Question

MODELING WITH MATHEMATICS The data pairs \((x, y)\) represent the average annual tuition \(y\) (in dollars) for public colleges in the United States \(x\) years after 2005 . Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in 2020. Interpret the slope and \(y\)-intercept in this situation. (See Example 4.) $$ (0,11,386),(1,11,731),(2,11,848) $$ \((3,12,375),(4,12,804),(5,13,297)\)

Short answer

The equation derived from linear regression will take the form \(y = mx + c\), where 'm' is the rate of tuition increase per year and 'c' is the initial cost in 2005. The estimated tuition fee in 2020 can be found by substituting x = 15 into the equation.

Step-by-step solution

Input Data into Calculator

To begin with, you need to enter the provided data into a graphing calculator. The years after 2005 will be represented as 'x', while the corresponding tuition fees in those years will be represented as 'y'.

Perform Linear Regression

Once the data has been entered into the calculator, you are able to carry out the linear regression function. The graphing calculator will generate a line of best fit for the input data and will also produce an equation of that line in the form \(y = mx + c\), where 'm' is the slope of the line and 'c' is the y-intercept.

Interpret the Results

The slope 'm' of the line represents the rate of change in tuition fee per year, while the y-intercept 'c' represents the tuition fee in the year 2005. The exact figures will come from the graphing calculator's linear regression function.

Estimate Tuition in 2020

To estimate the tuition in 2020, substitute 'x' with 15 (2020 is 15 years after 2005) in the line equation. The resultant 'y' will be an approximation of the average annual tuition in 2020.

Key concepts

Slope Interpretation

The slope in a linear regression context is a measure of how quickly or slowly a dependent variable changes with respect to changes in an independent variable. In the scenario of estimating college tuition, the slope represents the rate at which the average annual tuition fee changes per year.
The equation of a line is often expressed in the form \( y = mx + c \), where \( m \) is the slope. If the slope is positive, it means that as the years progress, tuition fees are increasing. Conversely, a negative slope would indicate a decrease in tuition fees over time.

• A steeper slope signifies a rapid increase or decrease in tuition.
• A smaller slope implies a more gradual change.

Understanding the slope is crucial for evaluating how quickly tuition is rising and can guide students and policymakers in future financial planning.

Y-Intercept Interpretation

The y-intercept in a linear equation like \( y = mx + c \) represents the value of \( y \) when \( x \) is zero. In our example, the y-intercept reflects the average annual tuition fee in the year 2005, which is when \( x = 0 \).
This point can be particularly significant as it provides a baseline from which we can observe the change in tuition over the years. It's essentially the starting point of our data timeline.

• Enables comparison of past tuition costs to current or future costs.
• Helps identify trends when combined with the slope.

Recognizing the y-intercept helps us contextualize how tuition fees evolve from their initial value back in 2005.

Line of Best Fit

The line of best fit is a straight line that best represents the data on a scatter plot. This line is used in linear regression to illustrate the relationship between the independent and dependent variables. In our case of college tuition, it highlights how tuition prices have changed over time from the dataset starting in 2005 onwards.
The line is determined using methods like the least squares method, which minimizes the sum of the squares of the vertical distances of the points from the line.

• Serves as the graphical representation of the relationship between variables.
• Highly effective for predicting data tendencies.

By analyzing the line of best fit, we can readily make predictions and gain insights into tuition trends, helping in financial forecasting.

Graphing Calculator

A graphing calculator is a tool used to plot data, perform calculations, and execute functions such as linear regression. These calculators are vital for converting complex datasets into comprehensible graphs and equations.
When handling data for tuition analysis, a graphing calculator simplifies the process by:

• Allowing direct data entry for immediate processing.
• Automatically generating the equation of the line of best fit.
• Facilitating visualization of data trends and predictions.

By using a graphing calculator, students can more accurately and efficiently analyze data, derive equations, and visualize linear trends.

Tuition Estimation

Estimating future tuition costs is crucial for planning and budgeting educational expenses. By applying the equation of the line of best fit derived through linear regression, students and families can predict future tuition.
For the year 2020, substituting \( x = 15 \) into the equation allows us to estimate the tuition 15 years after 2005. This predicted value gives a quantitative expectation, aiding in financial readiness.

• Ensures informed financial decisions for future educational planning.
• Helps institutions set appropriate pricing strategies.

Tuition estimation based on historical trends allows for better financial forecasting and awareness among stakeholders.