8.1 Linear Regression
Linear Regression:
Analytic technique for determining the relationship between a dependent
variable and an independent variable.
Process used only if the two variables have a strong linear correlation
Finding the line of best fit to create a mathematical model: y = ax + b
Referred to as the least-squares method
Use this equation of line best fit to predict the value of one of the two
variables given the value of the other variable.
To find the equation of the line of best fit you use the following formulas:
The equation is
a
n
xy
[n
x2 (
x
y = ax + b
y
x) 2 ]
and
b
where:
y
ax
Note: The slope “a” indicates only how “y” varies with “x” on the line of best fit, it does
not tell you anything about the strength of the correlation “r”.
From Lesson 3.1, we set up a table to find the correlation coefficient (r) for the
data of a small company. Since we know the correlation coefficient is strong,
we can use linear regression to find the equation of the line of best fit.
Example 1: Using the data from Lesson 8.1:
a) Find the equation of the line of best fit using linear regression.
Using the equation of the line of best fit to make predictions:
b) Predict the annual income of an employee if they are 58 years old.
c) Predict the age of the employee if their annual income is $32 600.
Outliers and small sample sizes:
can seriously affect the equation of the line of best fit
If the outlier is the result of measurement error, eliminate the outlier to
get a useful line of best fit.
Example 2: A driving school tabulates the number of hours of instruction and
the driving test scores for the instructor’s students to evaluate their instructor’s
performance.
Analyse both graphs.
After the outlier has removed, the correlation coefficient, r, is 0.93. This indicates a
strong positive linear correlation between the two variables.