The following table shows the average total cost, measured as dollars expended per pupil, of operating a high school with the given enrollment. This table is based on a study of economies of scale in high school operations from the 1960s.

Enrollment: 170 250 450 650
Average total cost: 538 473 411 415

Estimate the average cost per student for a high school with an enrollment of 350 students.
The interpolated value for an enrollment of 350 students is $ ------
per student.

There are a number of ways the cost table might be interpolated. The one used here is to use quadratic regression to write a formula for the cost as a function of enrollment. Then the function value for 350 students is found. This is all accomplished using a graphing calculator, as shown in the attachment. The interpolated value for an enrollment of 350 students is $433 per student.

Another method of interpolation could use linear interpolation between the two nearest points. Our enrollment of interest, 350, is halfway between 250 and 450, so the cost per student could be estimated to be halfway between 473 and 411:

cost per student = (473 +411)/2 = 442 . . . dollars per student

We know this estimate is a bit high, because the cost function is concave upward. A straight line between two points on the curve will lie above the curve.

Other methods of interpolation will give other values. A cubic interpolating function gives a value of $428 per student.