Math for the People

Section 9.5 Changes in College Tuition over Time

Use the Excel spreadsheet on average undergraduate tuition and fees and room and board rates charged for full-time students in degree-granting institutions, by level and control of institution: 1969-70 through 2020-21¹ [9.11.165] to do the following:

1. Create a graph, trendline, and interpret the \(R^{2}\) value for each type of institution: public nonprofit, private nonprofit, private for-profit. What does it mean?

2. Use it to predict average tuition and fees across the country in the current year. How accurate is your prediction?

3. What trends over time do you notice in each the three graphs? Describe trends in words, using phrases like increasing, decreasing,increasing slope, decreasing slope. What overall trend do we notice that's common to all three graphs over time?

4. The \(R\)-squared value that I asked you to show in Item 1 measures what proportion of the variation in the \(y\) variable is explained by the trendline. For instance, \(R^{2}=1\) means that \(100\%\) of the variation in \(y\) is explained by the trendline, so the trendline is a perfect fit. Conversely, \(R^{2}=0\) means that the trendline explains \(0\%\) of the variation in the \(y\) variable, so this trendline doesn't fit the data at all. What percentage of the variation in two-year public college tuition (\(y\)) is explained by the variation in year (\(x\))? What about when \(y=\) four-year private college tuition? Four-year public college tuition?

5. Outliers are data that fall far away from the trendline. What outliers do you notice, if any? Interpret these outliers in context: what do they mean for tuition and fees over time, and what possible historical explanations could there be for these outliers?

6. Polynomials are a special type of mathematical function, or process for turning one number into another. Polynomials are functions of the form

   \begin{equation*} f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{1}x^{1}+a_{0} \end{equation*}

   sums of powers of \(x\text{,}\) maybe with constant terms in front. For example, \(f(x)=x+1\text{,}\) \(g(x)=17x^{3}-234x+7\text{,}\) and \(h(x)=-x^{142}+1\) are all polynomials. The polynomial \(a(x)=x^{2}+2x+3\) takes the input \(1\) and turns it into the output \(1^{2}+2(1)+3=6\text{.}\) All numbers which make sense to input are called the domain of the function, and the possible sensical outputs you could get are called the range.

   What is the domain and range of each of the polynomial trendlines?

7. Use this model to predict the tuition cost for a student entering each type of institution in the current academic year. How accurate do you think your prediction is?

8. Use the model to predict the tuition cost for a student entering a public two-year college in the academic years \(2050\)-\(2051\text{,}\) \(2100\)-\(2101\text{,}\) and \(1900\)-\(1901\text{.}\) How accurate do you think your predictions are?

9. Within (roughly) what domain do you think your trendline can make fairly accurate predictions? Explain.