The table lists the average tuition and fees (in constant...

The table lists the average tuition and fees (in constant 2010 dollars) at public colleges and universities for selected years. $$\begin{array}{|l|l|l|l|l|l|}\hline \text { Year } & 1980 & 1990 & 2000 & 2005 & 2010 \\\hline \begin{array}{l}\text { Tuition and Fees } \\\text { (in 2010 dollars) }\end{array} & 5938 & 7699 & 9390 & 11,386 & 13,297 \\\hline\end{array}$$ (a) Find the equation of the least-squares regression line that models the data. (b) Graph the data and the regression line in the same viewing window. (c) Estimate tuition and fees in 2007 . (d) Use the model to predict tuition and fees in 2016 .

The table lists the average tuition and fees (in constant 2010 dollars) at public colleges and universities for selected years.
$$\begin{array}{|l|l|l|l|l|l|}\hline \text { Year } & 1980 & 1990 & 2000 & 2005 & 2010 \\\hline \begin{array}{l}\text { Tuition and Fees } \\\text { (in 2010 dollars) }\end{array} & 5938 & 7699 & 9390 & 11,386 & 13,297 \\\hline\end{array}$$
(a) Find the equation of the least-squares regression line that models the data.
(b) Graph the data and the regression line in the same viewing window.
(c) Estimate tuition and fees in 2007 .
(d) Use the model to predict tuition and fees in 2016 .

Key Concepts

Interpolation and Extrapolation

Interpolation and extrapolation are methods used to estimate values from a regression model. Interpolation refers to making predictions within the range of the observed data, while extrapolation involves predicting values outside the observed data range. While interpolation is usually more reliable, extrapolation can introduce additional uncertainty since it extends the trend beyond the available data.

Prediction Using Regression

Prediction using regression involves substituting new values for the independent variable into the regression equation to estimate the corresponding value of the dependent variable. This technique is widely used in statistics and data analysis to forecast future outcomes or estimate intermediate values based on the modeled trend.

Least Squares Regression

Least squares regression is a statistical method used to determine the best-fitting straight line through a set of data points by minimizing the sum of the squared differences (residuals) between the observed values and the values predicted by the line. This method provides a simple model to understand the linear relationship between an independent and a dependent variable, ensuring that the overall error is minimized.

Linear Regression Model

A linear regression model describes the relationship between a dependent variable and one (or more) independent variables by fitting a linear equation to the observed data. The model is typically expressed in the slope-intercept form, where the slope represents the rate of change in the dependent variable per unit change in the independent variable, while the intercept represents the value of the dependent variable when the independent variable is zero.

Graphical Representation of Data

Graphical representation involves plotting the data points on a scatter plot and overlaying the regression line to visually analyze the relationship between variables. This visual tool helps in identifying trends, outliers, and the overall fit of the model, making it easier to understand and communicate the underlying patterns and variations in the data.

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