Download You may recall that to find the equation of the LSRL, your calculator

Tuesday December 10
•
•
•
•
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
Hours
GPA
4
2.9
5
3.3
11
3.9
1
2.2
15
4.1
2
1.8
10
4.6
6
2.9
7
2.2
0
3
7
3.3
9
4.5
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.
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.
Your Task: With your
team, use your
calculators to explore the
interpretation of the
correlation coefficient r.
– 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.
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.
a) 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.
b) Each member of your team should choose two
new points that have a negative slope between
them. Compare your results with your team.
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?
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?
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.
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 = 1.66 + 0.13x.
a)
b)
c)
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).
a)
b)
c)
Go to
http://illuminations.nctm.org/LessonDetail.a
spx?ID=L456#qs .
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:
a)
b)
c)
d)
d)
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?
6-72. Extension: A
computer will help us
explore the
correlation coefficient
further.