Linear Regression I:Scatterplots and Regression Lines • SAT scores and graduation rates • Looking at a scatterplot • Fitting a regression line • What does it mean? • Other factors that affect graduation rates • Confound: reputation • Other independent variables
SAT Scores and Graduation Rates • To test the hypothesis that colleges with smarter students have higher graduation rates, I will look at data from 148 colleges in the United States. SAT Scores Graduation Rates
SAT Scores and Graduation Rates • We can summarize the direction and strength of a relationship between two variables by calculating “r,” the correlation. (Pollock, p. 171-174) • If we want to know more about the relationship, we can fit a “regression line” to the scatterplot.
SAT Scores and Graduation Rates • A regression line summarizes how much the dependent variable (graduation rates) changes when the independent variable (SAT scores) increases. • The line gives you a “predicted value” of graduation rates for a college with a given average SAT score. • To make this prediction as good as possible, the line minimizes vertical distances between the line and the data points.
SAT Scores and Graduation Rates • Like all lines, a regression line can be summarized with this formula: y = a + b•x where: b = slope of line or “regression coefficient” a = the intercept, or the value of y when x=0
SAT Scores and Graduation Rates • The regression coefficient for the effect of SAT scores on graduation rates is 0.06. • This is the predicted effect on graduation rates when SAT scores go up by one unit. • This means that comparing one college to another college where average SATs were 100 points higher should lead to a graduation rate that is 6 percentage points higher.
Other Factors Affecting Grad Rates • The school’s reputation could be a confound: a school with a good rep might attract smart students and want to keep its reputation high by graduating them.
Other Factors Affecting Grad Rates Year of Founding SAT Scores Graduation Tuition Rates Student/Faculty Ratio