15.5 Linear Correlation
Objectives:
• Be able to construct a scatterplot to show the
relationship between two variables.
• Understand the properties of the linear
correlation coefficient.
• Be able to use your calculator to compute the
linear correlation coefficient.
Scatterplots
• To determine whether there is a correlation
between two variables, we obtain pairs of
data, called data points, relating the first
variable to the second.
• To understand such data, we plot data points
in a graph called a scatterplot.
Scatterplots cont. (2)
Example:
An instructor wants to know whether there is a correlation
between the number of times students attended tutoring sessions during
the semester and their grades on a 50-point examination. The table
shows the data for 10 students. Represent these data points by a
scatterplot and interpret the graph.
Solution:
We plot the points (18, 42), (6, 31), (16, 46), and so on.
As the number of tutoring sessions increases, the grades also generally
increase. So the correlation coefficient is positive.
Linear Correlation
• Linear correlation exists between two
variables if, when graphed, the points in
the graph tend to lie in a straight line.
• The linear correlation coefficient allows us to
compute to what degree the points of a
scatterplot lie along a straight line.
• r = 1 means that the points all lie on a straight
line and y increases as x increases.
• r = -1 means that the points all lie on a straight
line and y decreases as x increases.
Linear Correlation cont. (2)
• There is a positive correlation between the variables x and y if
whenever x increases or decreases, then y changes in the
same way.
• We will say that there is a negative correlation between the
variables x and y if whenever x increases or decreases,
then y changes in the opposite way.
Linear correlation coefficient, r
Using your Calculator to Compute r
1) Enter the x-values in List1 and the y-values in List2.
You must have the same number of entries in List1 and List2!
2) Go to
STAT
and then
3)
ENTER

4:
CALC
LinReg(ax+b)
Confidence of Correlation
1. Compute r for n pairs of data.
2.
If the absolute value of r exceeds the number
in the column labeled α = .05 on line n, then
we are 95% confident that there is a significant
linear correlation. (There is less than a 0.05
(5%) chance that the variables do not have
significant linear correlation. )
3.
If the absolute value of r exceeds the number
in the column labeled α = .01 on line n, then
we are 99% confident that there is a significant
linear correlation. (There is less than a 0.01
(1%) chance that the variables do not have
significant linear correlation.
Correlation Applications
• Example A: You wish to determine if there is a relationship between the
weight of the car and gas mileage. The table shows data regarding
gas mileage of cars with various weights. Find the correlation
coefficient for these data and determine whether there is linear
correlation at the 95% or 99% level.
• Example B:
Find the correlation coefficient for the number of times
students attended tutoring sessions during the semester and their
grades on the 50-point examination. The table shows the data for 10
students. Determine whether there is linear correlation at the 95%
or 99% level.
Correlation Applications cont. (2)
• Solution A: We will represent the weight by x and the gas mileage by y.
So enter the weight in List1 and the gas mileage in List2.
We now compute
There are 10 pairs of data, so we look at line 10 of the table and find the
value 0.765 under the α = 0.01 column. The absolute value of r is 0.85, so
we are 99% confident that there is significant negative linear correlation
between car weight and gas mileage.
• Solution B:
Enter the number of times students attended tutoring
sessions during the semester in List1 and their grades on the
50-point examination in List2.
• Compute the correlation coefficient, r .
• There are again 10 pairs of data, so we look at line 10 of the table
to find the confidence of correlation level.