We're looking for a way to estimate the weight (y) of a mystery person. Sometimes we'll guess to high, other
times too low. We seek a method that will minimize the
errors we make in the long run. Absent any information
other than this set of weights, what's the best weight to
guess?

You're asking in the ap section, so i'll give the ap calculus-style answer. alternatively, you can find the vertex of the parabola and use the "y"-coordinate as the answer.

the maximum height can occur at a critical number, that is, a number  such that  .

the derivative of  can be calculated as follows:



find a value of  that is a zero of the derivative:


this is parabola that is opening downward; we can assume there is a maximum at  .

to find that maximum, calculate  .