In a chemical experiment involving two metals heated to high temperatures, it has been conjectured that there is a critical change in the behavior of the mixture when the temperature reaches Celsius. The results of the experiment are in the file p2dat.m, with errors of at most one in the last digit.

Some weeks after the data were collected, another scientist at the same lab noticed subtle inconsistencies in the results, and the lab records for the experiment were examined closely. It was learned that each result was collected by one of two separate lab technicians (depending on which one was less busy). A further complication was that the two technicians used different thermometers (one of which had a bias) to collect the results. The technicians cannot remember who made each measurement. Hence, the data set is flawed.

Should the experiment be conducted again (at considerable time and expense)? Or can useful results be extracted from the existing data?

The scientist in charge of the lab has decided that, if it can be determined, "with reasonable precision," when the mixture reached the critical temperature, then the results are of value, and the experiment need not be repeated.

 C Grade:  Suppose that the first technician collected data points

Fit these data points (in the least-squares sense) using a model of the form

Use Newton's method to determine when the critical temperature was reached. Repeat this calculation for data points

Comment on the results. Is one of these subsets more "reasonable" than the other? In addition, use all the data points, and fit the data (in the least-squares sense) using the same model; again determine the critical temperature.

 B Grade:  For each of the polynomials used in the C-grade section, determine the scaling factor for the time when the critical temperature was reached, with respect to each data point. Given the accuracy of the data points, are the scaling factors "satisfactory".

 A Grade:  Suppose that the data were less accurate: that the times had a standard deviation of 15 seconds and the temperatures had a standard deviation of Celsius. Analyze (through simulation) the sensitivity of your results for the C-grade section. Present both graphical and numerical evidence supporting your conclusions.

 Preliminary deadline: Wednesday, September 30

 Final deadline: Thursday, October 8, 9:30am

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