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Recall that we presented some examples that used variables about characteristics of automobiles. We showed the relationship between two of these variables, the Weight (in Tons) of automobiles and the Horsepower of their engines. Once again, here is the scatterplot relating the two variables:
The Regression Line
Equation for a Straight Line
ViSta Regres: The regression analysis is done using ViSta's Regression Analysis module, which can be done by clicking on the Regres button on the workmap. The workmap and report that result are:
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Report: The regression analysis report has three major sections, each containing important information about the analysis:
Under the "Estimate" column the report presents the values for the intercept and slope of the line that regression analysis estimates produces the best fit to the points.
The intercept and slope are often called the "coefficients", because they are the coefficients of the regression line. They are called "estimates" (short for "estimated coefficients") because they are estimates of what the coefficients are in the population.
This means that if we had someone with a Verbal SAT of zero, we would estimate that person's GPA to be 1.29.
Notice that this value doesn't make much sense! In fact, the Intercept is usually not interpreted, especially if a value of zero for the predictor variable can't really be obtained in practice.
This means that for every point change in VerbSAT/100 (which corresponds to 100 points change in Verbal SAT) we expect a change in GPA of 0.32.
Thus, for a person whose SAT is 100 points higher than another person's, we would predict that the first person's GPA would be .32 points higher than the second person's. This makes good sense, and is an important part of the results of regression analysis.
Note: This is not what the book calls the "Standard Error of Estimate". That value is presented below by the name "Sigma Hat (RMS Error)".
Note that the question of whether the slope is zero gets at the question of the nature of the relationship between the two variables. This is important because the question is: Does one variable change when the other does? (Note that zero intercept makes little interpretive sense and the test is usually ignored).