[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.
b. Determine the correlation coefficient between “4 Year Cost” and “Grad Rate.”

Verified step by step guidance

1. Identify the two variables involved: "4 Year Cost" (the four-year cost of attending the college) and "Grad Rate" (the percentage of incoming freshmen who graduate within six years).
2. Organize the data into pairs \((x_i, y_i)\), where \(x_i\) represents the "4 Year Cost" for the \(i^{th}\) college and \(y_i\) represents the corresponding "Grad Rate."
3. Calculate the mean of each variable: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) and \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\), where \(n\) is the number of colleges.
4. Compute the covariance between the two variables using the formula: \[\text{Cov}(X,Y) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})\]
5. Calculate the standard deviations of each variable: \[s_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2} \quad \text{and} \quad s_y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2}\] Finally, find the correlation coefficient \(r\) using the formula: \[r = \frac{\text{Cov}(X,Y)}{s_x s_y}\] This value measures the strength and direction of the linear relationship between "4 Year Cost" and "Grad Rate."

Key Concepts

Correlation Coefficient

The correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables. It ranges from -1 to 1, where values close to 1 indicate a strong positive relationship, values close to -1 indicate a strong negative relationship, and values near 0 suggest no linear association.

Variables and Data Interpretation

Understanding the variables involved is crucial: '4 Year Cost' represents the total cost to attend a college for four years, and 'Grad Rate' is the percentage of students graduating within six years. Interpreting these correctly helps in analyzing how cost might relate to graduation success.

Data Analysis Using Statistical Software or Tools

Calculating the correlation coefficient typically requires statistical software or tools that can handle large datasets. Familiarity with importing data, selecting variables, and running correlation functions is essential to accurately compute and interpret the relationship.