The table shows the average cost of tuition and fees y in dollars at 4-year public colleges. In this table x = 0 represents 1980 and x = 20 corresponds to 2000.

These data can be modeled by using linear regression. Ideally, we would like f(x) = ax + b to satisfy the following five equations.

Since the data points are not collinear, it is impossible for the graph of a line to pass through all five points. These five equations can be written as

The least-squares solution is found by solving the normal equations

for X. The solution is X= (A^TA)^-1A^TB. Using technology, we find a = 138.16 and b = 693.8. Thus f is given by the formula f(x) = 138.16x + 693.8. The function f and the data can be graphed. See the figure.

Solve the normal equations to model the data with the line determined by f(x) = ax + b. Plot the data and f in the same viewing rectangle.

The table shows average cost of tuition and fees y in dollars at private 4-year colleges. In this table x = 0 corresponds to 1980 and x = 20 to 2000.

Transcribed Image Text:

x 0 y 804 5 1318 10 1908 15 2860 20 3487