the value of r from this investigation.
Your Task: With your team, use your calculators to explore the interpretation of the correlation coefficient r.

5. 6-68. This investigation will help you learn more about the correlation coefficient r.
For parts (a) through (e), use your calculator to make a scatterplot and find the LSRL. Make a sketch of each graph in your notebook, and record the value of the correlation coefficient r. Your teacher will show you how to use your calculator to find r.
Start by graphing any two points that have integer coordinates and a positive slope between them. Each member in your team should choose a different pair of points. Compare your results with your team.
Each member of your team should choose two new points that have a negative slope between them. Compare your results with your team.

6. c) What happens when you have more than two data points
c) What happens when you have more than two data points? Use your original two points from part (a), and add an additional third point that results in r = 1. How can you describe the location of all possible points that result in r = 1?
d) Start again with your original two points from part (a). Enter a third point that makes the slope of the LSRL negative. What happens to r?
e) Start again with your original two points from part (a). Choose a third point that makes r close to zero (say, r between –0.2 and 0.2).
f) Work with your team to discuss the following questions, and record all of your conclusions about the value of r.
What is the largest r can be? The smallest?
What do the scatterplot and LSRL look like if r = 1? r = –1? r = 0?
What does a value of r close to 1 mean, compared to a value of r close to zero?

7. 6-69. The following scatterplots have correlation r = −0. 9, r = −0
6-69. The following scatterplots have correlation r = −0.9, r = −0.6, r = 0.1, and r = 0.6. Which scatterplot has which correlation coefficient, r?

8. 6-70. LEARNING LOG
Work with your team to discuss how the value of r helps you numerically describe the strength and direction of an association. When you have come to an agreement, write your ideas as a Learning Log entry. Title this entry “Correlation Coefficient, r” and label it with today’s date.

9. 6-71. In problem 6-1, you completed an investigation that helped Robbie use a viewing tube to see a football game. Typical data is shown in the table below. The LSRL is y = x.
Find the correlation coefficient. Is the association strong or weak?
Describe the form, direction, strength, and outliers of the association.
You already know a graphical way to determine if the “form” is linear by looking at the residual plot for the data. A mathematical description of “direction” is the slope. A mathematical description of “strength” is the correlation coefficient. Mathematical descriptions for outliers will be dealt with in a later course. Describe the form, direction, and strength of the viewing tube data in more mathematical terms than you did in part (b).

10. 6-72. Extension: A computer will help us explore the correlation coefficient further.
Go to .
Add some points to the graph by clicking on the graph. Press “Show Line” to plot the LSRL line and calculate the correlation coefficient, r. Press Ctrl-click to delete a point. Hold Shift-click to drag a point. Your screen should look something like this:
Create scatterplots with the following associations and record r:
Strong positive linear association
Weak positive linear association
Strong negative linear association
No linear association (random scatter)
Use just five points to make a strong negative linear association (say r < −0.95). Drag one of the points around to observe the effect on the slope and correlation coefficient. Can you make the slope positive by dragging just one point?