2. In this problem we want to compute the average population of our study insect from a collection of observations. The population of a particular insect is gathered over the summer, and the table below lists the population and the survey times.

Week #	Population	Week #	Population
0	134	9	904
1	211	12	883
2	267	14	775
5	654	15	704

It is clear that the data were not collected uniformly, so a question arises as to how to compute the average of the population over the summer.

a. Compute the average in the usual way that you compute averages by adding up the data points and dividing by the number of points.

b. Use Excel's trendline to fit a 4^th order polynomial through the data. Plot both the data points and the 4^th order polynomial on a graph. Find the least squares sum of the error between this model and the data. Use this polynomial to find when the population is at a maximum and what that maximum population is. If the polynomial is given by p(t), then the average population P_ave is given by the following:

c. In an earlier lab, we saw how Fourier series also fit data very well. Let

be a Fourier series that we want to approximate the data above. An Excel spreadsheet is provided with this model and initial guesses for the parameters w, a₀, a₁, a₂, f₁, and f₂. Use the Solver routine in Excel (twice) to find the best parameters fitting the data. List the parameter values and the least square sum of the error between this model and the data. Plot this model (with at least 50 evenly spaced points) and the data. Use S(t) to find when the population is at a maximum and what that maximum population is. Find the average population by computing

Compare your answers for the least squares sum of error, maximum, time of maximum, and average to your answers in Part b.

d. Give a reason why the averages given in Parts b and c. is better than the average from Part a. Can you think of a situation where the average in Part a. would be the better approximation?