In this part we consider how to measure the quality of our fit by the model function to the given data.

1. Overlay a semilog or loglog plot (whichever is appropriate) of your model function on a semilog or loglog plot of the given data. [The commands for doing one type of plot are already in your worksheet. If you need the other type, modify the commands accordingly.] If you have constructed your function $f\left(t\right)$ correctly, its graph should appear as a straight line, and the line should come reasonably close to most of the data points. If this is not the case, go back now to the previous page, and rethink your construction of the function.

2. Next, overlay an ordinary Cartesian plot of your model function on the initial plot of the data points. [Again, the commands are already in your worksheet.] Describe the extent to which you think your model function represents the given data.

3. The visual information in steps 1 and 2 can help you decide whether you followed the right modeling strategy and whether you have reasonable values for the parameters `a` and `b` in the model function. However, these graphs don't really measure the accuracy of our fit to the data. We can get a better sense of that by calculating residuals, that is, the numerical differences between the numbers of cases and the numbers predicted by model at corresponding times. Specifically, the `j`-th residual is the difference ${\mathrm{Cases}}_j-f\left({\mathrm{Months}}_j\right)$. Use the commands in your worksheet to plot the residuals as a function of time. If you have a good fit, the residuals should be relatively small – are they?

4. Even with a very good fit, you will probably see that the residuals tend to grow in magnitude as time goes on, some positive and some negative. There is a trick word in the preceding question: “relatively.” If the large residuals are differences in very large numbers, they may still be relatively small. We can “scale” the residuals by dividing each one by the number of Cases at the corresponding time – this gives us relative residuals. Use the commands in your worksheet to plot the relative residuals, and then comment (again) on whether your residuals are relatively small.