# Analyzing Accuracy

In most situations, it is not good enough merely to know what kind of model function is likely to give the best results. Usually it is important to have some idea of how accurate those results are as well.

## Experiment #3

Since there is no way to discover what the true value is, one has to deduce what inaccuracies might have been introduced from the properties of the model function and the data.

## Experiment #4

We are now in a position to formulate an estimate of how close the results we have obtained by numerical integration are to the actual CO2 concentration. Using what we discovered about over- and under-estimates for monotone and convex data, we will attempt to find upper and lower bounds for the actual net change in CO2 during the 24-hour period for which we have data.

For this experiment, use data set 1 again. Carefully record the intervals and model functions used as well as the results. You will need this information for Question 11.

• Split the interval [0,23] into subintervals on which you assume that the underlying rate function is monotone. (There is no right answer, but choosing "large" intervals will make the subsequent analysis easier. For example, you may assume the underlying function is monotone on three intervals.)

For each interval on which the rate function is monotone, use the piecewise constant model (restricted to those intervals) to compute upper and lower bounds for the integral of the rate function.

• Now split the interval [0,23] into subintervals on which you assume that the underlying rate function is concave. For each interval, choose models whose integrals give upper (and lower) bounds.

### Question 11

At dawn, there were 2.600 mmol of CO2 per liter of water. Use your experimental findings to give upper and lower bounds for the actual CO2 concentration in the San Marcos River after 24 hours. Explain what you did to obtain your answer. Make sure that your final answer makes sense by checking that the values you obtained from Experiment #1 all lie within your estimates. (Hint: assume that the true function that generated the data is monotone and/or concave up/down on certain time intervals, then use the ideas from Experiments #3 and #4.)

### Question 12

This is an open-ended question, so feel free to formulate conjectures provided you carefully explain how you attempted to answer it.

Of the values you recorded in Experiment #1, which would you give as the best approximation to the total change in CO2 for the river over this 24-hour period? Specify the model you prefer, and give reasons for your preferences. Based on your answer, do you think the river is healthy? Explain.

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