Module 3: Examining Relationships: Quantitative Data
Module 3: Examining Relationships: Quantitative Data
Linear Regression (3 of 4)
Linear Regression (3 of 4)
Learning OUTCOMES
- For a linear relationship, use the least squares regression line to model the pattern in the data and to make predictions.
Let’s quickly revisit the list of our data analysis tools for working with linear relationships:
- Use a scatterplot and r to describe direction and strength of the linear relationship.
- Find the equation of the least-squares regression line to summarize the relationship.
- Use the equation and the graph of the least-squares line to make predictions.
- Avoid extrapolation when making predictions.
Now we focus on the equation of a line in more detail. Our goal is to understand what the numbers in the equation tell us about the relationship between the explanatory variable and the response variable.
Here are some of the equations of lines that we have used in our discussion of linear relationships:
Predicted distance = 576 − 3 * Age
Predicted height = 39 + 2.7 * forearm length
Predicted monthly car insurance premium = 97 − 1.45 * years of driving experience
Notice that the form of the equations is the same. In general, each equation has the form
Predicted y = a + b * x
When we find the least-squares regression line, a and b are determined by the data. The values of a and b do not change, so we refer to them as constants.
In the equation of the line, the constant a is the prediction when x = 0. It is called initial value. In a graph of the line, a is the y-intercept.
In the equation of the line, the constant b is the rate of change, called the slope. In a graph of the least-squares line, b describes how the predictions change when x increases by one unit. More specifically, b describes the average change in the response variable when the explanatory variable increases by one unit.
We can write the equation of the line to reflect the meaning of a and b:
Predicted y = a + b * x
Predicted y-value = (initial value) + (rate of change)*x
Predicted y-value = (y-intercept) + (slope)*x
The constants a and b are shown in the graph of the line below.
Algebra review
The algebra of a line
The general form for the equation of a line is Y = a + bX. The constants “a” and “b” can be either positive or negative. The constant “a” is the y-intercept where the line crosses the y-axis. The constant “b” is the slope. It describes the steepness of the line. In algebra we describe the slope as “rise over run”. The slope is the amount that Y increases (or decreases) for each 1-unit increase in X.
EXAMPLE
1
Consider the line [latex]Y = 1 + \frac{1}{3}X[/latex].
The intercept is 1. The slope is 1/3, and the graph of this line is, therefore:
EXAMPLE
2
Consider the line [latex]Y = 1 - \frac{1}{3}X[/latex]. The intercept is 1. The slope is -1/3, and the graph of this line is, therefore:
The simulation below allows you to see how changing the values of the slope and y-intercept changes the line. The slider on the left controls the y-i