Scientific Research The following chart shows some the data shown in the g...

Scientific Research The following chart shows some the data shown in the graph in Exercise 23 (t = 0 represents

1983).

a. What is the logistic regression model for the data? (Round all coefficients to two significant digits.) At what value does the model predict that the number of research articles will level off? HINT [See Example 2.]

b. According to the model, how many Physical Review articles were published by U.S. researchers in 2000 (t = 17)? (The actual number was about 5,500 articles.)

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.