use technology to find a logistic regression curve approximating the given data. Dra...

use technology to find a logistic regression curve approximating the given data. Draw a graph showing the data points and regression curve. (Round b to three significant digits and A and N to two significant digits.) HINT [See Example 2.]

Example 2:

Here are the data graphed in Figure 17:

Find a logistic regression curve of the form

In the long term, what percentage of Internet-connected households with broadband does the model predict?

Solution We can use technology to obtain the following regression model:

Its graph and the original data are shown in Figure 19.

Because N≈ 104, the model “predicts” that 104% of all Internet-connected households will have broadband in the long term. Clearly, this makes no sense; a reasonable value of N is 100, as it seems reasonable that, in the long term, all Internet connections will be broadband. There are various curve-fitting procedures that permit one to fix one of the parameters (N in this case) beforehand and adjust the others (A and b here) to obtain the best possible fit. In the Excel Technology Guide at the end of the chapter we show how to do this using Excel’s “Solver” Add-in. Here is the best-fit model with N fixed at 100:

Although this second model is a slightly poorer fit to the data (SSE ≈ 31) than the first one (SSE ≈ 26), it has the advantage that it better predicts the long-term behavior we expect.