Linear Regression is a process by which the equation
of a line
is found that “best fits” a given set of data. The line
of best fit approximates the best linear representation for your data. One very important aspect of a regression line
is the relationship between the equation
and the “science quantity” often represented by the slope
of the line.
Let's look at an example of linear regression by examining the data
in the following table
to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. [Remember that lines are named using the convention y vs. x
tables are constructed as x | y
Based on this data:
- interpolate the equivalent temperature in degrees Celsius of our body temperature, 98.6 °F
- relate the linear equation of your model with its associated science formula to determine the "physical" meaning of the slope of this data's trend line
First we will plot the data
using a TI-83 graphing calculator. We will enter the data
measured in degrees Fahrenheit in L1 and the temperatures measured in degrees Celsius in L2. Once the data
is entered, your screen should look like the following:
After entering the data
into the calculator, graph
the data. The Fahrenheit data, listed in L1, represents the x-axis, and the Celsius data, listed in L2, represents the y-axis. Your screen should look like the following:
Now we need to find a linear equation
that models the data
we have plotted. According to the calculator, our equation
has the following properties:
Based on the graph
and the equation
information listed above, our correlation coefficient
(r) is equal to 1. That means that our data
perfectly models a linear function.
Using the model from step two and the graph
on our calculator from step three, we can trace along the graph
and determine what temperature in degrees Celsius equals 98.6 °F, our body temperature.
This screen capture shows us that 98.6 °F (x-value) is equivalent to 37 °C (y-value).
Consider our equation
. The accepted formula used to convert Fahrenheit degrees to Celsius degrees is typically written as
Expressed in this form we can clearly see that our model's equation
is indeed the same equation
conventionally used to convert temperatures between these two measuring scales. Since our line's slope
] is a decimal, we know that the size of a "degree" on the two temperature scales
is not the same; that is, these scales
are not in a one-to-one correspondence -- 1 Celsius degree (Cº) does not equal 1 Fahrenheit degree (Fº).
Examining the slope
] we can clearly see that the relationship between the two scales
is such that from a given point
on the line, you move up five degrees on the Celsius scale
and right nine degrees on the Fahrenheit scale
to arrive at the next point
on the line. Or equivalently, when the temperature changes 9 Fº it only changes 5 Cº.