9.1 Correlation
• Key Concepts:
– Scatter Plots
– Correlation
– Sample Correlation Coefficient, r
– Hypothesis Testing for the Population
Correlation Coefficient, ρ
9.1 Correlation
• What exactly do we mean by correlation?
– If two variables are correlated, it means a
relationship exists between them.
– Examples of correlated variables:
•
•
•
•
•
Job Satisfaction and Job Attendance
Number of Cows per Square Mile and Crime Rate
Height and Weight
High School GPA and College GPA
Square Footage and Price (of a house)
9.1 Correlation
• Two questions we need to answer:
1. Does a linear (or straight line) correlation
exist between the two variables?
2. If the variables appear linearly correlated,
how strong is the correlation?
– We can answer (1) using a scatter plot
• The independent (explanatory) variable is x
• The dependent (response) variable is y
– Example: How well does High School GPA, x, “explain”
College GPA, y?
– See section 2.2 for a review of scatter plots
9.1 Correlation
• Once the scatter plot is complete, we should be
able to see if a linear relationship exists between
the two variables.
– See p. 470 for what we mean by Negative Linear
Correlation, Positive Linear Correlation, No
Correlation, and Nonlinear Correlation.
• Next, we need a way to quantify or measure the
strength of the linear relationship between the
two variables.
9.1 Correlation
• The Correlation Coefficient measures the
strength and the direction of the linear
relationship between two variables. The sample
correlation coefficient, r, is defined as:
r
n xy    x   y 
n x    x 
2
2
n y    y 
where n is the number of pairs of data
2
2
9.1 Correlation
• Things we need to know about the sample
correlation coefficient, r :
– r will always lie between -1 and 1, inclusive: -1 ≤ r ≤ 1
– If r = -1, we say there is a perfect negative linear correlation
between the two variables.
– If r = 1, there is a perfect positive linear correlation between the
two variables.
– The strength of the linear relationship between the variables is
determined by r ’s proximity to 1 or -1. In other words, the closer
r is to 1 or -1, the stronger the linear relationship. The closer r is
to 0, the weaker the linear relationship.
• Practice:
#22 p. 482 (Age and Vocabulary)
9.1 Correlation
• Once we have the sample linear correlation
coefficient, r, we can use it in a t-Test to make
an inference about the population linear
correlation coefficient, ρ (Greek letter “rho”).
– Why bother?
• Remember we found r using a limited set of data. What
about the rest of the population? Do we have enough
evidence from the sample data to claim that a significant
linear correlation exists between our two variables?
– Example: If we have analyzed the High School GPA and College GPA
of 25 students, is there enough evidence to claim that a significant
linear correlation exists between the High School GPA and College
GPA of all students?
9.1 Correlation
• t-Test for the Population Correlation Coefficient
– We will use the two-tailed version of this test:
H0: ρ = 0 (no significant correlation exists)
Ha: ρ ≠ 0 (a significant correlation exists)
– The test statistic is r and the standardized test statistic
is given by:
t
r
r

r
1 r2
n2
Note: t follows a t-distribution with n – 2 degrees of freedom
9.1 Correlation
• Practice using the t-Test:
#32 p. 484 (Braking Distances: Wet Surface)