http://www.ck12.org Chapter 9. Regression and Correlation
9.3 Inferences about Regression
Learning Objectives
- Make inferences about the regression models including hypothesis testing for linear relationships.
- Make inferences about regression and predicted values including the construction of confidence intervals.
- Check regression assumptions.
Introduction
In the previous section, we learned about the least-squares or the linear regression model. The linear regression
model uses the concept of correlation to help us predict a variable based on our knowledge of scores on another
variable. As we learned in the previous section, this concept is used quite frequently in statistical analysis to predict
variables such as IQ, test performance, etc. In this section, we will investigate several inferences and assumptions
that we can make about the linear regression model.
Hypothesis Testing for Linear Relationships
Let’s think for a minute about the relationship between correlation and the linear regression model. As we learned,
if there is no correlation between two variables (XandY), then it would be near impossible to fit a meaningful
regression line to the points in the scatterplot graph. If there was no correlation and our correlation(r)value was
0, we would always come up with the same predicted value which would be the mean of all the predicted variables
(Y). The figure below shows an example of what a regression line fit to variables with no relationship(r= 0 )would
look like. As you can see for any value ofX, we always get the same predicted value.
Using this knowledge, we can determine that if there is no relationship betweenYandXconstructing a regression
line or model doesn’t help us very much because the predicted score would always be the same. In other words, a