The table lists the average annual costs (in dollars) of...

The table lists the average annual costs (in dollars) of tuition and fees at public four-year colleges for selected years. $$\begin{array}{|c|c|} \hline \text { Year } & \text { Tuition and Fees (in dollars) } \\ \hline 2000 & 3505 \\ 2003 & 4632 \\ 2005 & 5491 \\ 2008 & 6532 \\ 2010 & 7605 \end{array}$$ (a) Use a calculator to find the least-squares regression line for these data, where $x$ is the number of years after 2000 (b) Based on your result from part (a), write an equation that yields the same $y$ -values when the actual year is entered. (c) Estimate the cost of tuition and fees in 2009 to the nearest hundred dollars.

The table lists the average annual costs (in dollars) of tuition and fees at public four-year colleges for selected years.
$$\begin{array}{|c|c|} \hline \text { Year } & \text { Tuition and Fees (in dollars) } \\ \hline 2000 & 3505 \\ 2003 & 4632 \\
2005 & 5491 \\ 2008 & 6532 \\ 2010 & 7605 \end{array}$$
(a) Use a calculator to find the least-squares regression line for these data, where $x$ is the number of years after 2000
(b) Based on your result from part (a), write an equation that yields the same $y$ -values when the actual year is entered.
(c) Estimate the cost of tuition and fees in 2009 to the nearest hundred dollars.

Key Concepts

Prediction and Rounding

Prediction is the process of using a regression equation to estimate the value of the dependent variable based on a given independent variable. When predictions are made, especially in real-world scenarios like estimating tuition costs, it is often necessary to round the results to a specified level of precision, such as to the nearest hundred dollars, to make the estimate practical and meaningful.

Variable Transformation

This concept involves adjusting the variable used in a model to better suit the context or simplify computations. In this case, defining x as the number of years after a base year (2000) simplifies the interpretation of the intercept and makes the calculations more straightforward. Transforming variables in this way is common practice when the original variable has a large magnitude or an inconvenient reference point.

Least-Squares Regression

This concept refers to the statistical method used to determine the best-fitting straight line through a set of points in the least-squares sense. It minimizes the sum of the squares of the vertical distances between the observed values and the values predicted by the line. This method is fundamental in many fields for making predictions and understanding trends over time or other variables.

Slope and Intercept

In a linear regression equation of the form y = mx + b, the slope (m) represents the change in the dependent variable for each unit change in the independent variable, while the intercept (b) is the value of y when x is zero. Understanding these components provides insight into the underlying relationship between the variables, including the rate of change and the starting point.

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