[DATA] Graduation Rates PayScale reports statistics on colleges and universities. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 11_3_24 using the file format of your choice for the version of the text you are using. The data contain the four-year cost and graduation rate for over 1300 colleges and universities. Do schools that charge more have higher graduation rates? The variable “4 Year Cost” represents the four-year cost of attending the college or university. The variable “Grad Rate” represents the percentage of incoming freshman who graduate within six years.


f. What proportion of the variability in graduation rates is explained by the cost of attending?

Verified step by step guidance

1. Identify the two variables involved: the explanatory variable is the "4 Year Cost" (cost of attending college), and the response variable is the "Grad Rate" (graduation rate).
2. Calculate the correlation coefficient, denoted as \(r\), between the "4 Year Cost" and "Grad Rate". This measures the strength and direction of the linear relationship between the two variables.
3. Square the correlation coefficient to find \(r^2\), which is called the coefficient of determination.
4. Interpret \(r^2\) as the proportion of the variability in the graduation rates that can be explained by the cost of attending the college or university.
5. Express the result as a percentage by multiplying \(r^2\) by 100 to understand how much of the variation in graduation rates is accounted for by the variation in cost.

Key Concepts

Coefficient of Determination (R²)

The coefficient of determination, denoted R², measures the proportion of variance in the dependent variable explained by the independent variable in a regression model. It ranges from 0 to 1, where higher values indicate a better fit. In this context, R² tells us how much of the variability in graduation rates is explained by the cost of attending college.

Simple Linear Regression

Simple linear regression models the relationship between two quantitative variables by fitting a straight line. Here, it helps analyze how the four-year cost (predictor) relates to graduation rates (response). The regression line summarizes this relationship and is used to calculate R².

Variability and Explained Variance

Variability refers to how spread out data points are around their mean. Explained variance is the portion of this variability accounted for by the regression model. Understanding these concepts helps interpret how well cost predicts graduation rates and the strength of their association.